DOCSLIB.ORG

Gegenbauer Polynomials and Positive Definiteness

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Gegenbauer Polynomials and Positive Definiteness Gegenbauer polynomials and positive deniteness Christian Berg University of Copenhagen, Denmark Orsay, Paris November 27, 2015 Based on joint work with Emilio Porcu, University Federico Santa Maria Valparaíso, Chile Christian Berg Gegenbauer polynomials Overview 1. Presentation of the problem and main results 2. Reminder about Gegenbauer polynomials 3. The connection to spherical harmonics 4. Positive denite functions on locally compact groups G d d 5. The class P(S ; G) of positive denite functions on S × G 6. The class P(S1; G) and its representations 7. Some indications of proof 8. Applications to some homogeneous spaces Christian Berg Gegenbauer polynomials Presentation of the problem In Geostatistics one examines measurements depending on the location on the earth and on time. This leads to Random Fields of 2 stochastic variables Z(ξ; u) indexed by (ξ; u) belonging to S × R, 2 where S the 2-dimensional sphereis a model for the earth and R is a model for time. If the variables are real-valued, one considers a basic probability space (Ω; F; P), where all the random variables Z(ξ; u) are dened as measurable mappings from Ω to R. The covariance of two stochastic variables X ; Y is by denition cov(X ; Y ) = E((X − E(X ))(Y − E(Y )): For n variables X1;:::; Xn the covariance matrix cov X X n [ ( k ; l )]k;l=1 is symmetric and positive semi-denite. Christian Berg Gegenbauer polynomials Isotropic and stationary covariance kernels One is interested in isotropic and stationary random elds 2 Z(ξ; u); (ξ; u) 2 S × R, i.e., the situation where there exists a continuous function f :[−1; 1] × R ! R such that the covariance kernel is given as 2 cov(Z(ξ; u); Z(η; v)) = f (ξ · η; v − u); ξ; η 2 S ; u; v 2 R: Here ξ · η = cos(θ(ξ; η)) is the scalar product equal to cosine of the length of the geodesic arc (=angle) between ξ and η. We require with other words that the covariance kernel only depends on the geodesic distance between the points on the sphere and on the time dierence. Christian Berg Gegenbauer polynomials First main result 2 We shall characterize the class P(S ; R) of continuous functions f :[−1; 1] × R ! R which are positive denite in the following sense: 2 For any n 2 N and any (ξ1; u1);:::; (ξn; un) 2 S × R the matrix f u u n [ (ξk · ξl ; l − k )]k;l=1 is symmetric and positive semi-denite. Theorem (B-Porcu 2015) 2 The functions f 2 P(S ; R) are precisely the functions 1 1 X X f (x; u) = 'n(u)Pn(x); 'n(0) < 1; n=0 n=0 where ('n) is a sequence of real-valued continuous positive denite functions on R and Pn are the Legendre polynomials on [−1; 1] normalized as Pn(1) = 1. The series is uniformly convergent. Christian Berg Gegenbauer polynomials Generalizations This Theorem can be generalized in various ways: 2 d The sphere S can be replaced by S ; d = 1; 2;:::. The additive group R can be replaced by any locally compact group G. 2 1 The sphere S can be replaced by the Hilbert sphere S . d We shall characterize the set P(S ; G) of continuous functions f :[−1; 1] × G ! C such that the kernel f (ξ · η; u−1v) d 2 is positive denite on (S × G) . Here d = 1; 2;:::; 1: If G = feg is the trivial group we get a classical Theorem of Schoenberg from 1942 about positive denite functions on spheres. Christian Berg Gegenbauer polynomials Gegenbauer polynomials To formulate these generalizations we need to recall: (λ) The Gegenbauer polynomials Cn for λ > 0 are given by the generating function 1 2 −λ X (λ) n (1 − 2xr + r ) = Cn (x)r ; jrj < 1; x 2 C: (1) n=0 For λ > 0, we have the classical orthogonality relation: Z 1 n 2 21−2λ 1 x2 λ−1=2C (λ) x C (λ) x dx πΓ( + λ) (2) ( − ) n ( ) m ( ) = 2 δm;n: −1 Γ (λ)(n + λ)n! Christian Berg Gegenbauer polynomials Chebyshev polynomials For λ = 0 we use the generating function 1 1 − xr X (0) n = Cn (x)r ; jrj < 1; x 2 : (3) 1 − 2xr + r 2 C n=0 It is well-known that (0) Cn (x) = Tn(x) = cos(n arccos x); n = 0; 1;::: are the