A Formula for Inserting Point Masses

TL;DR

This paper derives a formula for updating Verblunsky coefficients when a point mass is added to a probability measure on the unit circle, facilitating analysis of measure perturbations.

Contribution

It provides a new explicit formula for Verblunsky coefficients after adding a point mass, extending previous theoretical results.

Findings

Derived a formula for Verblunsky coefficients after measure perturbation

Extended Simon’s results to include point mass insertions

Facilitated analysis of measure modifications on the unit circle

Abstract

Let mu be a probability measure on the unit circle and nu be the measure formed by adding a pure point to mu. We give a formula for the Verblunsky coefficients of the perturbed measure, based on a result of Simon.