Asymptotics of the Chebyshev Polynomials of General Sets

Home , Chebyshev polynomials

Joint Work with Jacob Christiansen and Maxim Zinchenko Part 1: Inv. Math, 208 (2017), 217-245 Part 2: Submitted (also with Peter Yuditskii) Part 3: Pavlov Memorial Volume, to appear.

Asymptotics of the Chebyshev Polynomials of Chebyshev Polynomials General Sets Alternation Theorem Potential Theory Barry Simon Lower Bounds and Special Sets IBM Professor of Mathematics and Theoretical Physics, Emeritus

Faber Fekete California Institute of Technology Szegő Theorem Pasadena, CA, U.S.A. Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Part 2: Submitted (also with Peter Yuditskii) Part 3: Pavlov Memorial Volume, to appear.

Faber Fekete California Institute of Technology Szegő Theorem Pasadena, CA, U.S.A. Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Joint Work with Jacob Christiansen and Maxim Zinchenko Bounds Part 1: Inv. Math, 208 (2017), 217-245 Szegő–Widom Asymptotics Part 3: Pavlov Memorial Volume, to appear.

Faber Fekete California Institute of Technology Szegő Theorem Pasadena, CA, U.S.A. Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Joint Work with Jacob Christiansen and Maxim Zinchenko Bounds Part 1: Inv. Math, 208 (2017), 217-245 Szegő–Widom Part 2: Submitted (also with Peter Yuditskii) Asymptotics Asymptotics of the Chebyshev Polynomials of Chebyshev Polynomials General Sets Alternation Theorem Potential Theory Barry Simon Lower Bounds and Special Sets IBM Professor of Mathematics and Theoretical Physics, Emeritus

Faber Fekete California Institute of Technology Szegő Theorem Pasadena, CA, U.S.A. Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Joint Work with Jacob Christiansen and Maxim Zinchenko Bounds Part 1: Inv. Math, 208 (2017), 217-245 Szegő–Widom Part 2: Submitted (also with Peter Yuditskii) Asymptotics Part 3: Pavlov Memorial Volume, to appear. Chebyshev polynomials just replace L2 by L∞ (so only the support matters). Specifically, let e ⊂ C be a compact, infinite, set of points. For any function, f, define

kfke = sup {|f(z)| | z ∈ e} The Chebyshev polynomial of degree n is the monic polynomial, Tn, with

kTnke = inf {kP ke | deg(P ) = n and P is monic} The minimizer is unique (as we’ll see below in the case that e ⊂ R), so it is appropriate to speak of the Chebyshev polynomial rather than a Chebyshev polynomial.

Chebyshev Polynomials

It is well known that monic orthogonal polynomials minimize the L2(e, dµ) norm if µ is a measure with Chebyshev Polynomials compact support, e ⊂ C. Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Specifically, let e ⊂ C be a compact, infinite, set of points. For any function, f, define

kfke = sup {|f(z)| | z ∈ e} The Chebyshev polynomial of degree n is the monic polynomial, Tn, with

Chebyshev Polynomials

It is well known that monic orthogonal polynomials minimize the L2(e, dµ) norm if µ is a measure with Chebyshev Polynomials compact support, e ⊂ C. Chebyshev polynomials just 2 ∞ Alternation replace L by L (so only the support matters). Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics The Chebyshev polynomial of degree n is the monic polynomial, Tn, with

Chebyshev Polynomials

Lower Bounds For any function, f, deﬁne and Special Sets kfk = sup {|f(z)| | z ∈ e} Faber Fekete e Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics The minimizer is unique (as we’ll see below in the case that e ⊂ R), so it is appropriate to speak of the Chebyshev polynomial rather than a Chebyshev polynomial.

Chebyshev Polynomials

Lower Bounds For any function, f, deﬁne and Special Sets kfk = sup {|f(z)| | z ∈ e} Faber Fekete e Szegő Theorem

Widom’s Work The Chebyshev polynomial of degree n is the monic

Totik Widom polynomial, Tn, with Bounds Main Results kTnke = inf {kP ke | deg(P ) = n and P is monic} Totik Widom Bounds

Szegő–Widom Asymptotics (as we’ll see below in the case that e ⊂ R), so it is appropriate to speak of the Chebyshev polynomial rather than a Chebyshev polynomial.

Chebyshev Polynomials

Lower Bounds For any function, f, deﬁne and Special Sets kfk = sup {|f(z)| | z ∈ e} Faber Fekete e Szegő Theorem

Widom’s Work The Chebyshev polynomial of degree n is the monic

Totik Widom polynomial, Tn, with Bounds Main Results kTnke = inf {kP ke | deg(P ) = n and P is monic} Totik Widom Bounds The minimizer is unique

Szegő–Widom Asymptotics so it is appropriate to speak of the Chebyshev polynomial rather than a Chebyshev polynomial.

Chebyshev Polynomials

Lower Bounds For any function, f, deﬁne and Special Sets kfk = sup {|f(z)| | z ∈ e} Faber Fekete e Szegő Theorem

Widom’s Work The Chebyshev polynomial of degree n is the monic

Lower Bounds For any function, f, deﬁne and Special Sets kfk = sup {|f(z)| | z ∈ e} Faber Fekete e Szegő Theorem

Widom’s Work The Chebyshev polynomial of degree n is the monic

Totik Widom polynomial, Tn, with Bounds Main Results kTnke = inf {kP ke | deg(P ) = n and P is monic} Totik Widom Bounds The minimizer is unique (as we’ll see below in the case that Szegő–Widom e ⊂ R), so it is appropriate to speak of the Chebyshev Asymptotics polynomial rather than a Chebyshev polynomial. I assume that I am not the only one who does not understand the interest in and signiﬁcance of these strange problems on maxima and minima studied by Chebyshev in memoirs whose titles often begin with, “On functions deviating least from zero . . . ”. Could it be that one must have a Slavic soul to understand the great Russian Scholar? This quote is a little bizarre given that, as we’ll see, Borel (who was Lebesgue’s thesis advisor) made important contributions to the subject in 1905!

Chebyshev Polynomials

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Could it be that one must have a Slavic soul to understand the great Russian Scholar? This quote is a little bizarre given that, as we’ll see, Borel (who was Lebesgue’s thesis advisor) made important contributions to the subject in 1905!

Chebyshev Polynomials

Chebyshev invented his explicit polynomials which obey Qn(cos(θ)) = cos(nθ) not because of their functional Chebyshev Polynomials relation but because they are the best approximation on n Alternation [−1, 1] to x by polynomials of degree n − 1. In this regard, Theorem Sodin and Yuditskii unearthed the following quote from a Potential Theory 1926 report by Lebesgue on the work of S. N. Bernstein. Lower Bounds and Special Sets I assume that I am not the only one who does not Faber Fekete Szegő Theorem understand the interest in and signiﬁcance of these strange Widom’s Work problems on maxima and minima studied by Chebyshev in Totik Widom memoirs whose titles often begin with, “On functions Bounds deviating least from zero . . . ”. Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics This quote is a little bizarre given that, as we’ll see, Borel (who was Lebesgue’s thesis advisor) made important contributions to the subject in 1905!

Chebyshev Polynomials

Totik Widom have a Slavic soul to understand the great Russian Scholar? Bounds

Szegő–Widom Asymptotics Chebyshev Polynomials

Totik Widom have a Slavic soul to understand the great Russian Scholar? Bounds This quote is a little bizarre given that, as we’ll see, Borel Szegő–Widom Asymptotics (who was Lebesgue’s thesis advisor) made important contributions to the subject in 1905! We say that Pn, a degree n polynomial, has an alternating n set in e ⊂ R if there exists {xj}j=0 ⊂ e with

x0 < x1 < . . . < xn and so that

n−j Pn(xj) = (−1) kPnke While the basic idea of the following theorem goes back to Chebyshev, the result itself is due to Borel and Markov, independently, around 1905.

The Alternation Theorem

We will focus for most of this talk on the case e ⊂ R, in which case, Tn is real, since on R, |Re(Tn)| is smaller than Chebyshev Polynomials |Tn|.

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics and so that

n−j Pn(xj) = (−1) kPnke While the basic idea of the following theorem goes back to Chebyshev, the result itself is due to Borel and Markov, independently, around 1905.

The Alternation Theorem

We will focus for most of this talk on the case e ⊂ R, in which case, Tn is real, since on R, |Re(Tn)| is smaller than Chebyshev Polynomials |Tn|.

Alternation Theorem We say that Pn, a degree n polynomial, has an alternating n Potential Theory set in e ⊂ R if there exists {xj}j=0 ⊂ e with Lower Bounds and Special Sets x0 < x1 < . . . < xn Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics While the basic idea of the following theorem goes back to Chebyshev, the result itself is due to Borel and Markov, independently, around 1905.

The Alternation Theorem

We will focus for most of this talk on the case e ⊂ R, in which case, Tn is real, since on R, |Re(Tn)| is smaller than Chebyshev Polynomials |Tn|.

Totik Widom n−j Bounds Pn(xj) = (−1) kPnke Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics The Alternation Theorem

We will focus for most of this talk on the case e ⊂ R, in which case, Tn is real, since on R, |Re(Tn)| is smaller than Chebyshev Polynomials |Tn|.

Totik Widom n−j Bounds Pn(xj) = (−1) kPnke Main Results

Totik Widom While the basic idea of the following theorem goes back to Bounds Chebyshev, the result itself is due to Borel and Markov, Szegő–Widom Asymptotics independently, around 1905. Conversely, any monic polynomial with an alternating set is the Chebyshev polynomial.

If Tn is the Chebyshev polynomial, let y0 < y1 < . . . < yk be the set of all the points in e where its takes the value ±kTnke. If there are fewer than n sign changes among these ordered points we can ﬁnd a degree at most n − 1 polynomial, Q, non-vanishing at each yj and with the same sign as Tn at those points. For small and positive, Tn − Q will be a monic polynomial with smaller k·ke. Thus there must be at least n sign ﬂips and therefore an alternating set.

The Alternation Theorem

The Alternation Theorem The Chebyshev polynomial of degree n has an alternating set. Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics If Tn is the Chebyshev polynomial, let y0 < y1 < . . . < yk be the set of all the points in e where its takes the value ±kTnke. If there are fewer than n sign changes among these ordered points we can ﬁnd a degree at most n − 1 polynomial, Q, non-vanishing at each yj and with the same sign as Tn at those points. For small and positive, Tn − Q will be a monic polynomial with smaller k·ke. Thus there must be at least n sign ﬂips and therefore an alternating set.

The Alternation Theorem

The Alternation Theorem The Chebyshev polynomial of degree n has an alternating set. Conversely, any monic Chebyshev Polynomials polynomial with an alternating set is the Chebyshev

Alternation polynomial. Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics If there are fewer than n sign changes among these ordered points we can ﬁnd a degree at most n − 1 polynomial, Q, non-vanishing at each yj and with the same sign as Tn at those points. For small and positive, Tn − Q will be a monic polynomial with smaller k·ke. Thus there must be at least n sign ﬂips and therefore an alternating set.

The Alternation Theorem

The Alternation Theorem The Chebyshev polynomial of degree n has an alternating set. Conversely, any monic Chebyshev Polynomials polynomial with an alternating set is the Chebyshev

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics For small and positive, Tn − Q will be a monic polynomial with smaller k·ke. Thus there must be at least n sign ﬂips and therefore an alternating set.

The Alternation Theorem

The Alternation Theorem The Chebyshev polynomial of degree n has an alternating set. Conversely, any monic Chebyshev Polynomials polynomial with an alternating set is the Chebyshev

Alternation polynomial. Theorem Potential Theory If Tn is the Chebyshev polynomial, let y0 < y1 < . . . < yk Lower Bounds be the set of all the points in e where its takes the value and Special Sets ±kTnke. If there are fewer than n sign changes among Faber Fekete Szegő Theorem these ordered points we can ﬁnd a degree at most n − 1 Widom’s Work polynomial, Q, non-vanishing at each yj and with the same Totik Widom sign as T at those points. Bounds n

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Thus there must be at least n sign ﬂips and therefore an alternating set.

The Alternation Theorem

The Alternation Theorem The Chebyshev polynomial of degree n has an alternating set. Conversely, any monic Chebyshev Polynomials polynomial with an alternating set is the Chebyshev

Main Results Tn − Q will be a monic polynomial with smaller k·ke.

Totik Widom Bounds

Szegő–Widom Asymptotics The Alternation Theorem

The Alternation Theorem The Chebyshev polynomial of degree n has an alternating set. Conversely, any monic Chebyshev Polynomials polynomial with an alternating set is the Chebyshev

Main Results Tn − Q will be a monic polynomial with smaller k·ke. Thus

Totik Widom there must be at least n sign ﬂips and therefore an Bounds alternating set. Szegő–Widom Asymptotics Then at each point, xj, in the alternating set for Pn, Q ≡ Pn − Tn has the same sign as Pn, so Q has at least n zeros, which is impossible, since it is of degree at most n − 1. ` The alternation theorem implies uniqueness of the Chebyshev polynomial. For, if Tn and Sn are two 1 minimizers, so is Q ≡ 2 (Tn + Sn).

At the alternating points for Q, we must have Tn = Sn, so they must be equal polynomials since there are n + 1 points and their diﬀerence has degree at most n − 1.

The Alternation Theorem

Conversely, let Pn be a degree n monic polynomial with an alternating set and suppose that kTnke < kPnke. Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics so Q has at least n zeros, which is impossible, since it is of degree at most n − 1. ` The alternation theorem implies uniqueness of the Chebyshev polynomial. For, if Tn and Sn are two 1 minimizers, so is Q ≡ 2 (Tn + Sn).

At the alternating points for Q, we must have Tn = Sn, so they must be equal polynomials since there are n + 1 points and their diﬀerence has degree at most n − 1.

The Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics The alternation theorem implies uniqueness of the Chebyshev polynomial. For, if Tn and Sn are two 1 minimizers, so is Q ≡ 2 (Tn + Sn).

At the alternating points for Q, we must have Tn = Sn, so they must be equal polynomials since there are n + 1 points and their diﬀerence has degree at most n − 1.

The Alternation Theorem

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics 1 Q ≡ 2 (Tn + Sn).

At the alternating points for Q, we must have Tn = Sn, so they must be equal polynomials since there are n + 1 points and their diﬀerence has degree at most n − 1.

The Alternation Theorem

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics At the alternating points for Q, we must have Tn = Sn, so they must be equal polynomials since there are n + 1 points and their diﬀerence has degree at most n − 1.

The Alternation Theorem

Conversely, let Pn be a degree n monic polynomial with an alternating set and suppose that kTnke < kPnke. Then at Chebyshev Polynomials each point, xj, in the alternating set for Pn, Q ≡ Pn − Tn Alternation has the same sign as Pn, so Q has at least n zeros, which is Theorem impossible, since it is of degree at most n − 1. ` Potential Theory Lower Bounds The alternation theorem implies uniqueness of the and Special Sets Chebyshev polynomial. For, if Tn and Sn are two Faber Fekete 1 Szegő Theorem minimizers, so is Q ≡ 2 (Tn + Sn). Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics so they must be equal polynomials since there are n + 1 points and their diﬀerence has degree at most n − 1.

The Alternation Theorem

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics The Alternation Theorem

Szegő–Widom Asymptotics there must be at least one zero (in R, not necessarily in e) between xj−1 and xj because of the sign change. Since this accounts for all n zeros: Fact 1 All the zeros of the Chebyshev polynomials of a set e ⊂ R lie in R and all are simple and lie in cvh(e). Here, cvh(e) is the convex hull of e and that result follows from x0, xn ∈ e.

Alternation and Zeros

If Tn is the Chebyshev polynomial for e ⊂ R and x0 < x1 < . . . < xn is an alternating set for Tn, Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Since this accounts for all n zeros: Fact 1 All the zeros of the Chebyshev polynomials of a set e ⊂ R lie in R and all are simple and lie in cvh(e). Here, cvh(e) is the convex hull of e and that result follows from x0, xn ∈ e.