Chebyshev polynomials of the rst kind. Z 1 π if n 0 1 x2 −1=2T x T x dx 2 δm;n > (4) ( − ) n( ) m( ) = if n 0 −1 πδm;n = ; (λ) Warning: λ ! Cn (x) is discontinuous at λ = 0. Christian Berg Gegenbauer polynomials More on Gegenbauer polynomials Putting x = 1 in the generating functions yields (λ) Cn (1) = (2λ)n=n!; λ > 0; Tn(1) = 1: Recall that for a 2 C (a)n = a(a + 1) ··· (a + n − 1); n ≥ 1; (a)0 = 1: It is of fundamental importance that (λ) (λ) jCn (x)j ≤ Cn (1); x 2 [−1; 1]; λ ≥ 0: The special value λ = (d − 1)=2 is relevant for the d-dimensional sphere d d+1 S = fx 2 R j jjxjj = 1g; d 2 N (d−1)=2 because of the relation of Cn to spherical harmonics. Christian Berg Gegenbauer polynomials Spherical harmonics d d A spherical harmonic of degree n for S is the restriction to S of a d+1 real-valued harmonic homogeneous polynomial in R of degree n. d Hn(d) = fspherical harmonics of degree ng ⊂ C(S ) d is a nite dimensional subspace of the continuous functions on S . We have (d)n−1 Nn(d) := dim Hn(d) = (2n + d − 1); n ≥ 1; N0(d) = 1: n! The surface measure of the sphere is denoted !d , and it is of total mass 2π(d+1)=2 jj!d jj = : Γ((d + 1)=2) The spaces Hn(d) are mutual orthogonal subspaces of the Hilbert 2 d space L (S ;!d ), which they generate. Christian Berg Gegenbauer polynomials Ultraspherical polynomials 2 d This means that any F 2 L (S ;!d ) has an orthogonal expansion 1 1 X 2 X 2 F = Sn; Sn 2 Hn(d); jjF jj2 = jjSnjj2; n=0 n=0 2 d where the rst series converges in L (S ;!d ), and the second series is Parseval's equation. Here Sn is the orthogonal projection of F onto Hn(d) given as Nn(d) Z Sn(ξ) = cn(d; ξ · η)F (η) d!d (η); d jj!d jj S where ((d−1)=2) ((d−1)=2) cn(d; x) = Cn (x)=Cn (1): These polynomials are called ultraspherical polynomials or d-dimensional Legendre polynomials. The 2-dimensional Legendre polynomials are the classical Legendre polynomials previously denoted Pn. Christian Berg Gegenbauer polynomials Orthogonality relation for ultraspherical polynomials Specializing the orthogonality relation for the Gegenbauer polynomials to λ = (d − 1)=2: Z 1 2 d=2−1 jj!d jj (1 − x ) cn(d; x)cm(d; x) dx = δm;n: −1 jj!d−1jjNn(d) (Dene jj!0jj = 2). Note that jcn(d; x)j ≤ 1; x 2 [−1; 1]: Christian Berg Gegenbauer polynomials Schoenberg's Theorem from 1942 d Let P(S ) denote the class of continuous functions f :[−1; 1] ! R d such that for any n 2 N and for any ξ1; : : : ; ξn 2 S the n × n symmetric matrix f n [ (ξk · ξl )]k;l=1 is positive semi-denite. Theorem (Schoenberg 1942) d A function f :[−1; 1] ! R belongs to the class P(S ) if and only if 1 X f (x) = bn;d cn(d; x); x 2 [−1; 1]; n=0 for a non-negative summable sequence b 1 given as ( n;d )n=0 N d Z 1 jj!d−1jj n( ) 2 d=2−1 bn;d = f (x)cn(d; x)(1 − x ) dx: jj!d jj −1 Christian Berg Gegenbauer polynomials Positive denite functions on groups Consider an arbitrary locally compact group G, where we use the multiplicative notation, and in particular the neutral element of G is denoted e . In the representation theory of these groups the following functions play an crucial role. A continuous function f : G ! C is called positive denite if for any n 2 N and any u1;:::; un 2 G the n × n-matrix f u−1u n [ ( k l )]k;l=1 is hermitian and positive semi-denite. By P(G) we denote the set of continuous positive denite functions on G. Christian Berg Gegenbauer polynomials A generalization d We shall characterize the set P(S ; G) of continuous functions f :[−1; 1] × G ! C such that the kernel −1 d f (ξ · η; u v); ξ; η 2 S ; u; v 2 G (5) is positive denite in the sense that for any n 2 N and any d (ξ1; u1);::: (ξn; un) 2 S × G the n × n-matrix f u−1u n (6) (ξk · ξl ); k l ) k;l=1 is hermitian and positive semi-denite. d Note that for G = feg we can identify P(S ; G) with P(Sd ). Christian Berg Gegenbauer polynomials Simple properties Proposition d (i) For f1; f2 2 P(S ; G) and r ≥ 0 we have rf1; f1 + f2, and d f1 · f2 2 P(S ; G). d (ii) For a net of functions (fi )i2I from P(S ; G) converging pointwise to a continuous function f :[−1; 1] × G ! C, we d have f 2 P(S ; G).
