Multilevel symmetrized Toeplitz structures and spectral distribution results for the related 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 $\{Y_n{T}_n[f]{\}}_n$ 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 $T_{\mathbf{n}}[f]$ generated by $f\in L^1([-\pi ,\pi {]}^k)$, $\mathbf{n}$ being a multi-index identifying the matrix-size, and we prove spectral and singular value distribution results for the matrix-sequence $\{Y_{\mathbf{n}}{T}_{\mathbf{n}}[f]{\}}_{\mathbf{n}}$ with $Y_{\mathbf{n}}$ being the corresponding tensorization of the anti-identity matrix.