Alternation and Zeros

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Here, cvh(e) is the convex hull of e and that result follows from x0, xn ∈ e.

Alternation and Zeros

If Tn is the Chebyshev polynomial for e ⊂ R and x0 < x1 < . . . < xn is an alternating set for Tn, there must Chebyshev Polynomials be at least one zero (in R, not necessarily in e) between Alternation xj−1 and xj because of the sign change. Since this Theorem accounts for all n zeros: Potential Theory Lower Bounds Fact 1 All the zeros of the Chebyshev polynomials of a set and Special Sets e ⊂ lie in and all are simple and lie in cvh(e). Faber Fekete R R Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Alternation and Zeros

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics If there are only ﬁnitely many gaps and no component of e is a single point, we speak of a ﬁnite gap set. Between any two zeros of Tn, there is a point in the alternating set so

Fact 2 Each gap of e ⊂ R has at most one zero of Tn.

Above the top zero (resp. below the bottom zero) of Tn, |Tn(x)| is monotone increasing (resp. decreasing). It follows that xn = supy∈e y (resp x0 = infy∈e y) so Fact 3 At the end points of cvh(e) ⊂ R we have that |Tn(x)| = kTnke and −1 en ≡ Tn ([−kTnke, kTnke]) ⊂ cvh(e)

Alternation and Zeros

By a gap of e ⊂ R, we mean a bounded connected component of R \ e. Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Between any two zeros of Tn, there is a point in the alternating set so

Fact 2 Each gap of e ⊂ R has at most one zero of Tn.

Alternation and Zeros

By a gap of e ⊂ R, we mean a bounded connected component of R \ e. If there are only ﬁnitely many gaps and Chebyshev Polynomials no component of e is a single point, we speak of a ﬁnite gap

Alternation set. Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Fact 2 Each gap of e ⊂ R has at most one zero of Tn.

Alternation and Zeros

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Above the top zero (resp. below the bottom zero) of Tn, |Tn(x)| is monotone increasing (resp. decreasing). It follows that xn = supy∈e y (resp x0 = infy∈e y) so Fact 3 At the end points of cvh(e) ⊂ R we have that |Tn(x)| = kTnke and −1 en ≡ Tn ([−kTnke, kTnke]) ⊂ cvh(e)

Alternation and Zeros

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Fact 3 At the end points of cvh(e) ⊂ R we have that |Tn(x)| = kTnke and −1 en ≡ Tn ([−kTnke, kTnke]) ⊂ cvh(e)

Alternation and Zeros

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics −1 en ≡ Tn ([−kTnke, kTnke]) ⊂ cvh(e)

Alternation and Zeros

By a gap of e ⊂ R, we mean a bounded connected component of R \ e. If there are only ﬁnitely many gaps and Chebyshev Polynomials no component of e is a single point, we speak of a ﬁnite gap Alternation set. Between any two zeros of Tn, there is a point in the Theorem alternating set so Potential Theory Lower Bounds Fact 2 Each gap of e ⊂ R has at most one zero of Tn. and Special Sets Faber Fekete Above the top zero (resp. below the bottom zero) of Tn, Szegő Theorem |Tn(x)| is monotone increasing (resp. decreasing). It follows Widom’s Work that xn = sup y (resp x0 = infy∈e y) so Totik Widom y∈e Bounds Fact 3 At the end points of cvh(e) ⊂ we have that Main Results R |T (x)| = kT k and Totik Widom n n e Bounds

Szegő–Widom Asymptotics Alternation and Zeros

and we deﬁne the Robin constant, of a compact set e ⊂ C R(e) = inf{E(µ) | supp(µ) ⊂ e and µ(e) = 1} If R(e) = ∞, we say e is a polar set or has capacity zero. If something holds except for a polar set, we say it holds q.e. (for quasi-everywhere). The capacity, C(e), of e is deﬁned by C(e) = exp(−R(e)) R(e) = log(1/C(e))

Coulomb Energies and All That

Szegő realized that Chebyshev polynomials are intimately connected with two dimensional potential theory, so I want

Chebyshev to review some of the basics of that subject. Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics and we deﬁne the Robin constant, of a compact set e ⊂ C R(e) = inf{E(µ) | supp(µ) ⊂ e and µ(e) = 1} If R(e) = ∞, we say e is a polar set or has capacity zero. If something holds except for a polar set, we say it holds q.e. (for quasi-everywhere). The capacity, C(e), of e is deﬁned by C(e) = exp(−R(e)) R(e) = log(1/C(e))

Coulomb Energies and All That

Szegő realized that Chebyshev polynomials are intimately connected with two dimensional potential theory, so I want

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics If R(e) = ∞, we say e is a polar set or has capacity zero. If something holds except for a polar set, we say it holds q.e. (for quasi-everywhere). The capacity, C(e), of e is deﬁned by C(e) = exp(−R(e)) R(e) = log(1/C(e))

Coulomb Energies and All That

Szegő realized that Chebyshev polynomials are intimately connected with two dimensional potential theory, so I want

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics If something holds except for a polar set, we say it holds q.e. (for quasi-everywhere). The capacity, C(e), of e is deﬁned by C(e) = exp(−R(e)) R(e) = log(1/C(e))

Coulomb Energies and All That

Szegő realized that Chebyshev polynomials are intimately connected with two dimensional potential theory, so I want

Chebyshev to review some of the basics of that subject. Given a Polynomials probability measure, dµ, of compact support on C, we Alternation Theorem deﬁne its Coulomb energy, E(µ) by Potential Theory Z E(µ) = dµ(x) dµ(y) log |x − y|−1 Lower Bounds and Special Sets Faber Fekete and we deﬁne the Robin constant, of a compact set e ⊂ C Szegő Theorem Widom’s Work R(e) = inf{E(µ) | supp(µ) ⊂ e and µ(e) = 1} Totik Widom Bounds If R(e) = ∞, we say e is a polar set or has capacity zero. Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics The capacity, C(e), of e is deﬁned by C(e) = exp(−R(e)) R(e) = log(1/C(e))

Coulomb Energies and All That

Szegő realized that Chebyshev polynomials are intimately connected with two dimensional potential theory, so I want

Szegő–Widom Asymptotics Coulomb Energies and All That

Szegő realized that Chebyshev polynomials are intimately connected with two dimensional potential theory, so I want

Szegő–Widom by Asymptotics C(e) = exp(−R(e)) R(e) = log(1/C(e)) Since E(·) is strictly convex on the probability measures, this minimizer is unique. It is called the equilibrium measure or harmonic measure of e and denoted dρe. The second name comes from the fact that if f is a continuous function on e, there is a unique function, uf , harmonic on (C ∪ {∞}) \ e, which approaches f(x) for q.e. x ∈ e (i.e., solves the Dirichlet problem) and Z uf (∞) = f(x)dρe(x) e R −1 The function Φe(z) = e dρe(x) log |x − z| is called the equilibrium potential.

Equilibrium Measures and All That

If e is not a polar set, it follows from weak lower semicontinuity of E(·) and weak compactness of the family Chebyshev Polynomials of probability measures that there is a probability measure

Alternation whose Coulomb energy is R(e). Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics It is called the equilibrium measure or harmonic measure of e and denoted dρe. The second name comes from the fact that if f is a continuous function on e, there is a unique function, uf , harmonic on (C ∪ {∞}) \ e, which approaches f(x) for q.e. x ∈ e (i.e., solves the Dirichlet problem) and Z uf (∞) = f(x)dρe(x) e R −1 The function Φe(z) = e dρe(x) log |x − z| is called the equilibrium potential.

Equilibrium Measures and All That

If e is not a polar set, it follows from weak lower semicontinuity of E(·) and weak compactness of the family Chebyshev Polynomials of probability measures that there is a probability measure

Alternation whose Coulomb energy is R(e). Since E(·) is strictly convex Theorem on the probability measures, this minimizer is unique. Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics The second name comes from the fact that if f is a continuous function on e, there is a unique function, uf , harmonic on (C ∪ {∞}) \ e, which approaches f(x) for q.e. x ∈ e (i.e., solves the Dirichlet problem) and Z uf (∞) = f(x)dρe(x) e R −1 The function Φe(z) = e dρe(x) log |x − z| is called the equilibrium potential.

Equilibrium Measures and All That

If e is not a polar set, it follows from weak lower semicontinuity of E(·) and weak compactness of the family Chebyshev Polynomials of probability measures that there is a probability measure

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics and Z uf (∞) = f(x)dρe(x) e R −1 The function Φe(z) = e dρe(x) log |x − z| is called the equilibrium potential.

Equilibrium Measures and All That

If e is not a polar set, it follows from weak lower semicontinuity of E(·) and weak compactness of the family Chebyshev Polynomials of probability measures that there is a probability measure

Alternation whose Coulomb energy is R(e). Since E(·) is strictly convex Theorem on the probability measures, this minimizer is unique. It is Potential Theory called the equilibrium measure or harmonic measure of e Lower Bounds and Special Sets and denoted dρe. The second name comes from the fact Faber Fekete that if f is a continuous function on e, there is a unique Szegő Theorem function, u , harmonic on ( ∪ {∞}) \ e, which approaches Widom’s Work f C

Totik Widom f(x) for q.e. x ∈ e (i.e., solves the Dirichlet problem) Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics R −1 The function Φe(z) = e dρe(x) log |x − z| is called the equilibrium potential.

Equilibrium Measures and All That

If e is not a polar set, it follows from weak lower semicontinuity of E(·) and weak compactness of the family Chebyshev Polynomials of probability measures that there is a probability measure

Totik Widom f(x) for q.e. x ∈ e (i.e., solves the Dirichlet problem) and Bounds Z u (∞) = f(x)dρe(x) Main Results f e Totik Widom Bounds

Szegő–Widom Asymptotics Equilibrium Measures and All That

If e is not a polar set, it follows from weak lower semicontinuity of E(·) and weak compactness of the family Chebyshev Polynomials of probability measures that there is a probability measure

Totik Widom f(x) for q.e. x ∈ e (i.e., solves the Dirichlet problem) and Bounds Z u (∞) = f(x)dρe(x) Main Results f e Totik Widom The function Φ (z) = R dρ (x) log |x − z|−1 is called the Bounds e e e

Szegő–Widom equilibrium potential. Asymptotics It is the unique function harmonic on C \ e with q.e. boundary value 0 on e and so that Ge(z) − log |z| is harmonic at ∞. Moreover, Ge(z) ≥ 0 everywhere and near ∞

Ge(z) = log |z| + R(e) + O(1/|z|) equivalently,

|z| exp(G (z)) = + O(1) e C(e)

Green’s Function

The Green’s function, Ge(z), of a compact subset, e ⊂ C, is deﬁned by Chebyshev Polynomials Alternation Ge(z) = R(e) − Φe(z) Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Moreover, Ge(z) ≥ 0 everywhere and near ∞

Ge(z) = log |z| + R(e) + O(1/|z|) equivalently,

|z| exp(G (z)) = + O(1) e C(e)

Green’s Function

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics equivalently,

|z| exp(G (z)) = + O(1) e C(e)

Green’s Function

The Green’s function, Ge(z), of a compact subset, e ⊂ C, is deﬁned by Chebyshev Polynomials Alternation Ge(z) = R(e) − Φe(z) Theorem Potential Theory It is the unique function harmonic on C \ e with q.e. Lower Bounds boundary value 0 on e and so that Ge(z) − log |z| is and Special Sets harmonic at ∞. Moreover, Ge(z) ≥ 0 everywhere and near Faber Fekete Szegő Theorem ∞ Widom’s Work Totik Widom G (z) = log |z| + R(e) + O(1/|z|) Bounds e

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Green’s Function

C(e) ≤ C(fn) −1 The function, G(z) = n log (|Tn(z)|/kTnke) is the Green’s function of fn (check properties) so 1/n n C(fn) = kTnke ⇒ kTnke ≥ C(e) an inequality of Szegő with a new proof (not that his proof was complicated).

Szegő’s Lower Bound

Let e ⊂ C.

Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics so that e ⊂ fn and thus

Szegő’s Lower Bound

Let e ⊂ C. Deﬁne fn = {z | |Tn(z)| ≤ kTnke}

Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics and thus

Szegő’s Lower Bound

Let e ⊂ C. Deﬁne fn = {z | |Tn(z)| ≤ kTnke} so that e ⊂ fn

Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics −1 The function, G(z) = n log (|Tn(z)|/kTnke) is the Green’s function of fn (check properties) so 1/n n C(fn) = kTnke ⇒ kTnke ≥ C(e) an inequality of Szegő with a new proof (not that his proof was complicated).

Szegő’s Lower Bound

Let e ⊂ C. Deﬁne fn = {z | |Tn(z)| ≤ kTnke} so that e ⊂ fn and thus Chebyshev Polynomials C(e) ≤ C(fn) Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics 1/n n C(fn) = kTnke ⇒ kTnke ≥ C(e) an inequality of Szegő with a new proof (not that his proof was complicated).

Szegő’s Lower Bound

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics an inequality of Szegő with a new proof (not that his proof was complicated).

Szegő’s Lower Bound

Let e ⊂ C. Deﬁne fn = {z | |Tn(z)| ≤ kTnke} so that e ⊂ fn and thus Chebyshev Polynomials C(e) ≤ C(fn) Alternation −1 Theorem The function, G(z) = n log (|Tn(z)|/kTnke) is the Potential Theory Green’s function of fn (check properties) so Lower Bounds and Special Sets 1/n n C(fn) = kTnke ⇒ kTnke ≥ C(e) Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Szegő’s Lower Bound

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics en = {z ∈ C | Tn(z) ∈ [−kTnke, kTnke]} which the alternation theorem implies is a union of n intervals in which Tn is monotone (the intervals can touch). As in the complex case, we have that

C(e) ≤ C(en) but now the Green’s function is s ! 1 T (z) T (z)2 G (z) = log n + n − 1 en 2 n kTnke kTnke For z near ∞ the argument inside the log is close to n 2z /kTnke which leads to 1/n n C(en) = (kTnke/2) ⇒ kTnke ≥ 2C(e) an inequality of Schiefermayr with a new and simpler proof.

Schiefermayr’s Theorem

Similarly if e ⊂ R, we deﬁne

Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics which the alternation theorem implies is a union of n intervals in which Tn is monotone (the intervals can touch). As in the complex case, we have that

Schiefermayr’s Theorem

Similarly if e ⊂ R, we deﬁne

Chebyshev en = {z ∈ C | Tn(z) ∈ [−kTnke, kTnke]} Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics As in the complex case, we have that

Schiefermayr’s Theorem

Similarly if e ⊂ R, we deﬁne

Chebyshev en = {z ∈ C | Tn(z) ∈ [−kTnke, kTnke]} Polynomials which the alternation theorem implies is a union of n Alternation Theorem intervals in which Tn is monotone (the intervals can touch). Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics but now the Green’s function is s ! 1 T (z) T (z)2 G (z) = log n + n − 1 en 2 n kTnke kTnke For z near ∞ the argument inside the log is close to n 2z /kTnke which leads to 1/n n C(en) = (kTnke/2) ⇒ kTnke ≥ 2C(e) an inequality of Schiefermayr with a new and simpler proof.

Schiefermayr’s Theorem

Similarly if e ⊂ R, we deﬁne

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics s ! 1 T (z) T (z)2 G (z) = log n + n − 1 en 2 n kTnke kTnke For z near ∞ the argument inside the log is close to n 2z /kTnke which leads to 1/n n C(en) = (kTnke/2) ⇒ kTnke ≥ 2C(e) an inequality of Schiefermayr with a new and simpler proof.

