Generalized Bounded Variation and Inserting point masses

TL;DR

This paper derives formulas for Verblunsky coefficients when inserting point masses into measures on the unit circle, introduces a new concept of generalized bounded variation, and analyzes the asymptotic behavior of orthogonal polynomials.

Contribution

It provides a simple formula for Verblunsky coefficients after adding point masses and introduces the notion of generalized bounded variation for these measures.

Findings

Verblunsky coefficients have a specific asymptotic form after inserting point masses.

The measure becomes of (m+1)-generalized bounded variation.

The reversed orthogonal polynomials converge to the inverse of the Szegő function away from pure points.

Abstract

Let $d\mu$ be a probability measure on the unit circle and $d\nu$ be the measure formed by adding a pure point to $d\mu$. We give a simple formula for the Verblunsky coefficients of $d\nu$ based on a result of Simon. Then we consider $d\mu_0$, a probability measure on the unit circle with $\ell^2$ Verblunsky coefficients $(\alpha_n (d\mu_0))_{n=0}^{\infty}$ of bounded variation. We insert $m$ pure points to $d\mu$, rescale, and form the probability measure $d\mu_m$. We use the formula above to prove that the Verblunsky coefficients of $d\mu_m$ are in the form $\alpha_n(d\mu_0) + \sum_{j=1}^m \frac{\ol{z_j}^{n} c_j}{n} + E_n$, where the $c_j$'s are constants of norm 1 independent of the weights of the pure points and independent of $n$; the error term $E_n$ is in the order of $o(1/n)$. Furthermore, we prove that $d\mu_m$ is of $(m+1)$-generalized bounded variation - a notion that we shall introduce in the paper. Then we use this fact to prove that $\lim_{n \to \infty} \vp_n^*(z, d\mu_m)$ is continuous and is equal to $D(z, d\mu_m)^{-1}$ away from the pure points.