Load more

Recommended publications
  • Orthogonal Polynomials: an Illustrated Guide
    Orthogonal Polynomials: An Illustrated Guide Avaneesh Narla December 10, 2018 Contents 1 Definitions 1 2 Example 1 2 3 Three-term Recurrence Relation 3 4 Christoffel-Darboux Formula 5 5 Zeros 6 6 Gauss Quadrature 8 6.1 Lagrange Interpolation . .8 6.2 Gauss quadrature formula . .8 7 Classical Orthogonal Polynomials 11 7.1 Hermite Polynomials . 11 7.2 Laguerre Polynomials . 12 7.3 Legendre Polynomials . 14 7.4 Jacobi Polynomials . 16 7.5 Chebyshev Polynomials of the First Kind . 17 7.6 Chebyshev Polynomials of the Second Kind . 19 7.7 Gegenbauer polynomials . 20 1 Definitions Orthogonal polynomials are orthogonal with respect to a certain function, known as the weight function w(x), and a defined interval. The weight function must be continuous and positive such that its moments (µn) exist. Z b n µn := w(x)x dx; n = 0; 1; 2; ::: a The interval may be infinite. We now define the inner product of two polynomials as follows Z 1 hf; giw(x) := w(x)f(x)g(x) dx −∞ 1 We will drop the subscript indicating the weight function in future cases. Thus, as always, a 1 sequence of polynomials fpn(x)gn=0 with deg(pn(x)) = n are called orthogonal polynomials for a weight function w if hpm; pni = hnδmn Above, the delta function is the Kronecker Delta Function There are two possible normalisations: If hn = 1 8n 2 f0; 1; 2:::g, the sequence is orthonormal. If the coefficient of highest degree term is 1 for all elements in the sequence, the sequence is monic.
    [Show full text]
  • On Sieved Orthogonal Polynomials Ii: Random Walk Polynomials
    Can. J. Math., Vol. XXXVIII, No. 2, 1986, pp. 397-415 ON SIEVED ORTHOGONAL POLYNOMIALS II: RANDOM WALK POLYNOMIALS JAIRO CHARRIS AND MOURAD E. H. ISMAIL 1. Introduction. A birth and death process is a stationary Markov process whose states are the nonnegative integers and the transition probabilities (1.1) pmn(t) = Pr{X(t) = n\X(0) = m) satisfy Pmt + 0(/) n = m + 1 (1.2) Pm„(t) = 8mt + 0(0 n = m - 1 1 — 08m + Sm)t + o(0 n = m, as / —> 0. Here we assume /?n > 0, 8n + 1 > 0, « = 0, 1,. , but ô0 ^ 0. Karlin and McGregor [10], [11], [12], showed that each birth and death process gives rise to two sets of orthogonal polynomials. The first is the set of birth and death process polynomials {Qn(x) } generated by Ôo(*) = 1> Gi(*) = (A) + «b " ^Vi8o, « > 0. In this case there exists a positive measure da supported on [0, oo) such that oo Qn(x)Qm(x)da(x) = 8mn/iTn, m, n = 0, 1,. /; o holds where •un = /yj,... p„-l/{sls2... 8„), « > o, «o = l. The second set is the set of random walk polynomials. They arise when one studies a random walk on the state space. The random walk polynomials {Rn(x) } satisfy the recursion (1.3) xRn(x) = BnRn + x(x) + AA-i(*)> n > 0 and the initial conditions Received October 3, 1984 and in revised form February 14, 1985. This research was partially supported by NSF Grant MCS 8313931, Arizona State University and the National University of Colombia. 397 Downloaded from https://www.cambridge.org/core.