Schiefermayr’s Theorem

Similarly if e ⊂ R, we deﬁne

Faber Fekete Szegő Theorem but now the Green’s function is

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics For z near ∞ the argument inside the log is close to n 2z /kTnke which leads to 1/n n C(en) = (kTnke/2) ⇒ kTnke ≥ 2C(e) an inequality of Schiefermayr with a new and simpler proof.

Schiefermayr’s Theorem

Similarly if e ⊂ R, we deﬁne

Faber Fekete Szegő Theorem but now the Green’s function is s ! Widom’s Work 1 T (z) T (z)2 G (z) = log n + n − 1 Totik Widom en 2 Bounds n kTnke kTnke Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics 1/n n C(en) = (kTnke/2) ⇒ kTnke ≥ 2C(e) an inequality of Schiefermayr with a new and simpler proof.

Schiefermayr’s Theorem

Similarly if e ⊂ R, we deﬁne

Szegő–Widom Asymptotics an inequality of Schiefermayr with a new and simpler proof.

Schiefermayr’s Theorem

Similarly if e ⊂ R, we deﬁne

Faber Fekete Szegő Theorem but now the Green’s function is s ! Widom’s Work 1 T (z) T (z)2 G (z) = log n + n − 1 Totik Widom en 2 Bounds n kTnke kTnke Main Results For z near ∞ the argument inside the log is close to Totik Widom n Bounds 2z /kTnke which leads to Szegő–Widom 1/n n Asymptotics C(en) = (kTnke/2) ⇒ kTnke ≥ 2C(e) an inequality of Schiefermayr with a new and simpler proof. (a formula called the Thouless formula by physicists after the recent Nobel Laureate, David Thouless). That shows that each of the sets between two opposite sign extrema of Tn has en–harmonic measure 1/n. This, in turn implies each connected component of en has harmonic measure k/n for some integer k. Such a set is called a period–n set.

If e is a period–n set, one can prove that en = e so that all the period–n sets are precisely the possible en’s.

Schiefermayr’s Theorem

The harmonic measure of a set e ⊂ R is the boundary value of the harmonic conjugate of the Green’s function Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics That shows that each of the sets between two opposite sign extrema of Tn has en–harmonic measure 1/n. This, in turn implies each connected component of en has harmonic measure k/n for some integer k. Such a set is called a period–n set.

If e is a period–n set, one can prove that en = e so that all the period–n sets are precisely the possible en’s.

Schiefermayr’s Theorem

The harmonic measure of a set e ⊂ R is the boundary value of the harmonic conjugate of the Green’s function (a Chebyshev Polynomials formula called the Thouless formula by physicists after the

Alternation recent Nobel Laureate, David Thouless). Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics This, in turn implies each connected component of en has harmonic measure k/n for some integer k. Such a set is called a period–n set.

If e is a period–n set, one can prove that en = e so that all the period–n sets are precisely the possible en’s.

Schiefermayr’s Theorem

The harmonic measure of a set e ⊂ R is the boundary value of the harmonic conjugate of the Green’s function (a Chebyshev Polynomials formula called the Thouless formula by physicists after the

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics If e is a period–n set, one can prove that en = e so that all the period–n sets are precisely the possible en’s.

Schiefermayr’s Theorem

The harmonic measure of a set e ⊂ R is the boundary value of the harmonic conjugate of the Green’s function (a Chebyshev Polynomials formula called the Thouless formula by physicists after the

Alternation recent Nobel Laureate, David Thouless). That shows that Theorem each of the sets between two opposite sign extrema of Tn Potential Theory has en–harmonic measure 1/n. This, in turn implies each Lower Bounds and Special Sets connected component of en has harmonic measure k/n for Faber Fekete some integer k. Such a set is called a period–n set. Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Schiefermayr’s Theorem

The harmonic measure of a set e ⊂ R is the boundary value of the harmonic conjugate of the Green’s function (a Chebyshev Polynomials formula called the Thouless formula by physicists after the

Widom’s Work If e is a period–n set, one can prove that en = e so that all Totik Widom the period–n sets are precisely the possible e ’s. Bounds n

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Its Green’s function is log |z| so R(e) = 0 and C(e) = 1. Since Tn is monic R 2π 0 exp(−inθ) Tn(exp(iθ)) dθ/2π = 1

we see that kTnke ≥ 1 so that n n Tn(z) = z ; kTnke = 1 = C(e) Example ([−1, 1]) It is known (and follows from results 1 later) that C(e) = 2 . By the Alternation Theorem, the polynomials given by Qn(cos(θ)) = cos(nθ) (i.e. “the Chebyshev polynomials of the ﬁrst kind”) are multiples of Chebyshev polynomials as we’ve deﬁned them, so −n+1 −n+1 n Tn(cos(θ)) = 2 cos(nθ); kTnke = 2 = 2 C(e)

so one can have equality in both lower bounds.

Example

Example (∂D, the unit circle)

Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Since Tn is monic R 2π 0 exp(−inθ) Tn(exp(iθ)) dθ/2π = 1

so one can have equality in both lower bounds.

Example

Example (∂D, the unit circle) Its Green’s function is log |z| so R(e) = 0 and C(e) = 1. Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics we see that kTnke ≥ 1 so that n n Tn(z) = z ; kTnke = 1 = C(e) Example ([−1, 1]) It is known (and follows from results 1 later) that C(e) = 2 . By the Alternation Theorem, the polynomials given by Qn(cos(θ)) = cos(nθ) (i.e. “the Chebyshev polynomials of the ﬁrst kind”) are multiples of Chebyshev polynomials as we’ve deﬁned them, so −n+1 −n+1 n Tn(cos(θ)) = 2 cos(nθ); kTnke = 2 = 2 C(e)

so one can have equality in both lower bounds.

Example

Example (∂D, the unit circle) Its Green’s function is log |z| so R(e) = 0 and C(e) = 1. Since Tn is monic Chebyshev Polynomials R 2π 0 exp(−inθ) Tn(exp(iθ)) dθ/2π = 1 Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics so that n n Tn(z) = z ; kTnke = 1 = C(e) Example ([−1, 1]) It is known (and follows from results 1 later) that C(e) = 2 . By the Alternation Theorem, the polynomials given by Qn(cos(θ)) = cos(nθ) (i.e. “the Chebyshev polynomials of the ﬁrst kind”) are multiples of Chebyshev polynomials as we’ve deﬁned them, so −n+1 −n+1 n Tn(cos(θ)) = 2 cos(nθ); kTnke = 2 = 2 C(e)

so one can have equality in both lower bounds.

Example

Example (∂D, the unit circle) Its Green’s function is log |z| so R(e) = 0 and C(e) = 1. Since Tn is monic Chebyshev Polynomials R 2π 0 exp(−inθ) Tn(exp(iθ)) dθ/2π = 1 Alternation Theorem we see that kTnke ≥ 1 Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Example ([−1, 1]) It is known (and follows from results 1 later) that C(e) = 2 . By the Alternation Theorem, the polynomials given by Qn(cos(θ)) = cos(nθ) (i.e. “the Chebyshev polynomials of the ﬁrst kind”) are multiples of Chebyshev polynomials as we’ve deﬁned them, so −n+1 −n+1 n Tn(cos(θ)) = 2 cos(nθ); kTnke = 2 = 2 C(e)

so one can have equality in both lower bounds.

Example

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics It is known (and follows from results 1 later) that C(e) = 2 . By the Alternation Theorem, the polynomials given by Qn(cos(θ)) = cos(nθ) (i.e. “the Chebyshev polynomials of the ﬁrst kind”) are multiples of Chebyshev polynomials as we’ve deﬁned them, so −n+1 −n+1 n Tn(cos(θ)) = 2 cos(nθ); kTnke = 2 = 2 C(e)

so one can have equality in both lower bounds.

Example

Faber Fekete Example ([−1, 1]) Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics By the Alternation Theorem, the polynomials given by Qn(cos(θ)) = cos(nθ) (i.e. “the Chebyshev polynomials of the ﬁrst kind”) are multiples of Chebyshev polynomials as we’ve deﬁned them, so −n+1 −n+1 n Tn(cos(θ)) = 2 cos(nθ); kTnke = 2 = 2 C(e)

so one can have equality in both lower bounds.

Example

Faber Fekete Example ([−1, 1]) It is known (and follows from results Szegő Theorem 1 later) that C(e) = 2 . Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics −n+1 −n+1 n Tn(cos(θ)) = 2 cos(nθ); kTnke = 2 = 2 C(e)

so one can have equality in both lower bounds.

Example

Szegő–Widom Asymptotics so one can have equality in both lower bounds.

Example

Faber Fekete Example ([−1, 1]) It is known (and follows from results Szegő Theorem 1 later) that C(e) = 2 . By the Alternation Theorem, the Widom’s Work polynomials given by Qn(cos(θ)) = cos(nθ) (i.e. “the Totik Widom Bounds Chebyshev polynomials of the first kind”) are multiples of Main Results Chebyshev polynomials as we’ve defined them, so Totik Widom −n+1 −n+1 n Bounds Tn(cos(θ)) = 2 cos(nθ); kTnke = 2 = 2 C(e) Szegő–Widom Asymptotics so one can have equality in both lower bounds. Given Szegő’s lower bound, we get a lower bound on the 1/n lim inf by C(e). One can get an upper bound on kTnke by Y 1/n(n+1) sup |zj − zk| z ∈e j 1≤j6=k≤n+1 using suitable trial monic polynomials. Fekete proved that as n → ∞, this last quantity had a limit that he called the transfinite diameter. One can view this sup as the exponential of the negative of a discrete Coulomb energy of 1 n + 1 point charges, each of charge about n+1 , so Szegő’s proof that this is C(e) is natural from a Coulomb energy point of view.

FFS Theorem

Theorem (Faber–Fekete–Szegő Theorem) For any compact subset e ⊂ C, we have that Chebyshev 1/n Polynomials lim kTnke = C(e) n→∞ Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics 1/n One can get an upper bound on kTnke by Y 1/n(n+1) sup |zj − zk| z ∈e j 1≤j6=k≤n+1 using suitable trial monic polynomials. Fekete proved that as n → ∞, this last quantity had a limit that he called the transﬁnite diameter. One can view this sup as the exponential of the negative of a discrete Coulomb energy of 1 n + 1 point charges, each of charge about n+1 , so Szegő’s proof that this is C(e) is natural from a Coulomb energy point of view.

FFS Theorem

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Fekete proved that as n → ∞, this last quantity had a limit that he called the transﬁnite diameter. One can view this sup as the exponential of the negative of a discrete Coulomb energy of 1 n + 1 point charges, each of charge about n+1 , so Szegő’s proof that this is C(e) is natural from a Coulomb energy point of view.

FFS Theorem

Theorem (Faber–Fekete–Szegő Theorem) For any compact subset e ⊂ C, we have that Chebyshev 1/n Polynomials lim kTnke = C(e) n→∞ Alternation Theorem Given Szegő’s lower bound, we get a lower bound on the Potential Theory 1/n lim inf by C(e). One can get an upper bound on kTnke by Lower Bounds and Special Sets Y 1/n(n+1) sup |zj − zk| Faber Fekete zj ∈e Szegő Theorem 1≤j6=k≤n+1 Widom’s Work using suitable trial monic polynomials. Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics One can view this sup as the exponential of the negative of a discrete Coulomb energy of 1 n + 1 point charges, each of charge about n+1 , so Szegő’s proof that this is C(e) is natural from a Coulomb energy point of view.

FFS Theorem

Main Results transﬁnite diameter.

Totik Widom Bounds

Szegő–Widom Asymptotics FFS Theorem

Main Results transﬁnite diameter. One can view this sup as the Totik Widom exponential of the negative of a discrete Coulomb energy of Bounds n + 1 point charges, each of charge about 1 , so Szegő’s Szegő–Widom n+1 Asymptotics proof that this is C(e) is natural from a Coulomb energy point of view. It is more restrictive in that he only studied the special case where e is a single (closed) analytic Jordan curve. But in this case, he proved much more — ﬁrst he n proved that limn→∞kTnke/C(e) = 1. He also obtained asymptotics for the polynomials themselves. The unbounded component, Ω, of (C ∪ {∞}) \ e is simply connected, so Ge(z) has a single valued harmonic conjugate and thus, by exponentiating, there is a function, Be(z), on Ω with |Be(z)| = exp(−Ge(z)) with an overall phase determined by demanding that as z → ∞, we have that −1 z Be(z) = C(e) + O(1). Since the curve is analytic, Be(z) has an analytic continuation to a neighborhood of e.

History

Faber’s name is associated to this theorem because of a 1919 paper in which he proved a result that is both much

Chebyshev more restrictive and much stronger than what we call the Polynomials FFS Theorem. Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics But in this case, he proved much more — ﬁrst he n proved that limn→∞kTnke/C(e) = 1. He also obtained asymptotics for the polynomials themselves. The unbounded component, Ω, of (C ∪ {∞}) \ e is simply connected, so Ge(z) has a single valued harmonic conjugate and thus, by exponentiating, there is a function, Be(z), on Ω with |Be(z)| = exp(−Ge(z)) with an overall phase determined by demanding that as z → ∞, we have that −1 z Be(z) = C(e) + O(1). Since the curve is analytic, Be(z) has an analytic continuation to a neighborhood of e.

History

Faber’s name is associated to this theorem because of a 1919 paper in which he proved a result that is both much

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics He also obtained asymptotics for the polynomials themselves. The unbounded component, Ω, of (C ∪ {∞}) \ e is simply connected, so Ge(z) has a single valued harmonic conjugate and thus, by exponentiating, there is a function, Be(z), on Ω with |Be(z)| = exp(−Ge(z)) with an overall phase determined by demanding that as z → ∞, we have that −1 z Be(z) = C(e) + O(1). Since the curve is analytic, Be(z) has an analytic continuation to a neighborhood of e.

History

Faber’s name is associated to this theorem because of a 1919 paper in which he proved a result that is both much

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics so Ge(z) has a single valued harmonic conjugate and thus, by exponentiating, there is a function, Be(z), on Ω with |Be(z)| = exp(−Ge(z)) with an overall phase determined by demanding that as z → ∞, we have that −1 z Be(z) = C(e) + O(1). Since the curve is analytic, Be(z) has an analytic continuation to a neighborhood of e.

History

Faber’s name is associated to this theorem because of a 1919 paper in which he proved a result that is both much

Chebyshev more restrictive and much stronger than what we call the Polynomials FFS Theorem. It is more restrictive in that he only studied Alternation Theorem the special case where e is a single (closed) analytic Jordan Potential Theory curve. But in this case, he proved much more — ﬁrst he n Lower Bounds proved that limn→∞kTnke/C(e) = 1. and Special Sets Faber Fekete He also obtained asymptotics for the polynomials Szegő Theorem themselves. The unbounded component, Ω, of Widom’s Work ( ∪ {∞}) \ e is simply connected, Totik Widom C Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics and thus, by exponentiating, there is a function, Be(z), on Ω with |Be(z)| = exp(−Ge(z)) with an overall phase determined by demanding that as z → ∞, we have that −1 z Be(z) = C(e) + O(1). Since the curve is analytic, Be(z) has an analytic continuation to a neighborhood of e.

History

Faber’s name is associated to this theorem because of a 1919 paper in which he proved a result that is both much

Totik Widom Bounds

Szegő–Widom Asymptotics Since the curve is analytic, Be(z) has an analytic continuation to a neighborhood of e.

