PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 3, Pages 873–879 S 0002-9939(00)05821-4 Article electronically published on October 11, 2000
ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE ASSOCIATED WITH THE LAGUERRE POLYNOMIALS
LI-CHIEN SHEN
(Communicated by Hal L. Smith)
Abstract. Using the well-known fact that the Fourier transform is unitary, we obtain a class of orthogonal polynomials on the unit circle from the Fourier transform of the Laguerre polynomials (with suitable weights attached). Some related extremal problems which arise naturally in this setting are investigated.
1. Introduction This paper deals with a class of orthogonal polynomials which arise from an application of the Fourier transform on the Laguerre polynomials. We shall briefly describe the essence of our method. Let Π+ denote the upper half plane {z : z = x + iy, y > 0} and let Z ∞ H(Π+)={f : f is analytic in Π+ and sup |f(x + yi)|2 dx < ∞}. 0
Received by the editors June 1, 1999. 2000 Mathematics Subject Classification. Primary 33C47. Key words and phrases. Fourier transform, Laguerre polynomial.
c 2000 American Mathematical Society 873
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We thank the referee for many detailed suggestions in revising the original man- uscript as well as for drawing our attention to papers [1] and [2]. 2. Key results We assume that the reader is reasonably familiar with the basic properties of the Laguerre polynomials which can be found in Chapter 4 of [3]. We say a few words about our notation. The Fourier transform of an integrable function f is defined as Z ∞ fˆ(t)= eixt f(x) dx, −∞
(a)n = a(a+1)(a+2)...(a+n−1) and F (a, b; c; z) is the standard hypergeometric series. Let Xn − k α (α +1)n ( n)kx − Ln(x)= (α> 1) n! k!(α +1)k k=0 be the Laguerre polynomial of degree n associated with the weight e−xxα dx,and let ( Lα(x) e−x/2 xα/2,x≥ 0, lα(x)= n n 0,x<0. α We now compute the Fourier transform of ln (x). First, we observe that Z ∞ − α − 2 1 ˆα ixt −x/2 α/2 α 1 − (2.1) l0 (t)= e e x dx =Γ +1 it , 0 2 2
Z ∞ −n − n ˆα ixt −x/2 α/2+n α 1 − ˆα (2.2) ( iD) l0 (t)= e e x dx = +1 it l0 (t), 0 2 n 2 − n − n dn − 1 where ( iD) =( i) dtn and t is a complex number with imaginary part > 2 .It is easy to justify the differentiation behind the integral sign because of the absolute convergence of the integral. From (2.1) and (2.2), we obtain Z ∞ ˆα ixt α/2 −x/2 α α − ˆα ln (t)= e x e Ln(x) dx = Ln( iD)l0 (t) 0 Xn (−n) α +1 k (1 + α)n ˆα k 2 k 1 = l0 (t) 1 n! k!(α +1)k − it k=0 2 (1 + α) α 1 = n lˆα(t) F −n, +1;α +1; . 0 1 − n! 2 2 it Now let z =(2t − i)/(2t + i) and recall that (c − a) F (−n, a; c; x)= n F (−n, a; a − n − c +1;1− x). (c)n We can write α (1 + α)n α α ˆα − 2 +1 − − ln (t)=Γ +1 (1 z) F n, +1;α +1;1 z 2 n! 2 α α +1 α =Γ +1 (1 − z) 2 g (z), 2 n
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where (α/2) α α (2.3) gα(z)= n F −n, +1;−n − +1;z n n! 2 2 (1 + α) α = n F −n, +1;α +1;1− z n! 2 α is a polynomial of degree n. For brevity, we let gn(z)=gn (z). We choose the α − 2 +1 ˆα branch of (1 z) which takes the value 1 at z = 0. Therefore ln , as a function of z, is analytic in the complex plane cut along [1, ∞). Moreover, from the fact that the Fourier transform preserves the inner product (the Parseval’s identity), we deduce that Z Γ2 α +1 2π θ (2.4) 2α 2 g (eiθ) g (eiθ)sinα dθ 2π n m 2 0 Z 2 α Γ 2 +1 α dz = gn(z) gm(z) |1 − z| 2πi |z|=1 z Z 2 α Γ 2 +1 α +1 α +1 −2 = − g (z) g (z)(1− z) 2 (1 − z¯) 2 (1 − z) dz 2πi n m Z |z|=1 1 ∞ ˆα ˆα = ln (t) lm(t) dt 2π −∞ Z ∞ α α = ln (x) lm(x) dx (Parseval’s identity) 0 Z ∞ α α −x α = Ln(x) Lm(x) e x dx (0 Γ(n+α+1) n = m = n! ([3, p. 84]). 0 n =6 m.
