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Asymptotics of the Chebyshev Polynomials of General Sets

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Asymptotics of the Chebyshev Polynomials of General Sets 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. 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 Asymptotics 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 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 L1 (so only the support matters). Specifically, let e ⊂ C be a compact, infinite, set of points. For any function, f, define kfke = sup fjf(z)j j z 2 eg The Chebyshev polynomial of degree n is the monic polynomial, Tn, with kTnke = inf fkP ke j deg(P ) = n and P is monicg 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 fjf(z)j j z 2 eg The Chebyshev polynomial of degree n is the monic polynomial, Tn, with kTnke = inf fkP ke j deg(P ) = n and P is monicg 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. Chebyshev polynomials just 2 1 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 kTnke = inf fkP ke j deg(P ) = n and P is monicg 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. Chebyshev polynomials just 2 1 Alternation replace L by L (so only the support matters). Theorem Specifically, let e ⊂ be a compact, infinite, set of points. Potential Theory C Lower Bounds For any function, f, define and Special Sets kfk = sup fjf(z)j j z 2 eg 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 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 1 Alternation replace L by L (so only the support matters). Theorem Specifically, let e ⊂ be a compact, infinite, set of points. Potential Theory C Lower Bounds For any function, f, define and Special Sets kfk = sup fjf(z)j j z 2 eg 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 fkP ke j deg(P ) = n and P is monicg 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 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 1 Alternation replace L by L (so only the support matters). Theorem Specifically, let e ⊂ be a compact, infinite, set of points. Potential Theory C Lower Bounds For any function, f, define and Special Sets kfk = sup fjf(z)j j z 2 eg 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 fkP ke j deg(P ) = n and P is monicg 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 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 1 Alternation replace L by L (so only the support matters). Theorem Specifically, let e ⊂ be a compact, infinite, set of points. Potential Theory C Lower Bounds For any function, f, define and Special Sets kfk = sup fjf(z)j j z 2 eg 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 fkP ke j deg(P ) = n and P is monicg Totik Widom Bounds The minimizer is unique (as we’ll see below in the case that Szegő–Widom e ⊂ R), Asymptotics 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 1 Alternation replace L by L (so only the support matters). Theorem Specifically, let e ⊂ be a compact, infinite, set of points. Potential Theory C Lower Bounds For any function, f, define and Special Sets kfk = sup fjf(z)j j z 2 eg 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 fkP ke j deg(P ) = n and P is monicg 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 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 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.
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