Bilinear Generating Functions for Orthogonal Polynomials

Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U _q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials. July 6, 1997. Date accepted: September 23, 1998.     

On a Conjecture of E. A. Rakhmanov

It is shown that a conjecture of E. A. Rakhmanov is true concerning the zero distribution of orthogonal polynomials with respect to a measure having a discrete real support. We also discuss the case of extremal polynomials with respect to some discrete L _p -norm, 0 < p ≤∈fty , and give an extension to complex supports. Furthermore, we present properties of weighted Fekete points with respect to discrete complex sets, such as the weighted discrete transfinite diameter and a weighted discrete Bernstein—Walsh-like inequality. August 24, 1998. Date revised: March 26, 1999. Date accepted: April 27, 1999.     

Bernstein—Szegő's Theorem for Sobolev Orthogonal Polynomials

In this paper we obtain the strong asymptotics for the sequence of orthogonal polynomials with respect to the inner product where ρ ₀ and ρ ₁ are weights which satisfy Szegő's condition, supported on a smooth Jordan closed curve or arc. December 14, 1997. Date revised: September 21, 1998. Date accepted: November 16, 1998.     

Nonnegative Trigonometric Polynomials

An extremal problem for the coefficients of sine polynomials, which are nonnegative in [0,π] , posed and discussed by Rogosinski and Szegő is under consideration. An analog of the Fejér—Riesz representation of nonnegative general trigonometric and cosine polynomials is proved for nonnegative sine polynomials. Various extremal sine polynomials for the problem of Rogosinski and Szegő are obtained explicitly. Associated cosine polynomials k _n (θ) are constructed in such a way that { k _n (θ) } are summability kernels. Thus, the L _p , pointwise and almost everywhere convergence of the corresponding convolutions, is established. April 26, 2000. Date revised: December 28, 2000. Date accepted: February 8, 2001.     

Multiple Orthogonal Polynomials Associated with the Modified Bessel Functions of the First Kind

   Abstract. We consider polynomials which are orthogonal with respect to weight functions, which are defined in terms of the modified Bessel function I _ν and which are related to the noncentral χ ² -distribution. It turns out that it is the most convenient to use two weight functions with indices ν and ν+1 and to study orthogonality with respect to these two weights simultaneously. We show that the corresponding multiple orthogonal polynomials of type I and type II exist and give several properties of these polynomials (differential properties, Rodrigues formula, explicit formulas, recurrence relation, differential equation, and generating functions).     

Uniqueness of Markov-Extremal Polynomials on Symmetric Convex Bodies

For a compact set K\subset R ^d with nonempty interior, the Markov constants M _n (K) can be defined as the maximal possible absolute value attained on K by the gradient vector of an n -degree polynomial p with maximum norm 1 on K . It is known that for convex, symmetric bodies M _n (K) = n ² /r(K) , where r(K) is the ``half-width' (i.e., the radius of the maximal inscribed ball) of the body K . We study extremal polynomials of this Markov inequality, and show that they are essentially unique if and only if K has a certain geometric property, called flatness. For example, for the unit ball B ^d (\smallbf 0, 1) we do not have uniqueness, while for the unit cube [-1,1] ^d the extremal polynomials are essentially unique. September 9, 1999. Date revised: September 28, 2000. Date accepted: November 14, 2000.     

Generalized Bernstein Polynomials

We introduce polynomials B ⁿ _i (x;ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating B ⁿ _i (x;ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

Contracted Zero Distributions of Extremal Polynomials Associated with Slowly Decaying Weights

We consider the asymptotic zero behavior of polynomials that are extremal with respect to slowly decaying weights on [0, ∈fty) , such as the log-normal weight \exp(-γ ² log  ² x) . The zeros are contracted by taking the appropriate d _n th roots with d _n →∈fty . The limiting distribution of the contracted zeros is described in terms of the solution of an extremal problem in logarithmic potential theory with a circular symmetric external field. November 23, 1998. Date revised: February 8, 1999. Date accepted: March 2, 1999.     

