Asymptotic zero distribution of orthogonal polynomials with discontinously varying recurrence coefficients

Abstract

The zero distribution of orthogonal polynomials pn,N, n = 0, 1, ⋯ generated by recurrence coefficients an,N and bn,N depending on a parameter N has been recently considered by Kuijlaars and Van Assche under the assumption that an,N and bn,N behave like a(n/N) and b(n/N), respectively, where a(·) and b(·) are continuous functions. Here, we extend this result by allowing a(·) and b(·) to be measurable functions so that the presence of possible jumps is included. The novelty is also in the sense of the mathematical tools since, instead of applying complex analysis arguments, we use recently developed results from asymptotic matrix theory due to Tyrtyshnikov, Serra Capizzano, and Tilli. © 2001 Academic Press.