We summarize our result in { α }∞ Theorem 1. The polynomials gn (z) n=0 form a complete orthogonal basis for Z 2π 2 | | | iθ |2 ∞ Hµ = f : f is analytic in z < 1 and f(e ) dµ < , 0
1 α θ where dµ = 2π sin 2 dθ. α 2 ∞ The completeness follows immediately from the completeness of ln (x)inL [0, ) [5, p. 108], the Paley-Wiener Theorem and the fact that the Fourier transform preserves the basis. n It is interesting to note that for α =0,gn(z)=z .Andforα =2, Xn n gn(z)= (k +1)z k=0 which is the nth partial sum of the familiar power series X∞ (1 − z)−2 = (k +1)zk. k=0
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 876 LI-CHIEN SHEN > − − − α > > > Remark 1. We note that for α 0, ( n)k/ n 2 +1 k 0forn k 0. Hence, from (2.3), we conclude that
max |gn(z)| = gn(R). |z|6R Similarly, | (k) | (k) max gn (z) = gn (R). |z|6R In particular, (k) α − gn(1) = (α +1)n/n!andgn (1) = +1 (α +1)n/((α +1)k(n k)!). 2 k
Remark 2. The reproducing kernel KN (z, w) with respect to the measure dµ for the polynomials of degree 6 N is defined as XN KN (z, w)= cngn(z)gn(w), n=0 where α c = a n!/(α +1) and a =2αΓ2 +1 /Γ(α +1). n α n α 2 The significance of the term “reproducing kernel” will be made clear in the next section. Then, from the above remark, we have XN max KN (w, w)=KN (1, 1) = aα (α +1)n/n!. |w|61 n=0 More generally, let XN | (k) |2 MN,k(w)= cn gn (w) . n=k Then
max MN,k(w)=MN,k(R). |w|6R
3. Extremal problems
Let PN be the set of all polynomials of degree 6 N. It follows from the orthog- onality (2.4) that if p ∈ P ,then N Z
p(w)= KN (z, w)p(z) dµ. |z|=1 More generally, by differentiating the above identity k times, we have Z (k) k ∂ (3.1) p (w)= D KN (z,w)p(z) dµ D¯ = . |z|=1 ∂w¯ Applying the Schwarz inequality to (3.1), we deduce (with the help of orthogo- nality) (k) (3.2) |p (w)| 6 kpkµ MN,k(w)forp ∈ PN , R k k2 | |2 where p µ = |z|=1 p(z) dµ.
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Therefore, we have
Theorem 2. If p ∈ PN ,then (k) sup |p (w)| = MN,k(w), kpkµ=1 and the equality is achieved by the polynomial XN (k) p(z)= cngn(z) gn (w)/MN,k(w). n=k We comment that this extremal function p(z) is essentially unique. To see this, we recall that [2, p. 63] in Schwarz inequality |(f, g)|≤kfkkgk, the equality holds if and only if f = cg a.e. Hence, the equality in (3.2) holds if and k only if p(z)=c D¯ KN (z, w); and c = a/MN,k(w), |a| =1, if we further require that kpkµ =1.
Corollary. If p ∈ PN and kpkµ =1,then Γ α +1 XN (α +1) |p(w)| 5 2α 2 · n Γ(α +1) n! n=0 for |w| =1.
