A Korovkin-Based Approximation of MultilevelToeplitz Matrices (With Rectangular Unstructured Blocks) via Multilevel Trigonometric Matrix Spaces
Stefano Serra‐Capizzano · SIAM Journal on Numerical Analysis · 1999
In this paper we are concerned with the approximation of multilevel Toeplitz matrices generated by multivariate rectangular matrix-valued continuous (also L 2 ) 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].