Spectral asymptotics for Kac-Murdock-Szeg\H{o} matrices

TL;DR

This paper surveys the extension of Szegő's limit theorems from Toeplitz matrices to Kac-Murdock-Szegő matrices, which have diagonals modeled by functions, clarifying and expanding existing results.

Contribution

It extends Szegő's limit theorems to Kac-Murdock-Szegő matrices, providing new insights and resolving contradictions in the literature.

Findings

Extended the first and strong limit theorems to KMS matrices

Clarified existing results and resolved contradictions

Provided a comprehensive survey of the topic

Abstract

Szeg\H{o}'s First Limit Theorem provides the limiting statistical distribution (LSD) of the eigenvalues of large Toeplitz matrices. Szeg\H{o}'s Second (or Strong) Limit Theorem for Toeplitz matrices gives a second order correction to the First Limit Theorem, and allows one to calculate asymptotics for the determinants of large Toeplitz matrices. In this paper we survey results extending the first and strong limit theorems to Kac-Murdock-Szeg\H{o} (KMS) matrices. These are matrices whose entries along the diagonals are not necessarily constants, but modeled by functions. We clarify and extend some existing results, and explain some apparently contradictory results in the literature.