On the decay of elements of inverse triangular Toeplitz matrix

TL;DR

This paper investigates the decay properties of inverse triangular Toeplitz matrices with slow decay elements, establishing conditions under which the inverse and fundamental matrices decay to zero and providing quantitative decay descriptions.

Contribution

It introduces new decay results for inverse matrices under monotonicity and log-convexity conditions, extending classical decay theories.

Findings

Inverse and fundamental matrices decay to zero under certain conditions

Quantitative decay descriptions using p-norms are provided

Inverse matrices with slow log-convex decay exhibit fast decay and boundedness

Abstract

We consider half-infinite triangular Toeplitz matrices with slow decay of the elements and prove under a monotonicity condition that elements of the inverse matrix, as well as elements of the fundamental matrix, decay to zero. We also provide a quantitative description of the decay of the fundamental matrix in terms of p-norms. Finally, we prove that for matrices with slow log-convex decay the inverse matrix has fast decay, i.e. is bounded. The results are compared with the classical results of Jaffard and Veccio and illustrated by numerical example.