On Lp -Discrepancy of Signed Measures

We consider L ^p -discrepancy estimates between two Borel measures supported on the interval [-1, 1] and give applications to the distribution of zeros of orthogonal polynomials. Thereby we obtain a new interpretation to the different local zero behavior of Pollaczek polynomials. June 16, 2000. Date accepted: December 26, 2000.     

On Multivariate Adaptive Approximation

Recently, A. Cohen, R. A. DeVore, P. Petrushev, and H. Xu investigated nonlinear approximation in the space BV (R ² ). They modified the classical adaptive algorithm to solve related extremal problems. In this paper, we further study the modified adaptive approximation and obtain results on some extremal problems related to the spaces V _σ,p ^r (R ^d ) of functions of ``Bounded Variation" and Besov spaces B ^α (R ^d ). November 23, 1998. Date revised: June 25, 1999. Date accepted: September 13, 1999.     

Recurrence relations for the coefficients of the Fourier series expansions with respect to q-classical orthogonal polynomials

We propose an algorithm to construct recurrence relations for the coefficients of the Fourier series expansions with respect to the q-classical orthogonal polynomials p_k(x;q). Examples dealing with inversion problems, connection between any two sequences of q-classical polynomials, linearization of ϑ_m(x) p_n(x;q), where ϑ_m(x) is x^mor (x;q)_m, and the expansion of the Hahn-Exton q-Bessel function in the little q-Jacobi polynomials are discussed in detail. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Divergence of Polynomial Interpolation in the Complex Plane

We extend the results in [1] and [2] from the divergence of Hermite—Fejér interpolation in the complex plane to the divergence of arbitrary polynomial interpolation in the complex plane. In particular, we prove the following theorem: Let \D _n =-1≤ t ₁ ⁽ⁿ⁾ <⋅s<t _n ⁽ⁿ⁾ <1 . Let \v _k ⁽ⁿ⁾ be polynomials of arbitrary degree such that \v _k ⁽ⁿ⁾ (t _j ⁽ⁿ⁾ )=\d _kj . Then the Lebesgue function tends to infinity at every complex neighborhood of some point in [-1,1] . March 23, 2000. Date revised: September 28, 2000. Date accepted: October 10, 2000.     

Restricted partition functions as Bernoulli and Eulerian polynomials of higher order

Explicit expressions for restricted partition function W(s,d^m) and its quasiperiodic components W_j(s,d^m) (called Sylvester waves) for a set of positive integers d^m = {d₁, d₂, ..., d_m} are derived. The formulas are represented in a form of a finite sum over Bernoulli and Eulerian polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is established. Application to counting algebraically independent homogeneous polynomial invariants of finite groups is discussed. 2000 Mathematics Subject Classification Primary—11P81; Secondary—11B68, 11B37 The research was supported in part (LGF) by the Gileadi Fellowship program of the Ministry of Absorption of the State of Israel.     

A Pair of Explicitly Solvable Singular Stochastic Control Problems

We consider a general model of singular stochastic control with infinite time horizon and we prove a ``verification theorem' under the assumption that the Hamilton—Jacobi—Bellman (HJB) equation has a C <sup>2</sup> solution. In the one-dimensional case, under the assumption that the HJB equation has a solution in W <sub>loc</sub> <sup>2,p</sup>(R) with , we prove a very general ``verification theorem' by employing the generalized Meyer—Ito change of variables formula with local times. In what follows, we consider two special cases which we explicitly solve. These are the formal equivalent of the one-dimensional infinite time horizon LQG problem and a simple example with radial symmetry in an arbitrary Euclidean space. The value function of either of these problems is C ² and is expressed in terms of special functions, and, in particular, the confluent hypergeometric function and the modified Bessel function of the first kind, respectively. Accepted 21 February 1997     

The connection between self‐associated two‐dimensional vector functionals and third degree forms

We deal with the 2‐orthogonal, 2‐symmetric self‐associated sequence (2‐orthogonal Tchebychev polynomials) and its cubic components. We prove that all the forms (linear functionals) arising are third degree forms. Therefore, an introduction to third degree forms is provided. We look for the connection between these components which are 2‐orthogonal with respect to the functional vector ^t(w₀{^μ},w₁ ^μ) and orthogonal sequences with respect to w₀ ^μ, μ=0,1,2. Associated forms w₀ ^μ)¹) and their inverse w₀ ^μ)^-1 are also studied through the symmetrized w₀}₀ ^μ, μ=0,1,2. Further, we give integral representations for some of these forms. This revised version was published online in June 2006 with corrections to the Cover Date.     