History

Faber’s name is associated to this theorem because of a 1919 paper in which he proved a result that is both much

Szegő–Widom demanding that as z → ∞, we have that Asymptotics −1 z Be(z) = C(e) + O(1). History

Faber’s name is associated to this theorem because of a 1919 paper in which he proved a result that is both much

Szegő–Widom demanding that as z → ∞, we have that Asymptotics −1 z Be(z) = C(e) + O(1). Since the curve is analytic, Be(z) has an analytic continuation to a neighborhood of e. Faber didn’t mention Green’s functions −1 or capacities at all! In this case, Be(z) can be described as the Riemann map of Ω to (C ∪ {∞} \ D) (with positive “derivative” at ∞) and the capacity appears as inverse of the value of that “derivative”. Interestingly enough, for these polynomials, Faber had “Szegő asymptotics” three years before Szegő had his asymptotics (for OPUC, not Chebyshev polynomials). Fekete’s work on transﬁnite diameters and its connection to capacity for some special cases is from 1923. Szegő had the full theorem in a 1924 paper whose title started “Comments on a paper by Mr. M. Fekete”.

History

Faber proved that uniformly on Ω plus a neighborhood of e, n Tn(z)Be(z) → 1. Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics −1 In this case, Be(z) can be described as the Riemann map of Ω to (C ∪ {∞} \ D) (with positive “derivative” at ∞) and the capacity appears as inverse of the value of that “derivative”. Interestingly enough, for these polynomials, Faber had “Szegő asymptotics” three years before Szegő had his asymptotics (for OPUC, not Chebyshev polynomials). Fekete’s work on transﬁnite diameters and its connection to capacity for some special cases is from 1923. Szegő had the full theorem in a 1924 paper whose title started “Comments on a paper by Mr. M. Fekete”.

History

Faber proved that uniformly on Ω plus a neighborhood of e, n Tn(z)Be(z) → 1. Faber didn’t mention Green’s functions Chebyshev Polynomials or capacities at all!

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Interestingly enough, for these polynomials, Faber had “Szegő asymptotics” three years before Szegő had his asymptotics (for OPUC, not Chebyshev polynomials). Fekete’s work on transﬁnite diameters and its connection to capacity for some special cases is from 1923. Szegő had the full theorem in a 1924 paper whose title started “Comments on a paper by Mr. M. Fekete”.

History

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Fekete’s work on transﬁnite diameters and its connection to capacity for some special cases is from 1923. Szegő had the full theorem in a 1924 paper whose title started “Comments on a paper by Mr. M. Fekete”.

History

Faber proved that uniformly on Ω plus a neighborhood of e, n Tn(z)Be(z) → 1. Faber didn’t mention Green’s functions Chebyshev −1 Polynomials or capacities at all! In this case, Be(z) can be described Alternation as the Riemann map of Ω to (C ∪ {∞} \ D) (with positive Theorem “derivative” at ∞) and the capacity appears as inverse of Potential Theory the value of that “derivative”. Lower Bounds and Special Sets Interestingly enough, for these polynomials, Faber had Faber Fekete Szegő Theorem “Szegő asymptotics” three years before Szegő had his Widom’s Work asymptotics (for OPUC, not Chebyshev polynomials). Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Szegő had the full theorem in a 1924 paper whose title started “Comments on a paper by Mr. M. Fekete”.

History

Szegő–Widom Asymptotics History

Szegő–Widom on a paper by Mr. M. Fekete”. Asymptotics The cases with e ⊂ R are exactly the ﬁnite gap sets. As in the work of Faber, it is natural to look for an analytic function, Be(z), with |Be(z)| = exp(−Ge(z)) on Ω, the unbounded component of (C ∪ {∞}) \ e. The problem is that Ω is no longer simply connected so the magnitude of Be(z) is single valued but its phase is multivalued.

Put diﬀerently, Be(z) can be continued along any curve in Ω and there is a map from the fundamental group of Ω to ∂D, which is a character (i.e. group homomorphism), so that after continuation around a closed curve, Be(z) is multiplied by the character applied to that curve.

Character Automorphic Functions

In 1969, Widom published a 100+ page brilliant, seminal work on asymptotics of Chebyshev and orthogonal Chebyshev Polynomials polynomials. In his set up, e is a ﬁnite union of (closed)

Alternation analytic Jordan curves and/or (open) Jordan arcs. Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics As in the work of Faber, it is natural to look for an analytic function, Be(z), with |Be(z)| = exp(−Ge(z)) on Ω, the unbounded component of (C ∪ {∞}) \ e. The problem is that Ω is no longer simply connected so the magnitude of Be(z) is single valued but its phase is multivalued.

Character Automorphic Functions

In 1969, Widom published a 100+ page brilliant, seminal work on asymptotics of Chebyshev and orthogonal Chebyshev Polynomials polynomials. In his set up, e is a ﬁnite union of (closed)

Alternation analytic Jordan curves and/or (open) Jordan arcs. The Theorem cases with e ⊂ R are exactly the ﬁnite gap sets. Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics The problem is that Ω is no longer simply connected so the magnitude of Be(z) is single valued but its phase is multivalued.

Character Automorphic Functions

In 1969, Widom published a 100+ page brilliant, seminal work on asymptotics of Chebyshev and orthogonal Chebyshev Polynomials polynomials. In his set up, e is a ﬁnite union of (closed)

Alternation analytic Jordan curves and/or (open) Jordan arcs. The Theorem cases with e ⊂ R are exactly the ﬁnite gap sets. Potential Theory

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Put diﬀerently, Be(z) can be continued along any curve in Ω and there is a map from the fundamental group of Ω to ∂D, which is a character (i.e. group homomorphism), so that after continuation around a closed curve, Be(z) is multiplied by the character applied to that curve.

Character Automorphic Functions

In 1969, Widom published a 100+ page brilliant, seminal work on asymptotics of Chebyshev and orthogonal Chebyshev Polynomials polynomials. In his set up, e is a ﬁnite union of (closed)

Alternation analytic Jordan curves and/or (open) Jordan arcs. The Theorem cases with e ⊂ R are exactly the ﬁnite gap sets. Potential Theory

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Character Automorphic Functions

In 1969, Widom published a 100+ page brilliant, seminal work on asymptotics of Chebyshev and orthogonal Chebyshev Polynomials polynomials. In his set up, e is a ﬁnite union of (closed)

Alternation analytic Jordan curves and/or (open) Jordan arcs. The Theorem cases with e ⊂ R are exactly the ﬁnite gap sets. Potential Theory

Lower Bounds As in the work of Faber, it is natural to look for an analytic and Special Sets function, Be(z), with |Be(z)| = exp(−Ge(z)) on Ω, the Faber Fekete Szegő Theorem unbounded component of (C ∪ {∞}) \ e. The problem is Widom’s Work that Ω is no longer simply connected so the magnitude of Totik Widom Be(z) is single valued but its phase is multivalued. Bounds Main Results Put diﬀerently, Be(z) can be continued along any curve in Totik Widom Ω and there is a map from the fundamental group of Ω to Bounds ∂ , which is a character (i.e. group homomorphism), so Szegő–Widom D Asymptotics that after continuation around a closed curve, Be(z) is multiplied by the character applied to that curve. n −n If Tn(z)Be(z) C(e) had a limit, that limit cannot be n independent since the character is n dependent. Widom had the idea that there should be functions Fχ(z) deﬁned for each χ in the character group and continuous in χ so the limit is the Fχ, call it Fn, associated to the character of n Be(z) . As a function of n, the limit will be almost periodic!

Character Automorphic Functions

Indeed, if the curve loops around a subset g ⊂ e, the phase changes by −2πρe(g). Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Widom had the idea that there should be functions Fχ(z) deﬁned for each χ in the character group and continuous in χ so the limit is the Fχ, call it Fn, associated to the character of n Be(z) . As a function of n, the limit will be almost periodic!

Character Automorphic Functions

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Character Automorphic Functions

Indeed, if the curve loops around a subset g ⊂ e, the phase changes by −2πρe(g). Chebyshev Polynomials n −n If Tn(z)Be(z) C(e) had a limit, that limit cannot be n Alternation Theorem independent since the character is n dependent. Widom had Potential Theory the idea that there should be functions Fχ(z) deﬁned for Lower Bounds each χ in the character group and continuous in χ so the and Special Sets limit is the Fχ, call it Fn, associated to the character of Faber Fekete n Szegő Theorem Be(z) . As a function of n, the limit will be almost periodic! Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Widom proved uniqueness of the minimizer and found a formula for it (in terms of some theta functions and solutions of some implicit equations). He also proved that kFχkΩ is continuous in χ. Because of the uniqueness, one can prove that the functions, Fχ(z), deﬁned for z ∈ Ω, are continuous in χ on the compact set of characters, uniformly locally in z (but as functions on the covering space not uniformly in all z).

Widom’s Minimizers

He even found a candidate for the functions! Let Fχ(z) be that function among all character automorphic functions, Chebyshev Polynomials A(z), on Ω with character χ and with A(∞) = 1, that Alternation minimizes supz∈Ω{|A(z)|}. Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics He also proved that kFχkΩ is continuous in χ. Because of the uniqueness, one can prove that the functions, Fχ(z), deﬁned for z ∈ Ω, are continuous in χ on the compact set of characters, uniformly locally in z (but as functions on the covering space not uniformly in all z).

Widom’s Minimizers

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Because of the uniqueness, one can prove that the functions, Fχ(z), deﬁned for z ∈ Ω, are continuous in χ on the compact set of characters, uniformly locally in z (but as functions on the covering space not uniformly in all z).

Widom’s Minimizers

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Widom’s Minimizers

Potential Theory Widom proved uniqueness of the minimizer and found a Lower Bounds formula for it (in terms of some theta functions and and Special Sets solutions of some implicit equations). He also proved that Faber Fekete Szegő Theorem kFχkΩ is continuous in χ. Because of the uniqueness, one Widom’s Work can prove that the functions, Fχ(z), deﬁned for z ∈ Ω, are Totik Widom continuous in χ on the compact set of characters, uniformly Bounds

Main Results locally in z (but as functions on the covering space not

Totik Widom uniformly in all z). Bounds

Szegő–Widom Asymptotics By modifying his proof, in Part 2 we ﬁnd a simple proof of uniqueness for any compact set e ⊂ C. Theorem (Widom) Let e be a ﬁnite union of disjoint analytic Jordan curves. Let Fn(z) be as above for the n character of Be(z) . Then:

n kTnke Tn(z)Be(z) lim n = 1; lim n − Fn(z) = 0 n→∞ C(e) kFnkΩ n→∞ C(e) where the limit is uniform on compact subsets of Ω.

Since |Be(z)| → 1 and kFnkΩ is taken as z → e , the z asymptotics and norm limit ﬁt together.

Widom’s Theorems and Conjecture

The Widom minimizers are analogs of the Ahlfors function for which Fisher found a simple elegant proof of uniqueness. Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Theorem (Widom) Let e be a ﬁnite union of disjoint analytic Jordan curves. Let Fn(z) be as above for the n character of Be(z) . Then:

n kTnke Tn(z)Be(z) lim n = 1; lim n − Fn(z) = 0 n→∞ C(e) kFnkΩ n→∞ C(e) where the limit is uniform on compact subsets of Ω.

Since |Be(z)| → 1 and kFnkΩ is taken as z → e , the z asymptotics and norm limit ﬁt together.

Widom’s Theorems and Conjecture

The Widom minimizers are analogs of the Ahlfors function for which Fisher found a simple elegant proof of uniqueness. Chebyshev Polynomials By modifying his proof, in Part 2 we ﬁnd a simple proof of Alternation uniqueness for any compact set e ⊂ C. Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics n kTnke Tn(z)Be(z) lim n = 1; lim n − Fn(z) = 0 n→∞ C(e) kFnkΩ n→∞ C(e) where the limit is uniform on compact subsets of Ω.

Since |Be(z)| → 1 and kFnkΩ is taken as z → e , the z asymptotics and norm limit ﬁt together.

Widom’s Theorems and Conjecture

Potential Theory Theorem (Widom) Let e be a ﬁnite union of disjoint Lower Bounds analytic Jordan curves. Let Fn(z) be as above for the and Special Sets n character of Be(z) . Then: Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Since |Be(z)| → 1 and kFnkΩ is taken as z → e , the z asymptotics and norm limit ﬁt together.

Widom’s Theorems and Conjecture

Szegő–Widom Asymptotics Widom’s Theorems and Conjecture

Widom’s Work n kTnke Tn(z)Be(z) Totik Widom lim = 1; lim − F (z) = 0 Bounds n n n n→∞ C(e) kFnkΩ n→∞ C(e) Main Results Totik Widom where the limit is uniform on compact subsets of Ω. Bounds Szegő–Widom Since |Be(z)| → 1 and kFnkΩ is taken as z → e , the z Asymptotics asymptotics and norm limit ﬁt together. Conjecture (Widom) Let e be a ﬁnite gap subset of R. Let Fn(z) be as above. Then:

n Tn(z)Be(z) lim − Fn(z) = 0 n→∞ C(e)n uniformly on compact subsets of Ω.

The norm, kTnke is twice as large as one might expect! Note: This is Widom’s conjecture for e ⊂ R; he made the conjecture for more general cases of e ⊂ C.

Widom’s Theorems and Conjecture

Theorem (Widom) Let e be a ﬁnite gap subset of R. Let Fn(z) be as above. Then Chebyshev Polynomials kTnke Alternation lim n = 1 Theorem n→∞ 2C(e) kFnkΩ Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics The norm, kTnke is twice as large as one might expect! Note: This is Widom’s conjecture for e ⊂ R; he made the conjecture for more general cases of e ⊂ C.

Widom’s Theorems and Conjecture

Totik Widom Bounds

Szegő–Widom Asymptotics Note: This is Widom’s conjecture for e ⊂ R; he made the conjecture for more general cases of e ⊂ C.

Widom’s Theorems and Conjecture

Szegő–Widom Asymptotics Widom’s Theorems and Conjecture

Theorem (Widom) Let e be a ﬁnite gap subset of R. Let Fn(z) be as above. Then Chebyshev Polynomials kTnke Alternation lim n = 1 Theorem n→∞ 2C(e) kFnkΩ Potential Theory Conjecture (Widom) Let e be a ﬁnite gap subset of . Let Lower Bounds R and Special Sets Fn(z) be as above. Then: Faber Fekete Szegő Theorem n Tn(z)Be(z) Widom’s Work lim − Fn(z) = 0 n→∞ C(e)n Totik Widom Bounds uniformly on compact subsets of Ω. Main Results Totik Widom The norm, kTnke is twice as large as one might expect! Bounds Note: This is Widom’s conjecture for e ⊂ ; he made the Szegő–Widom R Asymptotics conjecture for more general cases of e ⊂ C. where Ω is simply connected√ so Fn(z) ≡ 1. We have that √ 2 −1 2 Be(z) = z − z − 1. Notice that Be (z) = z + z − 1. On [−1, 1], of course, Be(x) has magnitude 1 (since Ge(x) = 0) so Be(x) = exp(iθ) and 1 −1 cos(θ) = 2 [Be(x) + Be (x)] = x. −n+1 Thus, by Tn(cos(θ)) = 2 cos(nθ), we see that −n n −n Tn(z) = 2 [Be (z) + Be (z)]. For z ∈ [−1, 1], both terms contribute and at some points add to 2 and we get −n+1 n n kTnke = 2 = 2C(e) . On Ω, |Be(z)| < 1 so the B term is negligible as n → ∞ and we lose the factor of 2. It was this example that led Widom to his conjecture.

Back to [−1, 1]

Example We return to the case of [−1, 1]

Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics √ −1 2 Notice that Be (z) = z + z − 1. On [−1, 1], of course, Be(x) has magnitude 1 (since Ge(x) = 0) so Be(x) = exp(iθ) and 1 −1 cos(θ) = 2 [Be(x) + Be (x)] = x. −n+1 Thus, by Tn(cos(θ)) = 2 cos(nθ), we see that −n n −n Tn(z) = 2 [Be (z) + Be (z)]. For z ∈ [−1, 1], both terms contribute and at some points add to 2 and we get −n+1 n n kTnke = 2 = 2C(e) . On Ω, |Be(z)| < 1 so the B term is negligible as n → ∞ and we lose the factor of 2. It was this example that led Widom to his conjecture.

