During the past dozen years, a major focus of my research has been the spectral theory of orthogonal polynomials—both orthogonal polynomials on the real line (OPRL) and on the unit circle (OPUC). There has been a flowering of the subject in part because of a cross-fertilization of two communities of researchers. I will discuss some aspects of this subject here; for a lot more, see my recent boo...

A scheme for constructing orthogonal systems of bivariate polynomials in the Bernstein–Bézier form over triangular domains is formulated. The orthogonal basis functions have a hierarchical ordering by degree, facilitating computation of least-squares approximations of increasing degree (with permanence of coefficients) until the approximation error is subdued below a prescribed tolerance. The o...

Abstract We study the relation between certain non-degenerate lower Hessenberg infinite matrices $${\mathcal {G}}$$ G and existence of sequences orthogonal polynomials with respect to Sobolev inner products. In other words, we extend well-known Favard theorem for orthogonality. characterize structure matrix a...

We present a generalization of multiple orthogonal polynomials of type I and type II, which we call multiple orthogonal polynomials of mixed type. Some basic properties are formulated, and a Riemann-Hilbert problem for the multiple orthogonal polynomials of mixed type is given. We derive a Christoffel-Darboux formula for these polynomials using the solution of the Riemann-Hilbert problem. The m...

We survey the analytic theory of matrix orthogonal polynomials. MSC: 42C05, 47B36, 30C10 keywords: orthogonal polynomials, matrix-valued measures, block Jacobi matrices, block CMV matrices

Let V be a set of points in R. Define a linear functional L on the space of polynomials, Lf = ∑ x∈V f(x)ρ(x), where ρ is a nonzero function on V . The structure of discrete orthogonal polynomials of several variables with respect to the bilinear form 〈f, g〉 = L(fg) is studied. For a given V , the subspace of polynomials that will generate orthogonal polynomials on V is identified. One result sh...

In the above two papers Walter Gautschi, jointly with Henry J. Landau and Gradimir V. Milovanović, investigate polynomials that are orthogonal with respect to a non-Hermitian inner product defined on the upper half of the unit circle in the complex plane. For special choices of the weight function, these polynomials are related to Jacobi polynomials. Their recurrence relation and properties of ...

This paper defines the multivariate Krawtchouk polynomials, orthogonal on the multinomial distribution, and summarizes their properties as a review. The multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets of functions defined on each of N multinomial trials. The dual multivariate Krawtchouk polynomials, which also have a polynomial structure, are seen to occur natural...

For little q-Jacobi polynomials, q-Hahn polynomials and big q-Jacobi polynomials we give particular q-hypergeometric series representations in which the termwise q = 0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as decompositions into a lower triangular matrix times upper triangular matrix. We develop a general theory of such decompositions rela...

The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function Wγ(x) = x γ1 1 · · ·x γd d (1− |x|)d+1 when all γi > −1 and they are eigenfunctions of a second order partial differential operator Lγ . The singular cases that some, or all, γ1, . . . , γd+1 are −1 are studied in this paper. Firstly a complete basis of polynomials that are eigenfunctions of Lγ ...