Quadrature formula and zeros of para-orthogonal polynomials on the unit circle

Given a probability measure μ on the unit circle T, we study para-orthogonal polynomials B_n(^.,w) (with fixed w ∈ T) and their zeros which are known to lie on the unit circle. We focus on the properties of zeros akin to the well known properties of zeros of orthogonal polynomials on the real line, such as alternation, separation and asymptotic distribution. We also estimate the distance between the consecutive zeros and examine the property of the support of μ to attract zeros of para-orthogonal polynomials. This revised version was published online in June 2006 with corrections to the Cover Date.     

Polynomial Interpolation on the Unit Sphere and on the Unit Ball

The problem of interpolation on the unit sphere S ^d by spherical polynomials of degree at most n is shown to be related to the interpolation on the unit ball B ^d by polynomials of degree n. As a consequence several explicit sets of points on S ^d are given for which the interpolation by spherical polynomials has a unique solution. We also discuss interpolation on the unit disc of R ² for which points are located on the circles and each circle has an even number of points. The problem is shown to be related to interpolation on the triangle in a natural way.     

Shohat-Favard and Chebyshev’s methods in d-orthogonality

This paper is concerned with the Shohat-Favard, Chebyshev and Modified Chebyshev methods for d-orthogonal polynomial sequences d∈ℕ. Shohat-Favard’s method is presented from the concept of dual sequence of a sequence of polynomials. We deduce the recurrence relations for the Chebyshev and the Modified Chebyshev methods for every d∈ℕ. The three methods are implemented in the Mathematica programming language. We show several formal and numerical tests realized with the software developed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Approximation of infinite matrices by matricial Haar polynomials

The main goal of this paper is to extend the approximation theorem of contiuous functions by Haar polynomials (see Theorem A) to infinite matrices (see Theorem C). The extension to the matricial framework will be based on the one hand on the remark that periodic functions which belong toL ^∞ (T) may be one-to-one identified with Toeplitz matrices fromB(l ₂) (see Theorem 0) and on the other hand on some notions given in the paper. We mention for instance:ms—a unital commutative subalgebra ofl ^∞,C(l ₂) the matricial analogue of the space of all continuous periodic functionsC(T), the matricial Haar polynomials, etc. In Section 1 we present some results concerning the spacems—a concept important for this generalization, the proof of the main theorem being given in the second section. Partially supported by EUROMMAT ICA1-CT-2000-70022. Partially supported by V-Stabi-RUM/1022123. Partially supported by EUROMMAT ICA1-CT-2000-70022 and V-Stabi-RUM/1022123.     

Continuous and Discrete Tight Frames of Orthogonal Polynomials for a Radially Symmetric Weight

This paper considers tight frame decompositions of the Hilbert space ℘ _n of orthogonal polynomials of degree n for a radially symmetric weight on ℝ ^d , e.g., the multivariate Gegenbauer and Hermite polynomials. We explicitly construct a single zonal polynomial p∈℘ _n with the property that each f∈℘ _n can be reconstructed as a sum of its projections onto the orbit of p under SO(d) (symmetries of the weight), and hence of its projections onto the zonal polynomials p _ξ obtained from p by moving its pole to ξ∈S:={ξ∈ℝ ^d :|ξ|=1}. Furthermore, discrete versions of these integral decompositions also hold where SO(d) is replaced by a suitable finite subgroup, and S by a suitable finite subset. One consequence of our decomposition is a simple closed form for the reproducing kernel for ℘ _n .