Back to [−1, 1]

Example We return to the case of [−1, 1] where Ω is simply connected√ so Fn(z) ≡ 1. We have that Chebyshev 2 Polynomials Be(z) = z − z − 1.

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics On [−1, 1], of course, Be(x) has magnitude 1 (since Ge(x) = 0) so Be(x) = exp(iθ) and 1 −1 cos(θ) = 2 [Be(x) + Be (x)] = x. −n+1 Thus, by Tn(cos(θ)) = 2 cos(nθ), we see that −n n −n Tn(z) = 2 [Be (z) + Be (z)]. For z ∈ [−1, 1], both terms contribute and at some points add to 2 and we get −n+1 n n kTnke = 2 = 2C(e) . On Ω, |Be(z)| < 1 so the B term is negligible as n → ∞ and we lose the factor of 2. It was this example that led Widom to his conjecture.

Back to [−1, 1]

Example We return to the case of [−1, 1] where Ω is simply connected√ so Fn(z) ≡ 1. We have that √ Chebyshev 2 −1 2 Polynomials Be(z) = z − z − 1. Notice that Be (z) = z + z − 1. Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics −n+1 Thus, by Tn(cos(θ)) = 2 cos(nθ), we see that −n n −n Tn(z) = 2 [Be (z) + Be (z)]. For z ∈ [−1, 1], both terms contribute and at some points add to 2 and we get −n+1 n n kTnke = 2 = 2C(e) . On Ω, |Be(z)| < 1 so the B term is negligible as n → ∞ and we lose the factor of 2. It was this example that led Widom to his conjecture.

Back to [−1, 1]

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics −n n −n Tn(z) = 2 [Be (z) + Be (z)]. For z ∈ [−1, 1], both terms contribute and at some points add to 2 and we get −n+1 n n kTnke = 2 = 2C(e) . On Ω, |Be(z)| < 1 so the B term is negligible as n → ∞ and we lose the factor of 2. It was this example that led Widom to his conjecture.

Back to [−1, 1]

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics For z ∈ [−1, 1], both terms contribute and at some points add to 2 and we get −n+1 n n kTnke = 2 = 2C(e) . On Ω, |Be(z)| < 1 so the B term is negligible as n → ∞ and we lose the factor of 2. It was this example that led Widom to his conjecture.

Back to [−1, 1]

Example We return to the case of [−1, 1] where Ω is simply connected√ so Fn(z) ≡ 1. We have that √ Chebyshev 2 −1 2 Polynomials Be(z) = z − z − 1. Notice that Be (z) = z + z − 1. Alternation On [−1, 1], of course, Be(x) has magnitude 1 (since Theorem Ge(x) = 0) so Be(x) = exp(iθ) and Potential Theory 1 −1 cos(θ) = [Be(x) + Be (x)] = x. Lower Bounds 2 and Special Sets −n+1 Thus, by Tn(cos(θ)) = 2 cos(nθ), we see that Faber Fekete −n n −n Szegő Theorem Tn(z) = 2 [Be (z) + Be (z)]. Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics n On Ω, |Be(z)| < 1 so the B term is negligible as n → ∞ and we lose the factor of 2. It was this example that led Widom to his conjecture.

Back to [−1, 1]

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics It was this example that led Widom to his conjecture.

Back to [−1, 1]

Main Results term is negligible as n → ∞ and we lose the factor of 2.

Totik Widom Bounds

Szegő–Widom Asymptotics Back to [−1, 1]

Main Results term is negligible as n → ∞ and we lose the factor of 2.

Totik Widom Bounds It was this example that led Widom to his conjecture.

Szegő–Widom Asymptotics n Because kTnke = 2C(en) , this bound is equivalent to Theorem (Totik–Widom bounds in the finite gap case) If e is a finite gap set, then for a constant D we have that n kTnke ≤ DC(e) This complements the 2C(e)n lower bound. Because of his asymptotic result, Widom already had this bound in 1969 but Totik’s proof was much simpler. Neither proof has very explicit estimates for D. Even though they only had the result for finite gap sets, we will say that a general set e has Totik–Widom bounds, if there is an upper bound of the above form.

Totik–Widom bounds

Theorem (Totik’s 1/n bound) If e is a ﬁnite gap set, the period n sets ˜en ⊃ e can be chosen so that Chebyshev E Polynomials C(˜en) ≤ C(e) 1 + n for some constant E. Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Theorem (Totik–Widom bounds in the finite gap case) If e is a finite gap set, then for a constant D we have that n kTnke ≤ DC(e) This complements the 2C(e)n lower bound. Because of his asymptotic result, Widom already had this bound in 1969 but Totik’s proof was much simpler. Neither proof has very explicit estimates for D. Even though they only had the result for finite gap sets, we will say that a general set e has Totik–Widom bounds, if there is an upper bound of the above form.

Totik–Widom bounds

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics This complements the 2C(e)n lower bound. Because of his asymptotic result, Widom already had this bound in 1969 but Totik’s proof was much simpler. Neither proof has very explicit estimates for D. Even though they only had the result for ﬁnite gap sets, we will say that a general set e has Totik–Widom bounds, if there is an upper bound of the above form.

Totik–Widom bounds

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Because of his asymptotic result, Widom already had this bound in 1969 but Totik’s proof was much simpler. Neither proof has very explicit estimates for D. Even though they only had the result for ﬁnite gap sets, we will say that a general set e has Totik–Widom bounds, if there is an upper bound of the above form.

Totik–Widom bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Neither proof has very explicit estimates for D. Even though they only had the result for ﬁnite gap sets, we will say that a general set e has Totik–Widom bounds, if there is an upper bound of the above form.

Totik–Widom bounds

Szegő–Widom Asymptotics Even though they only had the result for ﬁnite gap sets, we will say that a general set e has Totik–Widom bounds, if there is an upper bound of the above form.

Totik–Widom bounds

Szegő–Widom Asymptotics Totik–Widom bounds

Theorem (Totik’s 1/n bound) If e is a finite gap set, the period n sets ˜en ⊃ e can be chosen so that Chebyshev E Polynomials C(˜en) ≤ C(e) 1 + n for some constant E. Alternation n Theorem Because kTnke = 2C(en) , this bound is equivalent to Potential Theory Theorem (Totik–Widom bounds in the finite gap case) If e Lower Bounds and Special Sets is a finite gap set, then for a constant D we have that Faber Fekete n Szegő Theorem kTnke ≤ DC(e) Widom’s Work This complements the 2C(e)n lower bound. Because of his Totik Widom Bounds asymptotic result, Widom already had this bound in 1969 Main Results but Totik’s proof was much simpler. Neither proof has very Totik Widom explicit estimates for D. Even though they only had the Bounds result for finite gap sets, we will say that a general set e has Szegő–Widom Asymptotics Totik–Widom bounds, if there is an upper bound of the above form. ∗ The character group, π1, is in general an infinite dimensional torus. It contains a distinguished element, χe, the character of Be. We will say that e ∈ R has a canonical generator if and only if the powers of χe are dense in the character group. This is equivalent to saying that if e is decomposed into ` closed disjoint sets, the only rational relation among their harmonic masses is that their total sum is 1. It seems to us likely that, in some sense, this condition holds generically. It follows from results of Totik that among the 2ν dimensional set of unions of exactly ν disjoint unions, the set where the condition fails is a countable union of varieties of dimension ν + 1 so the set where it fails is both of 2ν Lebesgue measure zero and a nowhere dense Fσ.

Canonical Generators

I now want to discuss the case where e might have inﬁnitely many components – in the real case, inﬁnitely many gaps. Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics It contains a distinguished element, χe, the character of Be. We will say that e ∈ R has a canonical generator if and only if the powers of χe are dense in the character group. This is equivalent to saying that if e is decomposed into ` closed disjoint sets, the only rational relation among their harmonic masses is that their total sum is 1. It seems to us likely that, in some sense, this condition holds generically. It follows from results of Totik that among the 2ν dimensional set of unions of exactly ν disjoint unions, the set where the condition fails is a countable union of varieties of dimension ν + 1 so the set where it fails is both of 2ν Lebesgue measure zero and a nowhere dense Fσ.

Canonical Generators

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics This is equivalent to saying that if e is decomposed into ` closed disjoint sets, the only rational relation among their harmonic masses is that their total sum is 1. It seems to us likely that, in some sense, this condition holds generically. It follows from results of Totik that among the 2ν dimensional set of unions of exactly ν disjoint unions, the set where the condition fails is a countable union of varieties of dimension ν + 1 so the set where it fails is both of 2ν Lebesgue measure zero and a nowhere dense Fσ.

Canonical Generators

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics It seems to us likely that, in some sense, this condition holds generically. It follows from results of Totik that among the 2ν dimensional set of unions of exactly ν disjoint unions, the set where the condition fails is a countable union of varieties of dimension ν + 1 so the set where it fails is both of 2ν Lebesgue measure zero and a nowhere dense Fσ.

Canonical Generators

I now want to discuss the case where e might have infinitely many components – in the real case, infinitely many gaps. Chebyshev ∗ Polynomials The character group, π1, is in general an infinite dimensional Alternation torus. It contains a distinguished element, χe, the character Theorem of Be. We will say that e ∈ R has a canonical generator if Potential Theory and only if the powers of χe are dense in the character Lower Bounds and Special Sets group. This is equivalent to saying that if e is decomposed Faber Fekete into ` closed disjoint sets, the only rational relation among Szegő Theorem their harmonic masses is that their total sum is 1. Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics It follows from results of Totik that among the 2ν dimensional set of unions of exactly ν disjoint unions, the set where the condition fails is a countable union of varieties of dimension ν + 1 so the set where it fails is both of 2ν Lebesgue measure zero and a nowhere dense Fσ.

Canonical Generators

Main Results generically.

Totik Widom Bounds

Szegő–Widom Asymptotics Canonical Generators

Main Results generically. It follows from results of Totik that among the

Totik Widom 2ν dimensional set of unions of exactly ν disjoint unions, Bounds the set where the condition fails is a countable union of Szegő–Widom Asymptotics varieties of dimension ν + 1 so the set where it fails is both of 2ν Lebesgue measure zero and a nowhere dense Fσ. We’ll act as if these are properties of e although they are really properties of Ω ≡ (C ∪ {∞}) \ e as an infinitely connected Riemann surface. Each has been heavily studied in the literature on such surfaces and each is known to be a family of many different looking equivalent conditions. I’ll define the properties via a condition that relates directly to the Widom minimizer problem. ∗ ∞ Let χ ∈ π1 and let H (Ω, χ) be the family of bounded character automorphic analytic functions on Ω. We say that a compact set, e ∈ C, has the PW property if and only if ∗ ∞ for every χ ∈ π1, we have that H (Ω, χ) contains non–constant functions.

PW and DCT sets

To state the main new results in part 2, I need to discuss PW (for Parreau–Widom) and DCT (for Direct Cauchy Chebyshev Polynomials Theorem) sets.

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Each has been heavily studied in the literature on such surfaces and each is known to be a family of many diﬀerent looking equivalent conditions. I’ll deﬁne the properties via a condition that relates directly to the Widom minimizer problem. ∗ ∞ Let χ ∈ π1 and let H (Ω, χ) be the family of bounded character automorphic analytic functions on Ω. We say that a compact set, e ∈ C, has the PW property if and only if ∗ ∞ for every χ ∈ π1, we have that H (Ω, χ) contains non–constant functions.

PW and DCT sets

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics I’ll deﬁne the properties via a condition that relates directly to the Widom minimizer problem. ∗ ∞ Let χ ∈ π1 and let H (Ω, χ) be the family of bounded character automorphic analytic functions on Ω. We say that a compact set, e ∈ C, has the PW property if and only if ∗ ∞ for every χ ∈ π1, we have that H (Ω, χ) contains non–constant functions.

PW and DCT sets

To state the main new results in part 2, I need to discuss PW (for Parreau–Widom) and DCT (for Direct Cauchy Chebyshev Polynomials Theorem) sets. We’ll act as if these are properties of e Alternation although they are really properties of Ω ≡ (C ∪ {∞}) \ e as Theorem an inﬁnitely connected Riemann surface. Each has been Potential Theory heavily studied in the literature on such surfaces and each is Lower Bounds and Special Sets known to be a family of many diﬀerent looking equivalent Faber Fekete conditions. Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics ∗ ∞ Let χ ∈ π1 and let H (Ω, χ) be the family of bounded character automorphic analytic functions on Ω. We say that a compact set, e ∈ C, has the PW property if and only if ∗ ∞ for every χ ∈ π1, we have that H (Ω, χ) contains non–constant functions.

PW and DCT sets

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics We say that a compact set, e ∈ C, has the PW property if and only if ∗ ∞ for every χ ∈ π1, we have that H (Ω, χ) contains non–constant functions.

PW and DCT sets

Main Results character automorphic analytic functions on Ω.

Totik Widom Bounds

Szegő–Widom Asymptotics PW and DCT sets

Main Results character automorphic analytic functions on Ω. We say that Totik Widom a compact set, e ∈ C, has the PW property if and only if Bounds ∗ ∞ for every χ ∈ π1, we have that H (Ω, χ) contains Szegő–Widom Asymptotics non–constant functions. For subsets of R, it is often but not always true. We’ll say more about when it fails later but if e ⊂ R is perfect, then e locally has positive 1D Lebesgue measure, so, for example, the classical Cantor set does not have the PW property. On the other hand, it is known that homogeneous sets in the sense of Carleson do. It is also known that e ⊂ C has the PW property iﬀ X PW (e) ≡ Ge(w) < ∞ w∈C

where C is the set of critical points of Ge (i.e. points in the 0 unbounded component of Ω where Ge(w) = 0). In particular, any connected set has PW.

PW and DCT sets

Alternation Poincaré. Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics We’ll say more about when it fails later but if e ⊂ R is perfect, then e locally has positive 1D Lebesgue measure, so, for example, the classical Cantor set does not have the PW property. On the other hand, it is known that homogeneous sets in the sense of Carleson do. It is also known that e ⊂ C has the PW property iﬀ X PW (e) ≡ Ge(w) < ∞ w∈C

where C is the set of critical points of Ge (i.e. points in the 0 unbounded component of Ω where Ge(w) = 0). In particular, any connected set has PW.

PW and DCT sets

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics so, for example, the classical Cantor set does not have the PW property. On the other hand, it is known that homogeneous sets in the sense of Carleson do. It is also known that e ⊂ C has the PW property iﬀ X PW (e) ≡ Ge(w) < ∞ w∈C

where C is the set of critical points of Ge (i.e. points in the 0 unbounded component of Ω where Ge(w) = 0). In particular, any connected set has PW.

PW and DCT sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics On the other hand, it is known that homogeneous sets in the sense of Carleson do. It is also known that e ⊂ C has the PW property iﬀ X PW (e) ≡ Ge(w) < ∞ w∈C

where C is the set of critical points of Ge (i.e. points in the 0 unbounded component of Ω where Ge(w) = 0). In particular, any connected set has PW.

PW and DCT sets

Lifted up to the universal cover, for χ ≡ 1, this question is equivalent to the existence of automorphic functions, a Chebyshev Polynomials problem solved in the ﬁnitely connected case by Klein and Alternation Poincaré. For subsets of R, it is often but not always true. Theorem We’ll say more about when it fails later but if e ⊂ R is Potential Theory perfect, then e locally has positive 1D Lebesgue measure, Lower Bounds and Special Sets so, for example, the classical Cantor set does not have the Faber Fekete PW property. Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics It is also known that e ⊂ C has the PW property iﬀ X PW (e) ≡ Ge(w) < ∞ w∈C

where C is the set of critical points of Ge (i.e. points in the 0 unbounded component of Ω where Ge(w) = 0). In particular, any connected set has PW.

PW and DCT sets

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics In particular, any connected set has PW.

