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