    [Show full text]
  • Moments of Classical Orthogonal Polynomials
    Moments of Classical Orthogonal Polynomials zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr.rer.nat) im Fachbereich Mathematik der Universität Kassel By Patrick Njionou Sadjang ????? Ph.D thesis co-supervised by: Prof. Dr. Wolfram Koepf University of Kassel, Germany and Prof. Dr. Mama Foupouagnigni University of Yaounde I, Cameroon October 2013 Tag der mündlichen Prüfung 21. Oktober 2013 Erstgutachter Prof. Dr. Wolfram Koepf Universität Kassel Zweitgutachter Prof. Dr. Mama Foupouagnigni University of Yaounde I Abstract The aim of this work is to find simple formulas for the moments mn for all families of classical orthogonal polynomials listed in the book by Koekoek, Lesky and Swarttouw [30]. The generating functions or exponential generating functions for those moments are given. To my dear parents Acknowledgments Foremost, I would like to express my sincere gratitude to my advisors Prof. Dr. Wolfram Koepf and Prof. Dr. Mama Foupouagnigni for the continuous support of my Ph.D study and research, for their patience, motivation, enthusiasm, and immense knowledge. Their guidance helped me in all the time of research and writing of this thesis. I could not have imagined having better advisors and mentors for my Ph.D study. I am grateful to Prof. Dr. Mama Foupouagnigni for enlightening me the first glance of re- search. My sincere thanks also go to Prof. Dr. Wolfram Koepf for offering me the opportunity to visit the University of Kassel where part of this work has been written. I acknowledge the financial supports of the DAAD via the STIBET fellowship which en- abled me to visit the Institute of Mathematics of the University of Kassel.
    [Show full text]
  • Bibliography
    Bibliography 1. N. ABDUL-HALIM and W.A. AL-SALAM, A characterization of the Laguerre polynomials. Rendiconti del Seminario Matematico della UniversitadiPadova` 34, 1964, 176–179. 2. M. ABRAMOWITZ and I.A. STEGUN (eds.), Handbook of mathematical functions (with for- mulas, graphs, and mathematical tables). Dover Publications, New York, 1970. 3. L.D. ABREU and J. BUSTOZ, Turan´ inequalities for symmetric Askey-Wilson polynomials. The Rocky Mountain Journal of Mathematics 30, 2000, 401–409. 4. S. AHMED,A.LAFORGIA and M.E. MULDOON, On the spacing of the zeros of some classi- cal orthogonal polynomials. Journal of the London Mathematical Society (2) 25, 1982, 246– 252. 5. S. AHMED,M.E.MULDOON and R. SPIGLER, Inequalities and numerical bounds for ze- ros of ultraspherical polynomials. SIAM Journal on Mathematical Analysis 17, 1986, 1000– 1007. 6. K. ALLADI and M.L. ROBINSON, Legendre polynomials and irrationality. Journal fur¨ die Reine und Angewandte Mathematik 318, 1980, 137–155. 7. WM.R. ALLAWAY, The representation of orthogonal polynomials in terms of a differential operator. Journal of Mathematical Analysis and Applications 56, 1976, 288–293. 8. WM.R. ALLAWAY, Some properties of the q-Hermite polynomials. Canadian Journal of Mathematics 32, 1980, 686–694. 9. WM.R. ALLAWAY, Convolution orthogonality and the Jacobi polynomials. Canadian Math- ematical Bulletin 32, 1989, 298–308. 10. WM.R. ALLAWAY, Convolution shift, c-orthogonality preserving maps, and the Laguerre polynomials. Journal of Mathematical Analysis and Applications 157, 1991, 284–299. 11. N.A. AL-SALAM, Orthogonal polynomials of hypergeometric type. Duke Mathematical Jour- nal 33, 1966, 109–121.
