# Spectral Distribution in the Eigenvalues Sequence of Products of g-Toeplitz Structures

## TL;DR

This paper characterizes the spectral distribution of products of g-Toeplitz matrices, extending classical Toeplitz results, and shows that for g≥2, these products tend to cluster around zero, supported by numerical examples.

## Contribution

It extends spectral distribution analysis from classical Toeplitz matrices to g-Toeplitz matrices and characterizes eigenvalue behavior for their products.

## Key findings

- Sequences with g≥2 cluster to zero in spectral distribution.
- The spectral distribution is fully characterized under certain assumptions.
- Numerical examples confirm the theoretical results.

## Abstract

Starting from the definition of an $n\times n$ $g$-Toeplitz matrix, $T_{n,g}(u)=\left[\widehat{u}_{r-gs}\right]_{r,s=0}^{n-1},$ where $g$ is a given nonnegative parameter, $\{\widehat{u}_{k}\}$ is the sequence of Fourier coefficients of the Lebesgue integrable function $u$ defined over the domain $\mathbb{T}=(-\pi,\pi]$, we consider the product of $g$-Toeplitz sequences of matrices, $\{T_{n,g}(f_{1})T_{n,g}(f_{2})\},$ which extends the product of Toeplitz structures, $\{T_{n}(f_{1})T_{n}(f_{2})\},$ in the case where the symbols $f_{1},f_{2}\in L^{\infty}(\mathbb{T}).$ Under suitable assumptions, the spectral distribution in the eigenvalues sequence is completely characterized for the products of $g$-Toeplitz structures. Specifically, for $g\geq2$ our result shows that the sequences $\{T_{n,g}(f_{1})T_{n,g}(f_{2})\}$ are clustered to zero. This extends the well-known result, which concerns the classical case (that is, $g=1$) of products of Toeplitz matrices. Finally, a large set of numerical examples confirming the theoretic analysis is presented and discussed.