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

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Paola Ferrari
Isabella Furci
Stefano Serra-Capizzano

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, the authors 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 they 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.

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