    [Show full text]
  • Change of Basis Between Classical Orthogonal Polynomials
    Change of Basis between Classical Orthogonal Polynomials D.A. Wolfram College of Engineering & Computer Science The Australian National University, Canberra, ACT 0200 [email protected] Abstract Classical orthogonal polynomials have widespread applications includ- ing in numerical integration, solving differential equations, and interpola- tion. Changing basis between classical orthogonal polynomials can affect the convergence, accuracy, and stability of solutions. We provide a general method for changing basis between any pair of classical orthogonal polynomials by using algebraic expressions called coefficient functions that evaluate to connection coefficients. The method builds directly on previous work on the change of basis groupoid. The scope has fifteen kinds of classical orthogonal polynomials including the classes of Jacobi, Gegenbauer and generalized Laguerre polynomials. The method involves the mappings to and from the monomials for these polynomial bases. Sixteen coefficient functions appear to be new for polynomials that do not have definite parity. We derive the remainder from seven sources in the literature. We give a complete summary of thirty coefficient functions. This enables change of basis to be defined algebraically and uniformly between any pair of classical orthogonal polynomial bases by using a vector dot product to compose two coefficient functions to give a third. A definition of Jacobi polynomials uses a basis of shifted monomials. We find a key new mapping between the monomials and Jacobi polynomi- arXiv:2108.13631v1 [math.CA] 31 Aug 2021 als by using a general mapping between shifted monomials and monomials. It yields new mappings for the Chebyshev polynomials of the third and fourth kinds and their shifted versions.
    [Show full text]
  • Determinant Inequalities for Sieved Ultraspherical Polynomials
    IJMMS 25:11 (2001) 745–751 PII. S0161171201004896 http://ijmms.hindawi.com © Hindawi Publishing Corp. DETERMINANT INEQUALITIES FOR SIEVED ULTRASPHERICAL POLYNOMIALS J. BUSTOZ and I. S. PYUNG (Received 28 March 2000) Abstract. Paul Turan first observed that the Legendre polynomials satisfy the inequality 2 Pn(x) − Pn−1(x)Pn(x) > 0, −1 <x<1. Inequalities of this type have since been proved for both classical and nonclassical orthogonal polynomials. In this paper, we prove such an inequality for sieved orthogonal polynomials of the second kind. 2000 Mathematics Subject Classification. Primary 33C47; Secondary 33E30. 1. Introduction. It was observed by Paul Turán [9] that the Legendre polynomials satisfy the determinant inequality 2 Pn(x)−Pn+1(x)Pn−1(x) > 0, −1 <x<1,n= 0,1,.... (1.1) G. Szegö [8] gave two very beautiful proofs of Turán’s inequality. In the years since Szegö’s paper appeared, it has been proved by various authors [5, 6, 7] that inequality (1.1) is satisfied by the classical orthogonal polynomials. In general, let {Pn(x)} be a sequence of polynomials orthogonal in an interval [a,b]. Then the polynomials must satisfy a recursion Pn+1(x) = Anx +Bn Pn(x)−CnPn−1(x), n = 0,1,..., (1.2) where we define P−1(x) = 0. We begin with a very simple result shows that, inequalities of Turán type are satis- fied by any sequence of orthogonal polynomials. Theorem 1.1. If the polynomials {Pn(x)} are orthogonal on a ≤ x ≤ b, then for each n there exists cn,a≤ cn ≤ b, such that P 2(x) P (x) P (x) n n+1 n−1 2 − ≥ 0,a≤ x ≤ b.
    [Show full text]
  • A New Property of a Class of Jacobi Polynomials
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0 A NEW PROPERTY OF A CLASS OF JACOBI POLYNOMIALS GEORGE CSORDAS, MARIOS CHARALAMBIDES, AND FABIAN WALEFFE Abstract. Polynomials whose coefficients are successive derivatives of a class of Jacobi polynomials evaluated at x = 1 are stable. This yields a novel and short proof of the known result that the Bessel polynomials are stable polyno- mials. Stability preserving linear operators are discussed. The paper concludes with three open problems involving the distribution of zeros of polynomials. 1. Introduction The new property referred to in the title was observed and conjectured (see Conjecture 1 below) while developing a numerical solution for the Navier-Stokes equations [23]. It is related to the fundamental problem of constructing discretiza- tion schemes of continuous problems (involving, for example, boundary value prob- lems for various systems of partial differential equations) in such a manner that the associated eigenvalue problems are free of “spurious eigenvalues”; that is, the eigenvalues are all negative. For the purpose of this paper, it is sufficient to consider the problem of constructing polynomial approximations to the eigenvalue problem d2u/dx2 = λu for 1 < x < 1 with u( 1) = 0. It is well-known that the solutions to this problem consist− of negative eigenvalues± λ with trigonometric eigenfunctions u(x). If un(x) is a polynomial approximation of degree n to u(x), then the residual d2 (1) R (x) := λu (x) u (x) n n − dx2 n is also a polynomial of degree n in x and this relationship can be inverted to obtain [n/2] d2k (2) u (x) = µk+1 R (x) n dx2k n kX=0 where µ = 1/λ and [n/2] denotes the greatest integer less than or equal to n/2.