PW and DCT sets

Totik Widom known that e ⊂ C has the PW property iﬀ Bounds X PW (e) ≡ G (w) < ∞ Main Results e w∈C Totik Widom Bounds where C is the set of critical points of Ge (i.e. points in the Szegő–Widom 0 Asymptotics unbounded component of Ω where Ge(w) = 0). In particular, any connected set has PW. Once one has the PW condition, one can prove there exists a unique Widom minimizer, Fχ, which minimizes kAk∞ among all character automorphic functions with that character and A(∞) = 1. One can also consider the dual Widom maximizer, the function, Qχ, which is character automorphic with norm 1, non–negative at ∞ that maximizes the value at ∞. It is easy to see that

Qχ = Fχ/kFχk∞,Fχ = Qχ/Qχ(∞),

Qχ(∞) = 1/kFχk∞

PW and DCT sets

In the case of e ⊂ R which are regular (i.e. Ge continuous on e), there is one critical point in each gap and the value Chebyshev Polynomials of Ge there is the maximum value in the gap.

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics One can also consider the dual Widom maximizer, the function, Qχ, which is character automorphic with norm 1, non–negative at ∞ that maximizes the value at ∞. It is easy to see that

Qχ = Fχ/kFχk∞,Fχ = Qχ/Qχ(∞),

Qχ(∞) = 1/kFχk∞

PW and DCT sets

In the case of e ⊂ R which are regular (i.e. Ge continuous on e), there is one critical point in each gap and the value Chebyshev Polynomials of Ge there is the maximum value in the gap.

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Qχ = Fχ/kFχk∞,Fχ = Qχ/Qχ(∞),

Qχ(∞) = 1/kFχk∞

PW and DCT sets

In the case of e ⊂ R which are regular (i.e. Ge continuous on e), there is one critical point in each gap and the value Chebyshev Polynomials of Ge there is the maximum value in the gap.

Alternation Theorem Once one has the PW condition, one can prove there exists Potential Theory a unique Widom minimizer, Fχ, which minimizes kAk∞ Lower Bounds among all character automorphic functions with that and Special Sets character and A(∞) = 1. One can also consider the dual Faber Fekete Szegő Theorem Widom maximizer, the function, Qχ, which is character Widom’s Work automorphic with norm 1, non–negative at ∞ that Totik Widom maximizes the value at ∞. It is easy to see that Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics PW and DCT sets

In the case of e ⊂ R which are regular (i.e. Ge continuous on e), there is one critical point in each gap and the value Chebyshev Polynomials of Ge there is the maximum value in the gap.

Main Results Qχ = Fχ/kFχk∞,Fχ = Qχ/Qχ(∞),

Totik Widom Bounds Qχ(∞) = 1/kFχk∞

Szegő–Widom Asymptotics we say that e obeys the DCT condition if and only if it is a PW set and the map ∗ χ 7→ Qχ(∞) is a continuous function of π1 to (0, 1). This n implies various functions like n 7→ Fχe (z) are almost periodic.

PW and DCT sets

As the name implies the basic DCT condition has something to do with validity of a Cauchy formula involving Chebyshev ∞ Polynomials boundary values of H functions but for us a more useful

Alternation deﬁnition is an equivalent condition: Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics This n implies various functions like n 7→ Fχe (z) are almost periodic.

PW and DCT sets

As the name implies the basic DCT condition has something to do with validity of a Cauchy formula involving Chebyshev ∞ Polynomials boundary values of H functions but for us a more useful

Alternation deﬁnition is an equivalent condition: we say that e obeys Theorem the DCT condition if and only if it is a PW set and the map Potential Theory ∗ χ 7→ Qχ(∞) is a continuous function of π to (0, 1). Lower Bounds 1 and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics PW and DCT sets

As the name implies the basic DCT condition has something to do with validity of a Cauchy formula involving Chebyshev ∞ Polynomials boundary values of H functions but for us a more useful

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics One of the (two) main results in Part 1 is

Theorem If e ⊂ R is a regular Parreau-Widom set, then n kTnke ≤ 2 exp(PW (e))C(e) Homogeneous sets are regular and obey a Parreau Widom condition (a theorem of Jones and Marshall). This explicit constant is interesting even for the ﬁnite gap case. We also proved a weak converse: Theorem If e ⊂ C is a regular set for which a TW bound holds and e has a canonical generator, then e is a PW set.

Totik Widom Bounds

Recall that we say that e ⊂ R obeys a Totik-Widom bound n if there is a D with kTnke ≤ DC(e) and that this was only Chebyshev Polynomials known for ﬁnite gap sets.

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Homogeneous sets are regular and obey a Parreau Widom condition (a theorem of Jones and Marshall). This explicit constant is interesting even for the ﬁnite gap case. We also proved a weak converse: Theorem If e ⊂ C is a regular set for which a TW bound holds and e has a canonical generator, then e is a PW set.

Totik Widom Bounds

Recall that we say that e ⊂ R obeys a Totik-Widom bound n if there is a D with kTnke ≤ DC(e) and that this was only Chebyshev Polynomials known for ﬁnite gap sets. One of the (two) main results in

Alternation Part 1 is Theorem Potential Theory Theorem If e ⊂ R is a regular Parreau-Widom set, then Lower Bounds n and Special Sets kTnke ≤ 2 exp(PW (e))C(e) Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics This explicit constant is interesting even for the ﬁnite gap case. We also proved a weak converse: Theorem If e ⊂ C is a regular set for which a TW bound holds and e has a canonical generator, then e is a PW set.

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics We also proved a weak converse: Theorem If e ⊂ C is a regular set for which a TW bound holds and e has a canonical generator, then e is a PW set.

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Totik Widom Bounds

Alternation Part 1 is Theorem Potential Theory Theorem If e ⊂ R is a regular Parreau-Widom set, then Lower Bounds n and Special Sets kTnke ≤ 2 exp(PW (e))C(e) Faber Fekete Szegő Theorem Homogeneous sets are regular and obey a Parreau Widom Widom’s Work condition (a theorem of Jones and Marshall). This explicit Totik Widom constant is interesting even for the finite gap case. We also Bounds proved a weak converse: Main Results Totik Widom Theorem If e ⊂ C is a regular set for which a TW bound Bounds holds and e has a canonical generator, then e is a PW set. Szegő–Widom Asymptotics For a time I suspected the answer was yes, then no, and now I’m unsure, but I lean towards yes. A key example is the solid Koch snowflake. Since it is simply connected, it is a PW set. On the other hand the fact that its boundary has dimension greater than 1 makes it a candidate for failure of TW bounds if the theorem does not extend to the general complex case so much so I suspect it was false. But, recently, Andrievskii proved a TW bound holds for a class of sets that includes the Koch snowflake, so I’ve oscillated back.

Totik Widom Bounds

Interesting Open Question Does potential theory regularity + Parreau-Widom ⇒ Totik-Widom bound for Chebyshev Polynomials general e ⊂ C (our proof is only for e ⊂ R). Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics A key example is the solid Koch snowﬂake. Since it is simply connected, it is a PW set. On the other hand the fact that its boundary has dimension greater than 1 makes it a candidate for failure of TW bounds if the theorem does not extend to the general complex case so much so I suspect it was false. But, recently, Andrievskii proved a TW bound holds for a class of sets that includes the Koch snowﬂake, so I’ve oscillated back.

Totik Widom Bounds

Interesting Open Question Does potential theory regularity + Parreau-Widom ⇒ Totik-Widom bound for Chebyshev Polynomials general e ⊂ C (our proof is only for e ⊂ R). Alternation Theorem For a time I suspected the answer was yes, then no, and

Potential Theory now I’m unsure, but I lean towards yes.

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Since it is simply connected, it is a PW set. On the other hand the fact that its boundary has dimension greater than 1 makes it a candidate for failure of TW bounds if the theorem does not extend to the general complex case so much so I suspect it was false. But, recently, Andrievskii proved a TW bound holds for a class of sets that includes the Koch snowﬂake, so I’ve oscillated back.

Totik Widom Bounds

Potential Theory now I’m unsure, but I lean towards yes. A key example is Lower Bounds the solid Koch snowﬂake. and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics On the other hand the fact that its boundary has dimension greater than 1 makes it a candidate for failure of TW bounds if the theorem does not extend to the general complex case so much so I suspect it was false. But, recently, Andrievskii proved a TW bound holds for a class of sets that includes the Koch snowﬂake, so I’ve oscillated back.

Totik Widom Bounds

Potential Theory now I’m unsure, but I lean towards yes. A key example is Lower Bounds the solid Koch snowﬂake. Since it is simply connected, it is and Special Sets a PW set. Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics But, recently, Andrievskii proved a TW bound holds for a class of sets that includes the Koch snowﬂake, so I’ve oscillated back.

Totik Widom Bounds

Potential Theory now I’m unsure, but I lean towards yes. A key example is Lower Bounds the solid Koch snowﬂake. Since it is simply connected, it is and Special Sets a PW set. On the other hand the fact that its boundary has Faber Fekete Szegő Theorem dimension greater than 1 makes it a candidate for failure of Widom’s Work TW bounds if the theorem does not extend to the general Totik Widom complex case so much so I suspect it was false. Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Totik Widom Bounds

Main Results recently, Andrievskii proved a TW bound holds for a class of

Totik Widom sets that includes the Koch snowﬂake, so I’ve oscillated Bounds back. Szegő–Widom Asymptotics Theorem Let e ⊂ R be a DCT set and have canonical generators. Then the set of limit points of the ratio rn is exactly the closed interval [2, 2 exp(PW (e))]. Thus the upper and lower bounds are asymptotically exact and cannot be improved. For example, in the ﬁnite gap case, this result holds if the harmonic measures of the bands are rational independent. But the limits can be less than the whole interval in the non-generic case. For example, if e has two bands of equal size, the the set limit points is the two point set {2, 2 exp(PW (e))}

Limit Points of the ratio

Paper 3 has the following result about the quantity n rn = kTnke/C(e) Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Thus the upper and lower bounds are asymptotically exact and cannot be improved. For example, in the ﬁnite gap case, this result holds if the harmonic measures of the bands are rational independent. But the limits can be less than the whole interval in the non-generic case. For example, if e has two bands of equal size, the the set limit points is the two point set {2, 2 exp(PW (e))}

Limit Points of the ratio

Paper 3 has the following result about the quantity n rn = kTnke/C(e) Chebyshev Polynomials Theorem Let e ⊂ R be a DCT set and have canonical Alternation Theorem generators. Then the set of limit points of the ratio rn is

Potential Theory exactly the closed interval [2, 2 exp(PW (e))].

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics For example, in the ﬁnite gap case, this result holds if the harmonic measures of the bands are rational independent. But the limits can be less than the whole interval in the non-generic case. For example, if e has two bands of equal size, the the set limit points is the two point set {2, 2 exp(PW (e))}

Limit Points of the ratio

Potential Theory exactly the closed interval [2, 2 exp(PW (e))].

Lower Bounds and Special Sets Thus the upper and lower bounds are asymptotically exact Faber Fekete and cannot be improved. Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics But the limits can be less than the whole interval in the non-generic case. For example, if e has two bands of equal size, the the set limit points is the two point set {2, 2 exp(PW (e))}

Limit Points of the ratio

Potential Theory exactly the closed interval [2, 2 exp(PW (e))].

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics For example, if e has two bands of equal size, the the set limit points is the two point set {2, 2 exp(PW (e))}

Limit Points of the ratio

Potential Theory exactly the closed interval [2, 2 exp(PW (e))].

Lower Bounds and Special Sets Thus the upper and lower bounds are asymptotically exact Faber Fekete and cannot be improved. For example, in the ﬁnite gap Szegő Theorem case, this result holds if the harmonic measures of the bands Widom’s Work are rational independent. But the limits can be less than Totik Widom Bounds the whole interval in the non-generic case. Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Limit Points of the ratio

Potential Theory exactly the closed interval [2, 2 exp(PW (e))].

Szegő–Widom Asymptotics Theorem Widom’s conjecture on the almost periodic Szegő asymptotics outside e for the Chebyshev polynomials of ﬁnite gap sets is true. In Part 2, we extended this: Theorem Szegő–Widom asymptotics outside e holds for the Chebyshev polynomials of any e ⊂ R for any e that is both PW and DCT. The proof of this in Part 2 is simpler than the proof of Part 1 (and doesn’t require Widom’s result on the norm apriori but rather proves it).

Widom’s Conjecture

The other main result of Part 1 settled a 45 year old conjecture: Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics In Part 2, we extended this: Theorem Szegő–Widom asymptotics outside e holds for the Chebyshev polynomials of any e ⊂ R for any e that is both PW and DCT. The proof of this in Part 2 is simpler than the proof of Part 1 (and doesn’t require Widom’s result on the norm apriori but rather proves it).

Widom’s Conjecture

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Theorem Szegő–Widom asymptotics outside e holds for the Chebyshev polynomials of any e ⊂ R for any e that is both PW and DCT. The proof of this in Part 2 is simpler than the proof of Part 1 (and doesn’t require Widom’s result on the norm apriori but rather proves it).

Widom’s Conjecture

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics The proof of this in Part 2 is simpler than the proof of Part 1 (and doesn’t require Widom’s result on the norm apriori but rather proves it).

Widom’s Conjecture

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Widom’s Conjecture

The other main result of Part 1 settled a 45 year old conjecture: Chebyshev Polynomials Theorem Widom’s conjecture on the almost periodic Szegő Alternation asymptotics outside e for the Chebyshev polynomials of Theorem ﬁnite gap sets is true. Potential Theory Lower Bounds In Part 2, we extended this: and Special Sets Faber Fekete Theorem Szegő–Widom asymptotics outside e holds for the Szegő Theorem Chebyshev polynomials of any e ⊂ for any e that is both Widom’s Work R PW and DCT. Totik Widom Bounds The proof of this in Part 2 is simpler than the proof of Part Main Results 1 (and doesn’t require Widom’s result on the norm apriori Totik Widom Bounds but rather proves it). Szegő–Widom Asymptotics Theorem If e ⊂ R is a regular set PW set for which n n 7→ Aχe is almost periodic in n and which has a canonical generator, then e is a DCT set. Since almost periodicity of the limit is part of Szegő–Widom asymptotics, this is a kind of converse to DCT⇒Szegő–Widom asymptotics.

Szegő–Widom Asymptotics

We also proved that:

Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Since almost periodicity of the limit is part of Szegő–Widom asymptotics, this is a kind of converse to DCT⇒Szegő–Widom asymptotics.

Szegő–Widom Asymptotics

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Szegő–Widom Asymptotics

We also proved that: Theorem If e ⊂ is a regular set PW set for which Chebyshev R Polynomials n n 7→ Aχe is almost periodic in n and which has a canonical Alternation generator, then e is a DCT set. Theorem Potential Theory Since almost periodicity of the limit is part of Szegő–Widom Lower Bounds asymptotics, this is a kind of converse to and Special Sets

Faber Fekete DCT⇒Szegő–Widom asymptotics. Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Lemma Let e ⊂ en ⊂ R be a compact subset and its canonical period n superset. Let K be a gap of e and dρn, the equilibrium measure of en. Then ρn(K) ≤ 1/n.

Accepting this for the moment, let h(z) ≡ Ge(z) − Gen (z) which is harmonic at inﬁnity with

h C(en) i h(∞) = R(e) − R(en) = log C(e)

Since dρn is harmonic measure and h(x) = Ge(x) on en, if M {Kj}j=1 are the gaps for e, then, using the Lemma PM 1 PM h(∞) ≤ j=1 ρn(Kj) maxx∈Kj (Ge(x)) ≤ n j=1 Ge(wj) since regularity of e implies Ge vanishes at the ends of each gap so the maximum is taken a critical point wj. n Exponentiating and using kTnke ≤ 2C(en) we get the result. `

Proof modulo Lemma

We turn to the proof of TW bounds from Part 1:

Chebyshev Polynomials

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Let K be a gap of e and dρn, the equilibrium measure of en. Then ρn(K) ≤ 1/n.

