A mini-max problem for self-adjoint Toeplitz matrices

TL;DR

This paper investigates a minimization problem for self-adjoint Toeplitz matrices, identifying unique solutions and exploring related maximum problems, with some numerical insights into open questions.

Contribution

It introduces a unique minimal $L^{ abla}$-norm inducers for self-adjoint Toeplitz matrices and discusses the largely open associated maximum problem.

Findings

Unique minimal $L^{ abla}$-norm inducers identified

Description of the minimal inducers provided

Numerical evidence offered for the maximum problem

Abstract

We study a minimum problem and associated maximum problem for finite, complex, self-adjoint Toeplitz matrices. If $A$ is such a matrix, of size $(N+1)$-by-$(N+1)$, we identify $A$ with the operator it represents on $P_N$, the space of complex polynomials of degrees at most $N$, with the usual Hilbert space structure it inherits as a subspace of $L^2$ of the unit circle. The operator $A$ is the compression to $P_N$ of the multiplication operator on $L^2$ induced by any function in $L^{\infty}$ whose Fourier coefficients of indices between $-N$ and $N$ match the matrix entries of $A$. Our minimum problem is to minimize the $L^{\infty}$ norm of such inducers. We show there is a unique one of minimum norm, and we describe it. The associated maximum problem asks for the maximum of the ratio of the preceding minimum to the operator norm. That problem remains largely open. We present some suggestive numerical evidence.