    [Show full text]
  • Dual Addition Formula for Gegenbauer Polynomials
    Dual addition formula for Gegenbauer polynomials Tom Koornwinder Korteweg-de Vries Institute, University of Amsterdam [email protected] https://staff.fnwi.uva.nl/t.h.koornwinder/ Lecture on 20 March 2017 in the Mathematical Physics Seminar, Faculty of Mathematics, University of Vienna last modified : April 4, 2017 Tom Koornwinder Dual addition formula for Gegenbauer polynomials This talk based on: Tom H. Koornwinder, Dual addition formulas associated with dual product formulas, arXiv:1607.06053 [math.CA] Acknowledgement With regard to the history pages at the end I thank prof. Josef Hofbauer (Vienna) for information about Gegenbauer and Allé. Tom Koornwinder Dual addition formula for Gegenbauer polynomials Orthogonal polynomials µ positive measure on R such that 8n 2 R jxjn dµ(x) < 1. Z≥0 R For each n 2 Z≥0 let pn(x) be polynomial of degree n such that Z pm(x) pn(x) dµ(x) = 0 (m 6= n) (1) R (orthogonal polynomials with respect to µ). Then x pn(x) = An pn+1(x) + Bn pn(x) + Cn pn−1(x) (2) (put p−1(x) = 0) with AnCn > 0 (3-term recurrence relation). Conversely, the 3-term recurrence (2) with AnCn > 0 and p−1(x) = 0, p0(x) = 1 generates a sequence of polynomials fpn(x)gn=0;1;::: which satisfy (1) for some µ (Favard theorem). Tom Koornwinder Dual addition formula for Gegenbauer polynomials classical orthogonal polynomials We can write (2) as J pn(x) = x pn(x) where J is an operator (Jacobi matrix) acting on the variable n. Can we dualize this? So this would be L pn(x) = λn pn(x) where L is an operator acting on the variable x.
    [Show full text]
  • Pre-Publication Accepted Manuscript
    Masanori Sawa, Yukihiro Uchida Algebro-geometric aspects of the Christoffel-Darboux kernels for classical orthogonal polynomials Transactions of the American Mathematical Society DOI: 10.1090/tran/7998 Accepted Manuscript This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proofread, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. The Version of Record is accessible to everyone five years after publication in an issue. ALGEBRO-GEOMETRIC ASPECTS OF THE CHRISTOFFEL-DARBOUX KERNELS FOR CLASSICAL ORTHOGONAL POLYNOMIALS MASANORI SAWA AND YUKIHIRO UCHIDA Abstract. In this paper we study algebro-geometric aspects of the Christoffel- Darboux kernels for classical orthogonal polynomials with rational coefficients. We find a novel connection between a projective curve defined by the Christoffel- Darboux kernel and a system of Diophantine equations, which was originally designed by Hausdorff (1909) for applications to Waring's problem, and which is closely related to quadrature formulas in numerical analysis and Gaussian designs in algebraic combinatorics. We prove some nonsolvability results of such Hausdorff-type equations. 1. Introduction For real numbers a < b, let fΦlg be a sequence of orthogonal polynomials with respect to a given positive weight function w over the interval (a; b).
    [Show full text]
  • Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane
    Constructive Approximation https://doi.org/10.1007/s00365-020-09515-0 Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane Gernot Akemann1 · Taro Nagao2 · Iván Parra1 · Graziano Vernizzi3 Received: 24 May 2019 / Revised: 8 April 2020 / Accepted: 7 May 2020 © The Author(s) 2020 Abstract We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polyno- (1+α) mials Cn (z) for α>−1 containing the Legendre polynomials Pn(z) and the subset (α+ 1 ,± 1 ) 2 2 Pn (z) of the Jacobi polynomials. These polynomials provide an orthonormal basis and the corresponding weighted Bergman space forms a complete metric space. This leads to a certain family of Selberg integrals in the complex plane. We recover the known orthogonality of Chebyshev polynomials of the first up to fourth kind. The limit α →∞leads back to the known Hermite polynomials orthogonal in the entire complex plane. When the ellipse degenerates to a circle we obtain the weight func- tion and monomials known from the determinantal point process of the ensemble of truncated unitary random matrices. Keywords Planar orthogonal polynomials · Ellipse · Bergman space · Selberg integral Communicated by Peter Forrester. B Gernot Akemann [email protected] Taro Nagao [email protected] Iván Parra [email protected] Graziano Vernizzi [email protected] 1 Faculty of Physics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld, Germany 2 Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan 3 Siena College, 515 Loudon Road, Loudonville, NY 12211, USA 123 Constructive Approximation Mathematics Subject Classification 30C10 · 33C45 · 42C05 · 15B52 1 Introduction Orthogonal polynomials in the complex plane play an important role for non-Hermitian random matrix theory.