Accepting this for the moment, let h(z) ≡ Ge(z) − Gen (z) which is harmonic at inﬁnity with

h C(en) i h(∞) = R(e) − R(en) = log C(e)

Proof modulo Lemma

We turn to the proof of TW bounds from Part 1:

Lemma Let e ⊂ en ⊂ be a compact subset and its Chebyshev R Polynomials canonical period n superset. Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Then ρn(K) ≤ 1/n.

Accepting this for the moment, let h(z) ≡ Ge(z) − Gen (z) which is harmonic at inﬁnity with

h C(en) i h(∞) = R(e) − R(en) = log C(e)

Proof modulo Lemma

We turn to the proof of TW bounds from Part 1:

Lemma Let e ⊂ en ⊂ be a compact subset and its Chebyshev R Polynomials canonical period n superset. Let K be a gap of e and dρn, Alternation the equilibrium measure of en. Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Accepting this for the moment, let h(z) ≡ Ge(z) − Gen (z) which is harmonic at inﬁnity with

h C(en) i h(∞) = R(e) − R(en) = log C(e)

Proof modulo Lemma

We turn to the proof of TW bounds from Part 1:

Lemma Let e ⊂ en ⊂ be a compact subset and its Chebyshev R Polynomials canonical period n superset. Let K be a gap of e and dρn, Alternation the equilibrium measure of en. Then ρn(K) ≤ 1/n. Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics h C(en) i h(∞) = R(e) − R(en) = log C(e)

Proof modulo Lemma

We turn to the proof of TW bounds from Part 1:

Potential Theory Accepting this for the moment, let h(z) ≡ Ge(z) − Gen (z) Lower Bounds which is harmonic at inﬁnity with and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Since dρn is harmonic measure and h(x) = Ge(x) on en, if M {Kj}j=1 are the gaps for e, then, using the Lemma PM 1 PM h(∞) ≤ j=1 ρn(Kj) maxx∈Kj (Ge(x)) ≤ n j=1 Ge(wj) since regularity of e implies Ge vanishes at the ends of each gap so the maximum is taken a critical point wj. n Exponentiating and using kTnke ≤ 2C(en) we get the result. `

Proof modulo Lemma

We turn to the proof of TW bounds from Part 1:

Potential Theory Accepting this for the moment, let h(z) ≡ Ge(z) − Gen (z) Lower Bounds which is harmonic at inﬁnity with and Special Sets h C(en) i Faber Fekete h(∞) = R(e) − R(en) = log C(e) Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics PM 1 PM h(∞) ≤ j=1 ρn(Kj) maxx∈Kj (Ge(x)) ≤ n j=1 Ge(wj) since regularity of e implies Ge vanishes at the ends of each gap so the maximum is taken a critical point wj. n Exponentiating and using kTnke ≤ 2C(en) we get the result. `

Proof modulo Lemma

We turn to the proof of TW bounds from Part 1:

Widom’s Work Since dρn is harmonic measure and h(x) = Ge(x) on en, if M Totik Widom {Kj}j=1 are the gaps for e, then, using the Lemma Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics since regularity of e implies Ge vanishes at the ends of each gap so the maximum is taken a critical point wj. n Exponentiating and using kTnke ≤ 2C(en) we get the result. `

Proof modulo Lemma

We turn to the proof of TW bounds from Part 1:

Szegő–Widom Asymptotics Proof modulo Lemma

We turn to the proof of TW bounds from Part 1:

Widom’s Work Since dρn is harmonic measure and h(x) = Ge(x) on en, if M Totik Widom {Kj}j=1 are the gaps for e, then, using the Lemma Bounds PM 1 PM Main Results h(∞) ≤ j=1 ρn(Kj) maxx∈Kj (Ge(x)) ≤ n j=1 Ge(wj) Totik Widom since regularity of e implies Ge vanishes at the ends of each Bounds gap so the maximum is taken a critical point w . Szegő–Widom j n Asymptotics Exponentiating and using kTnke ≤ 2C(en) we get the result. ` Recall that any gap K has at most one zero.

Case 1 (Tn has no zero in K) Then there are zeros above and below K not in K. Thus K contains at most two half bands. (In fact, using the Alternation Theorem, one can show at most one half band).

Case 2 (Tn has a zero in K) By the Alternation Theorem, one of the two extreme points immediately below the zero must lie in e, so there is at most a half band below the zero. Similarly, at most a half band above, so no more than a full band. `

Proof of the Lemma

Because the integrated equilibrium measure of en is 1 Tn(x) 1 arccos , each band of en has ρen measure and Chebyshev πn kTnke n Polynomials the part of a band from a zero of Tn to a nearby band edge Alternation 1 Theorem has ρen measure 2n . Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Case 1 (Tn has no zero in K) Then there are zeros above and below K not in K. Thus K contains at most two half bands. (In fact, using the Alternation Theorem, one can show at most one half band).

Proof of the Lemma

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics (In fact, using the Alternation Theorem, one can show at most one half band).

Proof of the Lemma

Lower Bounds and Special Sets Case 1 (Tn has no zero in K) Then there are zeros above

Faber Fekete and below K not in K. Thus K contains at most two half Szegő Theorem bands. Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Case 2 (Tn has a zero in K) By the Alternation Theorem, one of the two extreme points immediately below the zero must lie in e, so there is at most a half band below the zero. Similarly, at most a half band above, so no more than a full band. `

Proof of the Lemma

Lower Bounds and Special Sets Case 1 (Tn has no zero in K) Then there are zeros above

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Proof of the Lemma

Lower Bounds and Special Sets Case 1 (Tn has no zero in K) Then there are zeros above

Faber Fekete and below K not in K. Thus K contains at most two half Szegő Theorem bands. (In fact, using the Alternation Theorem, one can Widom’s Work show at most one half band). Totik Widom Bounds Case 2 (Tn has a zero in K) By the Alternation Theorem, Main Results one of the two extreme points immediately below the zero Totik Widom Bounds must lie in e, so there is at most a half band below the zero. Szegő–Widom Similarly, at most a half band above, so no more than a full Asymptotics band. ` This implies that if K is a gap and nj is such as j → ∞

and any zeros of Tnj in K go to the edges, then

asymptotically there is none of enj in K in the sense that T∞ S∞ k=1 j=k(K ∩ enj ) = ∅.

Similarly if, for j large Tnj has a zero, xj, in K and

xj → x∞ ∈ K, then only x∞ is asymptotically in enj ∩ K T∞ S∞ in the sense that k=1 j=k(K ∩ enj ) = {x∞}.

Size of en \ e

In fact, one can prove if there is a zero not too close to a gap edge and n is large, then there is exactly a full Chebyshev Polynomials exponentially small (in Lebesgue measure) band of en

Alternation totally inside K. Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics then

asymptotically there is none of enj in K in the sense that T∞ S∞ k=1 j=k(K ∩ enj ) = ∅.

Similarly if, for j large Tnj has a zero, xj, in K and

xj → x∞ ∈ K, then only x∞ is asymptotically in enj ∩ K T∞ S∞ in the sense that k=1 j=k(K ∩ enj ) = {x∞}.

Size of en \ e

In fact, one can prove if there is a zero not too close to a gap edge and n is large, then there is exactly a full Chebyshev Polynomials exponentially small (in Lebesgue measure) band of en

Alternation totally inside K. Theorem Potential Theory This implies that if K is a gap and nj is such as j → ∞

Lower Bounds and any zeros of Tnj in K go to the edges, and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Similarly if, for j large Tnj has a zero, xj, in K and

xj → x∞ ∈ K, then only x∞ is asymptotically in enj ∩ K T∞ S∞ in the sense that k=1 j=k(K ∩ enj ) = {x∞}.

Size of en \ e

In fact, one can prove if there is a zero not too close to a gap edge and n is large, then there is exactly a full Chebyshev Polynomials exponentially small (in Lebesgue measure) band of en

Alternation totally inside K. Theorem Potential Theory This implies that if K is a gap and nj is such as j → ∞

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Size of en \ e

In fact, one can prove if there is a zero not too close to a gap edge and n is large, then there is exactly a full Chebyshev Polynomials exponentially small (in Lebesgue measure) band of en

Alternation totally inside K. Theorem Potential Theory This implies that if K is a gap and nj is such as j → ∞

Lower Bounds and any zeros of Tnj in K go to the edges, then and Special Sets asymptotically there is none of enj in K in the sense that Faber Fekete ∞ ∞ Szegő Theorem T S k=1 j=k(K ∩ enj ) = ∅. Widom’s Work Similarly if, for j large Tn has a zero, xj, in K and Totik Widom j Bounds xj → x∞ ∈ K, then only x∞ is asymptotically in enj ∩ K Main Results T∞ S∞ in the sense that k=1 j=k(K ∩ enj ) = {x∞}. Totik Widom Bounds

Szegő–Widom Asymptotics and form the product of Blaschke factors with those zeros. If e is PW, then one can prove this Blaschke product converges to a function vanishing precisely at the those lifts. This deﬁnes a character automorphic function, B(z, x), which is the unique character automorphic function with inﬁnity norm 1, positive at ∞ with zero’s precisely at x (and its images under the deck transformations). R \ e is a disjoint union of bounded open components (plus two unbounded components), K ∈ G. We’ll call these the gaps and G the set of gaps. A gap collection is a subset G0 ⊂ G. A gap set is a gap collection, G0, and for each Kk ∈ G0 a point xk ∈ Kk.

Overall Strategy

Finally, some remarks on the proof of SW asymptotics. For any x ∈ Ω, we can look at the lifts of x in the universal Chebyshev Polynomials cover, thought of as the unit disk

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics If e is PW, then one can prove this Blaschke product converges to a function vanishing precisely at the those lifts. This deﬁnes a character automorphic function, B(z, x), which is the unique character automorphic function with inﬁnity norm 1, positive at ∞ with zero’s precisely at x (and its images under the deck transformations). R \ e is a disjoint union of bounded open components (plus two unbounded components), K ∈ G. We’ll call these the gaps and G the set of gaps. A gap collection is a subset G0 ⊂ G. A gap set is a gap collection, G0, and for each Kk ∈ G0 a point xk ∈ Kk.

Overall Strategy

Finally, some remarks on the proof of SW asymptotics. For any x ∈ Ω, we can look at the lifts of x in the universal Chebyshev Polynomials cover, thought of as the unit disk and form the product of

Alternation Blaschke factors with those zeros. Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics This deﬁnes a character automorphic function, B(z, x), which is the unique character automorphic function with inﬁnity norm 1, positive at ∞ with zero’s precisely at x (and its images under the deck transformations). R \ e is a disjoint union of bounded open components (plus two unbounded components), K ∈ G. We’ll call these the gaps and G the set of gaps. A gap collection is a subset G0 ⊂ G. A gap set is a gap collection, G0, and for each Kk ∈ G0 a point xk ∈ Kk.

Overall Strategy

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics R \ e is a disjoint union of bounded open components (plus two unbounded components), K ∈ G. We’ll call these the gaps and G the set of gaps. A gap collection is a subset G0 ⊂ G. A gap set is a gap collection, G0, and for each Kk ∈ G0 a point xk ∈ Kk.

Overall Strategy

Alternation Blaschke factors with those zeros. If e is PW, then one can Theorem prove this Blaschke product converges to a function Potential Theory vanishing precisely at the those lifts. This deﬁnes a Lower Bounds and Special Sets character automorphic function, B(z, x), which is the Faber Fekete unique character automorphic function with inﬁnity norm 1, Szegő Theorem positive at ∞ with zero’s precisely at x (and its images Widom’s Work

Totik Widom under the deck transformations). Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics A gap collection is a subset G0 ⊂ G. A gap set is a gap collection, G0, and for each Kk ∈ G0 a point xk ∈ Kk.

Overall Strategy

Totik Widom under the deck transformations). Bounds R \ e is a disjoint union of bounded open components (plus Main Results two unbounded components), K ∈ G. We’ll call these the Totik Widom Bounds gaps and G the set of gaps. A gap collection is a subset Szegő–Widom G0 ⊂ G. A gap set is a gap collection, G0, and for each Asymptotics Kk ∈ G0 a point xk ∈ Kk. To prove Szegő–Widom asymptotics, it suﬃces, by Montel’s theorem and uniqueness of minimizers, to show that any n n limit point of the Ln(z) ≡ Tn(z)Be(z) /C(e) is a Widom minimizer. We’ll instead look at n n Mn(z) = B(z) /Bn(z) which obeys

|Mn(z)| = exp(−hn(z)); hn(z) ≡ Ge(z) − Gen (z)

so that on can write Ln in terms of Mn

Overall Strategy

For any gap set, S, we deﬁne the associated Blaschke product Chebyshev Y Polynomials BS(z) = Be(z, xk)

Alternation Kk∈G0 Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics We’ll instead look at n n Mn(z) = B(z) /Bn(z) which obeys

|Mn(z)| = exp(−hn(z)); hn(z) ≡ Ge(z) − Gen (z)

so that on can write Ln in terms of Mn

Overall Strategy

For any gap set, S, we deﬁne the associated Blaschke product Chebyshev Y Polynomials BS(z) = Be(z, xk)

Alternation Kk∈G0 Theorem

Potential Theory To prove Szegő–Widom asymptotics, it suﬃces, by Montel’s

Lower Bounds theorem and uniqueness of minimizers, to show that any and Special Sets n n limit point of the Ln(z) ≡ Tn(z)Be(z) /C(e) is a Widom Faber Fekete Szegő Theorem minimizer.

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics which obeys

|Mn(z)| = exp(−hn(z)); hn(z) ≡ Ge(z) − Gen (z)

so that on can write Ln in terms of Mn

Overall Strategy

For any gap set, S, we deﬁne the associated Blaschke product Chebyshev Y Polynomials BS(z) = Be(z, xk)

Alternation Kk∈G0 Theorem

Potential Theory To prove Szegő–Widom asymptotics, it suﬃces, by Montel’s

Lower Bounds theorem and uniqueness of minimizers, to show that any and Special Sets n n limit point of the Ln(z) ≡ Tn(z)Be(z) /C(e) is a Widom Faber Fekete Szegő Theorem minimizer. We’ll instead look at

Widom’s Work n n Mn(z) = B(z) /Bn(z) Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics so that on can write Ln in terms of Mn

Overall Strategy

For any gap set, S, we deﬁne the associated Blaschke product Chebyshev Y Polynomials BS(z) = Be(z, xk)

Alternation Kk∈G0 Theorem

Potential Theory To prove Szegő–Widom asymptotics, it suﬃces, by Montel’s

Widom’s Work n n Mn(z) = B(z) /Bn(z) Totik Widom Bounds which obeys Main Results

Totik Widom Bounds |Mn(z)| = exp(−hn(z)); hn(z) ≡ Ge(z) − Gen (z)

Szegő–Widom Asymptotics Overall Strategy

For any gap set, S, we deﬁne the associated Blaschke product Chebyshev Y Polynomials BS(z) = Be(z, xk)

Alternation Kk∈G0 Theorem

Potential Theory To prove Szegő–Widom asymptotics, it suﬃces, by Montel’s

Widom’s Work n n Mn(z) = B(z) /Bn(z) Totik Widom Bounds which obeys Main Results

Totik Widom Bounds |Mn(z)| = exp(−hn(z)); hn(z) ≡ Ge(z) − Gen (z)

Szegő–Widom Asymptotics so that on can write Ln in terms of Mn This means it suffices to prove that any limit point of the Mn is a dual Widom maximizer. Moreover, we can pass to subsequences and suppose that for the limit point is also true that the zeros in each gap have a limit or else leave the gap (through the edges). The limit points of zeros then define a gap set. The proof has two steps: first, prove that the limit is the Blaschke product of this gap set and, secondly, prove that any such product is a dual Widom maximizer. The proof of the second half follows, in part, ideas of Volberg–Yuditskii, who considered a related problem and uses some deep 1997 results of Sodin–Yuditskii on the Abel map in this setting.

