A First Szeg\H{o}'s Limit Theorem for a class of non-Toeplitz matrices

TL;DR

This paper extends Szeg\

Contribution

It establishes a first Szeg\

Findings

Eigenvalue distributions for new matrix classes

Extension of Szeg\

Application to Riemann integrable functions

Abstract

We compute the limiting statistical distribution of the eigenvalues of sequences of matrices whose entries satisfy what we call a vanishing mean variation condition and are $\mu$-distributed for some probability measure. As an application of our results, we extend the well-known class of Kac-Murdock-Szeg\H{o} generalized Toeplitz matrices to sequences of matrices whose diagonal entries are modeled by Riemann integrable functions.