    [Show full text]
  • Arxiv:1607.06196V2 [Math.CA] 28 Dec 2018 K = 3, J = 4, So a Natural Question Is: What Is This Representation? This Problem Has Been Solved in the Mean Time
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 071, 13 pages Report from the Open Problems Session at OPSFA13? Edited by Howard S. COHL Applied and Computational Mathematics Division, National Institute of Standards and Technology (NIST), Mission Viejo, CA, 92694 USA E-mail: [email protected] URL: http://www.nist.gov/itl/math/msg/howard-s-cohl.cfm Received January 28, 2016, in final form July 12, 2016; Published online July 21, 2016 https://doi.org/10.3842/SIGMA.2016.071 Abstract. These are the open problems presented at the 13th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA13), Gaithersburg, Maryland, on June 4, 2015. Key words: Schur's inequality; hypergeometric functions; orthogonal polynomials; lineariza- tion coefficients; connection coefficients; symbolic summation; multiple summation; numer- ical algorithms; Gegenbauer polynomials; multiple zeta values; distribution of zeros 2010 Mathematics Subject Classification: 31C12; 32Q10; 33C05; 33C45; 33C55; 33F10; 35J05 1 Schur's inequality Posed by Richard Askey Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA E-mail: [email protected] In Hardy, Littlewood and P´olya's book \Inequalities" [22, Problem 60 on p. 64], the following inequality (communicated by I. Schur) was stated: xn(x − y)(x − z) + yn(y − x)(y − z) + zn(z − x)(z − y) > 0; when x, y, z are positive and not all equal, and n ≥ 0. It is not hard to show that the inequality is true for all real x, y, z for n even when > 0 is replaced by ≥ 0. There is a strange theorem of Hilbert [23].
    [Show full text]
  • Associated with the Jacobi Polynomials
    mathematics of computation volume 47, number 176 october 1986,pages (369-682 Properties of the Polynomials Associated with the Jacobi Polynomials By S. Lewanowicz Abstract. Power forms and Jacobi polynomial forms are found for the polynomials W^"-^ associated with Jacobi polynomials. Also, some differential-difference equations and evalua- tions of certain integrals involving W¡¡"^'1 are given. 1. Introduction. Let w be a nonnegative function defined on an interval [a, b] for which all moments m„:= j x"w(x)dx, n = 0,1,..., exist and are finite, m0 > 0. Let {/>„} be the monic polynomials orthogonal on [a, b] with respect to w. The polynomials (1.1),, ,> q„(x)=/ N. [>> -rzTt—wU)ät,PÁX) -Pn({) (t\jt « = 0,1,...,n -, J a x ' are called the polynomials associated with the pn. As is well known, {pn(x)}, ( aÂx)} are linearly independent solutions of the recurrence formula (I-2) yn+i-(x~ a„)y„ + bny„_x = Q, « = 0,1,..., where (l3) an:= (xpn,p„)/(p„,pn), n>0, *o:=«o. b„:= (pn,pn)/(pn_x,pn_x), n>l, and (f,g)-= f f(x)g(x)w(x)dx. Ja The initial conditions are P-i(x):= 0, pa(x):= 1 and ?_i(x):= -1, q0(x):= 0, respectively. Received April 16, 1985; revised October 15, 1985. 1980 Mathematics Subject Classification.Primary 33A65, 39A10; Secondary 65D20. ©1986 American Mathematical Society 0025-5718/86 $1.00 + $.25 per page 669 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 670 S. LEWANOWICZ Let us consider the continued fraction b0 bx (1.4) x — a0 — x — ax — where an, b„ are the coefficients of (1.2) given in (1.3).
    [Show full text]