Bernstein-Szeg\H{o} measures, Banach algebras, and scattering theory

TL;DR

This paper characterizes the decay rates of Jacobi matrix entries via Bernstein-Szego measures, scattering functions, and Banach algebra techniques, revealing a deep connection between matrix decay and Fourier coefficients.

Contribution

It provides explicit descriptions of Bernstein-Szego measures for perturbed Jacobi matrices and links exponential decay of matrix entries to scattering function Fourier coefficients.

Findings

Decay of Jacobi matrix entries is determined by negative Fourier coefficients of scattering functions.

Explicit formulas for Bernstein-Szego measures related to finite perturbations of Chebyshev matrices.

Characterizations of decay rates of matrix entries for measures in the class M.

Abstract

We give a simple and explicit description of the Bernstein-Szego type measures associated with Jacobi matrices which differ from the Jacobi matrix of the Chebyshev measure in finitely many entries. We also introduce a class of measures M which parametrizes the Jacobi matrices with exponential decay and for each element in M we define a scattering function. Using Banach algebras associated with increasing Beurling weights, we prove that the exponential decay of the coefficients in a Jacobi matrix is completely determined by the decay of the negative Fourier coefficients of the scattering function. Combining this result with the Bernstein-Szego type measures we provide different characterizations of the rate of decay of the entries of the Jacobi matrices for measures in M.