Szeg˝ o's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [−1, 1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent polynomials Λ, and leads to a new orthogonality structure in the module Λ × Λ. This structure can be interpreted in terms of a 2 × 2 matrix measure on [−1, 1], a...

We consider Koornwinder’s method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder’s construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with ...

Using orthogonal polynomials, we give a different approach to some recent results on tridi-agonal matrices. 1. Introduction and preliminaries. Most of the results on tridiagonal matrices can be seen in the context of orthogonal polynomials. Here, we will see two examples. The aim of this note is to give simplified proofs of some recent results on tridiagonal matrices , using arguments from the ...

The LL anczos method for solving systems of linear equations is based on formal orthogonal polynomi-als. Its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials. A division by zero can occur in such recurrence relations, thus causing a breakdown in the algorithm...