From (3.2), we can also establish that if p ∈ PN ,then
inf kpkµ =1/MN,k(R), kp(k)k∞=1
(k) (k) where kp k∞ =max|z|=R |p (z)|. The proof is simple and it goes as follows. Choose a point w on the circle |z| = R, (k) (k) so that |p (w)| = kp k∞ = 1. Then, according to (3.2) and Remark 2,
kpkµ > 1/MN,k(w) > 1/MN,k(R), and again the equality holds for XN (k) p(z)= cngn(z)gn (R)/M N,k (R). n=k We remark that the extremal problems (the Bernstein-Markoff type inequalities) concerning the norm of the derivatives of a polynomial subject to certain given constraint have been studied by numerous authors and some of the more recent results of this nature have appeared in [6] and [7].
4. Connection with the Gegenbauer polynomials
α λ To see the connection of gn (z) with the Gegenbauer polynomial Cn (x), we write α k (−n − +1)k =(−1) (α/2)n /(α/2)n−k;then 2 Xn α α +1 − gα(z)= 2 k 2 n k zk. n k!(n − k)! k=0
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λ We recall that the definition for the generating function for Cn (cos θ)is X∞ λ n − 2 λ Cn (cos θ)r =1/(1 2r cos θ + r ) . n=0 Now write iθ −1 2 − α −iθ α iθ α +1 (1 − re ) (1 − 2r cos θ + r ) 2 =1/(1 − re ) 2 (1 − re ) 2 . Then the series expansion for the left-hand side is ! ! X∞ X∞ X∞ Xn in θ n α/2 n n in θ α/2 −ik θ e r Cn (cos θ)r = r e Ck (cos θ)e , n=0 n=0 n=0 k=0 and the series expansion for the right-hand side is ! ! ∞ ∞ X α +1 X α 2 n rnein θ 2 n rne−in θ n! n! n=0 n=0 X∞ Xn α α X∞ +1 − = rne−in θ 2 k 2 n k e2ik θ = rne−in θ gα(e2iθ). k!(n − k)! n n=0 k=0 n=0 We note that the series are absolutely convergent for |r| < 1. Equating the corresponding terms in both series, we have Xn α 2iθ α/2 (2n−k) iθ gn (e )= Ck (cos θ)e . k=0 We can derive a second relation with the Gegenbauer polynomials by writing (1 − reiθ )−1(1 − 2r cos θ + r2)−α/2 =(1− re−iθ)(1 − 2r cos θ + r2)−α/2−1 ∞ X α −iθ 2 +1 n =(1− re ) Cn (cos θ)r n=0 X∞ α +1 α +1 − α +1 2 2 − iθ 2 n = C0 (cos θ)+ (Cn (cos θ) e Cn−1 (cos θ))r . n=1 Then α +1 α +1 α 2iθ inθ 2 − iθ 2 gn (e )=e (Cn (cos θ) e Cn−1 (cos θ)), and α α 2iθ 2 +1 g0 (e )=C0 (cos θ).
References
1. H. T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc. 124 (1996), no.3, 887-898. MR 96f:33018 2. T. H. Koornwinder, Meixner-Pollaczek polynomials and Heisenberg algebra, J. Math. Phys. 30 (1989), no.4, 767-769. MR 90e:33037 3. N. N. Lebedev, Special Functions and Their Applications, Dover Publications, Inc., New York (1972). MR 50:2568 4. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York (1966). MR 35:1420 5. G. Szego, Orthogonal Polynomials, American Mathematical Society, New York (1959). MR 21:5029
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6. A. K. Varma, Markoff type inequalities for curved majorants is L2 norm in approximation theory (Kecskemet, 1990),689-697, Colloq. Math. Soc. Janos Bolyai, 58, North-Holland, Am- sterdam, 1991. MR 94e:26028 7. A. K. Varma, On some extremal properties of algebraic polynomials, J. Approx. Theory 69 (1992), 48-54. MR 92m:41042
Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105 E-mail address: [email protected]
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