Entries of the inverses of large positive definite Toeplitz matrices

Albrecht Böttcher

doi:10.1007/s44146-022-00001-6

Abstract

AbstractThis is an expository paper embarking on the asymptotic behavior of the entries of the inverses of positive definite symmetric Toeplitz matrices as the matrix dimension goes to infinity. We consider the behavior of the entries in neighborhoods of the four corners as well as the density of the distribution of the entries over all of the inverse matrix.

Highlights

Gabor Szegő’s two papers [20] of 1920 and 1921 contain the roots of an enormous development that is lasting until the present. This development includes Toeplitz determinants and orthogonal polynomials, and there are many recent works which present these topics in expository style; see, for example [4, 7, 18]
Another aspect of Szegő’s two papers concerns the analysis of positive definite Toeplitz matrices, and as I am not aware of an expository article devoted exclusively to this issue, I decided to write this paper when receiving the honorable invitation to submit a contribution to the 100th anniversary of the journal Acta Scientiarum Mathematicarum, which was founded by Szegő’s contemporaries Alfréd Haar and Frigyes Riesz in 1922
Given a real-valued function a in L1 over the complex unit circle, we denote by ak = ∫− a(ei )e−ik d ∕(2 ) ( k ∈ Z ) its Fourier coefficients and by Tn(a) the n × n Toeplitz matrix

Summary

Introduction

Gabor Szegő’s two papers [20] of 1920 and 1921 contain the roots of an enormous development that is lasting until the present This development includes Toeplitz determinants and orthogonal polynomials, and there are many recent works which present these topics in expository style; see, for example [4, 7, 18]. We confine ourselves to the case where the matrix is real and symmetric This is equivalent to the requirement that the function ↦ a(ei ) is even on (− , ). Throughout this paper we assume that a ∈ L1 is real-valued and not identically zero, that ak = a−k for all k, and that a ≥ 0 a.e. on , so that Tn(a) is (symmetric) positive definite and invertible for all n ≥ 1.

An illuminating example

Convergence of corner entries

Global limiting distribution