Note on Error Bound for Trace Approximation of Products of Toeplitz Matrices

TL;DR

This paper revisits the error bounds for trace approximations of products of Toeplitz matrices generated by functions with singularities, providing an alternative proof to a previously established theorem.

Contribution

It offers a new proof for the error order of trace approximation of Toeplitz matrix products, correcting previous inaccuracies.

Findings

Confirmed the error order for integral limit approximations

Provided an alternative proof to a key theorem

Clarified the conditions under which the approximation holds

Abstract

We investigate error orders for integral limit approximations to traces of products of Toeplitz matrices generated by integrable functions on $[-\pi,\pi]$ having some singularities at the origin. Even though a sharp error order of the above approximation is derived in Theorem~2 of \cite{Lieberman-Phillips-2004}, its proof contains an inaccuracy as pointed out by \cite{Ginovyan-Sahakyan-2013}. In the present paper, we reinvestigate the claim given in Theorem~2 of \cite{Lieberman-Phillips-2004} and give an alternative proof of their claim.