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

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