Multilevel symmetrized Toeplitz structures and spectral distribution results for the related matrix-sequences

TL;DR

This paper extends spectral and singular value distribution results to multilevel symmetrized Toeplitz matrices generated by functions in multiple dimensions, broadening understanding of their spectral properties for computational applications.

Contribution

It generalizes previous spectral distribution results from one-level to multilevel Toeplitz matrices with symmetrization, considering functions in multiple dimensions.

Findings

Spectral distribution results for multilevel symmetrized Toeplitz matrices.

Singular value distribution results for the same class of matrices.

Extension of spectral analysis to higher-dimensional matrix sequences.

Abstract

In recent years, motivated by computational purposes, the singular value and spectral features of the symmetrization of Toeplitz matrices generated by a Lebesgue integrable function have been studied. Indeed, under the assumptions that $f$ belongs to $L^1([-\pi,\pi])$ and it has real Fourier coefficients, the spectral and singular value distribution of the matrix-sequence $\{Y_nT_n[f]\}_n$ has been identified, where $n$ is the matrix-size, $Y_n$ is the anti-identity matrix, and $T_n[f]$ is the Toeplitz matrix generated by $f$. In this note, we consider the multilevel Toeplitz matrix $T_{\bf n}[f]$ generated by $f\in L^1([-\pi,\pi]^k)$, $\bf n$ being a multi-index identifying the matrix-size, and we prove spectral and singular value distribution results for the matrix-sequence $\{Y_{\bf n}T_{\bf n}[f]\}_{\bf n}$ with $Y_{\bf n}$ being the corresponding tensorization of the anti-identity matrix.