A Korovkin-Based Approximation of Multilevel Toeplitz Matrices (With Rectangular Unstructured Blocks) via Multilevel Trigonometric Matrix Spaces

In this paper we are concerned with the approximation of multilevel Toeplitz matrices generated by multivariate rectangular matrix-valued continuous (also L2) functions $f: I^p\rightarrow {\cal C}^{s\times t}$, with $I=[-\pi,\pi]$. We first define a class ${\cal A}_T$ of multilevel trigonometric matrix spaces with unstructured $s\times t$ blocks. Second, we introduce a (multi) sequence of linear approximation operators ${\cal P}_n(X)$, $n=(n_1,n_2,\ldots,n_p)$ acting on complex matrices of dimensions $(n_1\cdots n_p s)\times (n_1\cdots n_p t)$, $n_j\in {\cal N}_+$ and giving the minimizer of the distance of X from a fixed representative of ${\cal A}_T$ with regard to the Frobenius norm. Then, by stating and proving new versions of the Korovkin theorem for $f$ ranging in the Banach space $({\bf C}(I^p,{\cal C}^{s\times t}), \|\cdot\|_\infty)$ and by linking these theorems to the operators ${\cal P}_n$, we prove the ``goodness'' of the proposed approximation in terms of the strong and weak equidistribution of the eigenvalues/singular values of $\{A_n(f)\}$ and $\{{\cal P}_n(A_n(f))\}$. The discussed results provide a very uniform tool for treating the preconditioning problem with applications to a wide variety of structures as analyzed in the twin paper [S. Serra Capizzano, { Calcolo, in press].