Overall Strategy

2n Ln(z) = (1 + Bn(z) )Hn(z) n n C(en) B(z) Mn(z) Chebyshev Hn(z) = n n = Polynomials C(e) Bn(z) Mn(∞) Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Moreover, we can pass to subsequences and suppose that for the limit point is also true that the zeros in each gap have a limit or else leave the gap (through the edges). The limit points of zeros then deﬁne a gap set. The proof has two steps: ﬁrst, prove that the limit is the Blaschke product of this gap set and, secondly, prove that any such product is a dual Widom maximizer. The proof of the second half follows, in part, ideas of Volberg–Yuditskii, who considered a related problem and uses some deep 1997 results of Sodin–Yuditskii on the Abel map in this setting.

Overall Strategy

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics The limit points of zeros then deﬁne a gap set. The proof has two steps: ﬁrst, prove that the limit is the Blaschke product of this gap set and, secondly, prove that any such product is a dual Widom maximizer. The proof of the second half follows, in part, ideas of Volberg–Yuditskii, who considered a related problem and uses some deep 1997 results of Sodin–Yuditskii on the Abel map in this setting.

Overall Strategy

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics The proof has two steps: ﬁrst, prove that the limit is the Blaschke product of this gap set and, secondly, prove that any such product is a dual Widom maximizer. The proof of the second half follows, in part, ideas of Volberg–Yuditskii, who considered a related problem and uses some deep 1997 results of Sodin–Yuditskii on the Abel map in this setting.

Overall Strategy

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics and, secondly, prove that any such product is a dual Widom maximizer. The proof of the second half follows, in part, ideas of Volberg–Yuditskii, who considered a related problem and uses some deep 1997 results of Sodin–Yuditskii on the Abel map in this setting.

Overall Strategy

Szegő–Widom Asymptotics The proof of the second half follows, in part, ideas of Volberg–Yuditskii, who considered a related problem and uses some deep 1997 results of Sodin–Yuditskii on the Abel map in this setting.

Overall Strategy

2n Ln(z) = (1 + Bn(z) )Hn(z) n n C(en) B(z) Mn(z) Chebyshev Hn(z) = n n = Polynomials C(e) Bn(z) Mn(∞) Alternation Theorem This means it suffices to prove that any limit point of the Potential Theory Mn is a dual Widom maximizer. Moreover, we can pass to Lower Bounds subsequences and suppose that for the limit point is also and Special Sets true that the zeros in each gap have a limit or else leave the Faber Fekete Szegő Theorem gap (through the edges). The limit points of zeros then Widom’s Work define a gap set. Totik Widom Bounds The proof has two steps: first, prove that the limit is the Main Results Blaschke product of this gap set and, secondly, prove that Totik Widom any such product is a dual Widom maximizer. Bounds

Szegő–Widom Asymptotics Overall Strategy

Kk∈G0

where Ge(x, z) is the Green’s function with pole at x (so that Ge(∞, z) is what we called Ge(z)). Part 1, we proved Totik–Widom bounds for PW sets, e ⊂ R by using that when z = ∞, we have that Z hn(∞) = Ge(x)dρn(x) S Kj Kj ∈G

Limits of Mn are Gap Blaschke Products

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Part 1, we proved Totik–Widom bounds for PW sets, e ⊂ R by using that when z = ∞, we have that Z hn(∞) = Ge(x)dρn(x) S Kj Kj ∈G

Limits of Mn are Gap Blaschke Products

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Limits of Mn are Gap Blaschke Products

Rather than control Mn globally, it suﬃces, by a little complex analysis, to prove convergence of the absolute Chebyshev Polynomials values and only for z in a small neighborhood of ∞. Taking Alternation into account the formula we had for |Mn(z)| and taking Theorem logs, it suﬃces to prove that, for z near ∞, we have that Potential Theory X Lower Bounds nhn(z) → Ge(xk, z) and Special Sets Kk∈G0 Faber Fekete Szegő Theorem where Ge(x, z) is the Green’s function with pole at x (so Widom’s Work that Ge(∞, z) is what we called Ge(z)). Totik Widom Bounds Part 1, we proved Totik–Widom bounds for PW sets, e ⊂ R Main Results by using that when z = ∞, we have that Totik Widom Z Bounds hn(∞) = Ge(x)dρn(x) Szegő–Widom S K ∈G Kj Asymptotics j If we wrote the analog of this for general z, we’d get Z H(z) = H(x)dρn(x, z) en varying the harmonic measure. Instead we use Z hn(z) = Ge(x, z)dρn(x) S Kj Kj ∈G

which follows from noting that hn vanishes on e and, on (C ∪ {∞}) \ e obeys, ∆hn = dρn (en \ e) so this is just the Poisson formula (proven by checking that the diﬀerence is harmonic on Ω and vanishes on e).

A Poisson Formula

We proved this by thinking of dρn as harmonic measure at ∞, i.e. if H is harmonic on (C ∪ {∞}) \ en with boundary Chebyshev values H(x) on e , then H(∞) = R H(x)dρ (x). Polynomials n en n Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Z H(z) = H(x)dρn(x, z) en varying the harmonic measure. Instead we use Z hn(z) = Ge(x, z)dρn(x) S Kj Kj ∈G

A Poisson Formula

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Instead we use Z hn(z) = Ge(x, z)dρn(x) S Kj Kj ∈G

A Poisson Formula

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics which follows from noting that hn vanishes on e and, on (C ∪ {∞}) \ e obeys, ∆hn = dρn (en \ e) so this is just the Poisson formula (proven by checking that the diﬀerence is harmonic on Ω and vanishes on e).

A Poisson Formula

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics so this is just the Poisson formula (proven by checking that the diﬀerence is harmonic on Ω and vanishes on e).

A Poisson Formula

We proved this by thinking of dρn as harmonic measure at ∞, i.e. if H is harmonic on (C ∪ {∞}) \ en with boundary Chebyshev values H(x) on e , then H(∞) = R H(x)dρ (x). If we Polynomials n en n Alternation wrote the analog of this for general z, we’d get Theorem Z Potential Theory H(z) = H(x)dρn(x, z) Lower Bounds en and Special Sets varying the harmonic measure. Instead we use Faber Fekete Szegő Theorem Z hn(z) = Ge(x, z)dρn(x) Widom’s Work S K ∈G Kj Totik Widom j Bounds which follows from noting that hn vanishes on e and, on Main Results ( ∪ {∞}) \ e obeys, ∆h = dρ (e \ e) Totik Widom C n n n Bounds

Szegő–Widom Asymptotics A Poisson Formula

If the zeros in a gap have a limit, xk, in the gap, there is a single narrow band of ρn–weight 1/n near the point so the ﬁrst case is handled. If there is a zero that approaches an edge, the limit is 0 since regularity implies the Green’s function Ge(x, z) = Ge(z, x) vanishes as x approaches e. `

Limits of Mn are Gap Blaschke Products

By using the PW bound and the fact that R Ge(x, z)δ(x − xk) = Ge(xk, z), it suﬃces to prove that, Chebyshev Polynomials for all K ∈ G, we have that

Alternation Theorem

Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics If the zeros in a gap have a limit, xk, in the gap, there is a single narrow band of ρn–weight 1/n near the point so the ﬁrst case is handled. If there is a zero that approaches an edge, the limit is 0 since regularity implies the Green’s function Ge(x, z) = Ge(z, x) vanishes as x approaches e. `

Limits of Mn are Gap Blaschke Products

By using the PW bound and the fact that R Ge(x, z)δ(x − xk) = Ge(xk, z), it suﬃces to prove that, Chebyshev Polynomials for all K ∈ G, we have that Alternation δ(x − xk), if K ∈ G0 Theorem ndρ K → 0, if K/∈ G0 Potential Theory

Lower Bounds and Special Sets

Faber Fekete Szegő Theorem

Widom’s Work

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics If there is a zero that approaches an edge, the limit is 0 since regularity implies the Green’s function Ge(x, z) = Ge(z, x) vanishes as x approaches e. `

Limits of Mn are Gap Blaschke Products

Totik Widom Bounds

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics Limits of Mn are Gap Blaschke Products

Lower Bounds If the zeros in a gap have a limit, xk, in the gap, there is a and Special Sets single narrow band of ρ –weight 1/n near the point so the Faber Fekete n Szegő Theorem ﬁrst case is handled. If there is a zero that approaches an Widom’s Work edge, the limit is 0 since regularity implies the Green’s Totik Widom ` Bounds function Ge(x, z) = Ge(z, x) vanishes as x approaches e.

Main Results

Totik Widom Bounds

Szegő–Widom Asymptotics AMS

A Comprehensive Course in Analysis by Poincaré Prize Real Analysis winner Barry Simon is a fi ve-volume set that can serve as a graduate-level analysis textbook with a lot of additional Real Analysis bonus information, including hundreds of problems and numerous notes that extend the text and provide important A Comprehensive Course in Analysis, Part 1 historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis. Photo courtesy of Bob Paz, Caltech Photo courtesy of Bob Paz, Part 1 is devoted to real analysis. From one point of view, it presents the infi nitesimal calculus of the twentieth century with the ultimate integral calculus (measure theory) and the ultimate differential calculus (distribu- tion theory). From another, it shows the triumph of abstract spaces: topological Barry Simon spaces, Banach and Hilbert spaces, measure spaces, Riesz spaces, Polish spaces, locally convex spaces, Fréchet spaces, Schwartz space, and Lp spaces. Finally it is the study of big techniques, including the Fourier series and transform, dual spaces, the Baire category, fi xed point theorems, probability ideas, and Hausdorff dimension. Applications include the constructions of nowhere differentiable functions, Brownian motion, space-fi lling curves, solutions of the moment problem, Haar measure, and equilibrium measures in potential theory.

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Chebyshev

Polynomials A Comprehensive Course in Analysis by Poincaré Prize Real Analysis winner Barry Simon is a fi ve-volume set that can serve as a graduate-level analysis textbook with a lot of additional Real Analysis Alternation bonus information, including hundreds of problems and numerous notes that extend the text and provide important A Comprehensive Course in Analysis, Part 1 Theorem historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis. Photo courtesy of Bob Paz, Caltech Photo courtesy of Bob Paz, Potential Theory Part 1 is devoted to real analysis. From one point of view, it presents the infi nitesimal calculus of the twentieth century with the ultimate integral calculus (measure theory) and the ultimate differential calculus (distribu- tion theory). From another, it shows the triumph of abstract spaces: topological Barry Simon Lower Bounds spaces, Banach and Hilbert spaces, measure spaces, Riesz spaces, Polish spaces, and Special Sets locally convex spaces, Fréchet spaces, Schwartz space, and Lp spaces. Finally it is the study of big techniques, including the Fourier series and transform, dual spaces, the Baire category, fi xed point theorems, probability ideas, and Hausdorff dimension. Applications include the constructions of nowhere differentiable func- Faber Fekete tions, Brownian motion, space-fi lling curves, solutions of the moment problem, Szegő Theorem Haar measure, and equilibrium measures in potential theory. ANALYSIS Widom’s Work Part

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AMS Basic Complex Analysis Chebyshev Polynomials A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a ﬁ ve-volume set that can serve as a graduate-level analysis textbook with a lot of additional Basic Complex Analysis Alternation bonus information, including hundreds of problems and numerous notes that extend the text and provide important A Comprehensive Course in Analysis, Part 2A Theorem historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis. Photo courtesy of Bob Paz, Caltech Photo courtesy of Bob Paz, Potential Theory Part 2A is devoted to basic complex analysis. It inter- weaves three analytic threads associated with Cauchy, Riemann, and Weierstrass, respectively. Cauchy’s view focuses on the differential and integral calculus of functions of a complex variable, with the key topics being the Cauchy integral Barry Simon Lower Bounds formula and contour integration. For Riemann, the geometry of the complex plane is central, with key topics being fractional linear transformations and conformal and Special Sets mapping. For Weierstrass, the power series is king, with key topics being spaces of analytic functions, the product formulas of Weierstrass and Hadamard, and the Weierstrass theory of elliptic functions. Subjects in this volume that are often Faber Fekete missing in other texts include the Cauchy integral theorem when the contour is the boundary of a Jordan region, continued fractions, two proofs of the big Picard Szegő Theorem theorem, the uniformization theorem, Ahlfors’s function, the sheaf of analytic germs, and Jacobi, as well as Weierstrass, elliptic functions. ANALYSIS Widom’s Work Part Totik Widom 2A

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Chebyshev Polynomials A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a ﬁ ve-volume set that can serve as a graduate-level analysis textbook with a lot of additional Advanced Complex Analysis Alternation bonus information, including hundreds of problems and numerous notes that extend the text and provide important A Comprehensive Course in Analysis, Part 2B Theorem historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis. Photo courtesy of Bob Paz, Caltech Photo courtesy of Bob Paz, Part 2B provides a comprehensive look at a number of Potential Theory subjects of complex analysis not included in Part 2A. Presented in this volume are the theory of conformal metrics (including the Poincaré metric, the Ahlfors- Robinson proof of Picard’s theorem, and Bell’s proof of the Painlevé smoothness Barry Simon Lower Bounds theorem), topics in analytic number theory (including Jacobi’s two- and four- square theorems, the Dirichlet prime progression theorem, the prime number and Special Sets theorem, and the Hardy-Littlewood asymptotics for the number of partitions), the theory of Fuschian differential equations, asymptotic methods (including Euler’s method, stationary phase, the saddle-point method, and the WKB method), univa- Faber Fekete lent functions (including an introduction to SLE), and Nevanlinna theory. The chapters on Fuschian differential equations and on asymptotic methods can be Szegő Theorem viewed as a minicourse on the theory of special functions. ANALYSIS Widom’s Work Part Totik Widom 2B

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Chebyshev Harmonic Analysis Polynomials A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a ﬁ ve-volume set that can serve as a graduate-level analysis textbook with a lot of additional Harmonic Analysis Alternation bonus information, including hundreds of problems and numerous notes that extend the text and provide important A Comprehensive Course in Analysis, Part 3 Theorem historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis. Photo courtesy of Bob Paz, Caltech Photo courtesy of Bob Paz, Part 3 returns to the themes of Part 1 by discussing point- Potential Theory wise limits (going beyond the usual focus on the Hardy-Littlewood maximal function by including ergodic theorems and martingale convergence), harmonic functions and potential theory, frames and wavelets, H p spaces (including bounded Barry Simon Lower Bounds mean oscillation (BMO)) and, in the ﬁ nal chapter, lots of inequalities, including and Special Sets Sobolev spaces, Calderon-Zygmund estimates, and hypercontractive semigroups.

Faber Fekete Szegő Theorem ANALYSIS Widom’s Work Part Totik Widom 3

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Chebyshev Operator Theory Polynomials A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a ﬁ ve-volume set that can serve as a graduate-level analysis textbook with a lot of additional Operator Theory Alternation bonus information, including hundreds of problems and numerous notes that extend the text and provide important A Comprehensive Course in Analysis, Part 4 Theorem historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis. Photo courtesy of Bob Paz, Caltech Photo courtesy of Bob Paz, Potential Theory Part 4 focuses on operator theory, especially on a Hilbert space. Central topics are the spectral theorem, the theory of trace class and Fredholm determinants, and the study of unbounded self-adjoint operators. There is also an introduction to the theory of orthogonal polynomials and a long chapter Barry Simon Lower Bounds on Banach algebras, including the commutative and non-commutative Gel’fand- and Special Sets Naimark theorems and Fourier analysis on general locally compact abelian groups.

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