Asymptotic Analysis of Orthogonal Polynomials via the Transfer Matrix Approach

TL;DR

This paper introduces a transfer matrix method for deriving asymptotic formulas of orthogonal polynomials with bounded variation coefficients, and applies it to analyze measure perturbations outside the support.

Contribution

It develops a novel transfer matrix approach using hyperbolicity and Kooman's Theorem to analyze orthogonal polynomials with bounded variation coefficients.

Findings

Derived asymptotic formulas for orthogonal polynomials with bounded variation coefficients.

Proved stability of recurrence coefficients under point mass perturbations outside the support.

Applied the method to solve the point mass problem on the real line.

Abstract

In this paper, we present a new method via the transfer matrix approach to obtain asymptotic formulae of orthogonal polynomials with asymptotically identical coefficients of bounded variation. We make use of the hyperbolicity of the recurrence matrices and employ Kooman's Theorem to diagonalize them simultaneously. The method introduced in this paper allows one to consider products of matrices such that entries of consecutive matrices are of bounded variation. Finally, we apply the asymptotic formulae obtained to solve the point mass problem on the real line when the measure is essentially supported on an interval. We prove that if a point mass is added to such a measure outside its essential support, then the perturbed recurrence coefficients will also be asymptotically identical with the same limit and of bounded variation.