Korovkin tests, approximation, and ergodic theory

We consider sequences of s (.) k(n) x t (.) k(n) matrices {A(n)(f)} with a block structure spectrally distributed as an L-1 p-variate s x t matrix-val ued function f, and, for any n, we suppose that A(n)((.)) is a linear and p ositive operator. For every fixed n we approximate the matrix A(n)(f) in a suitable linear space M-n of s (.) k(n) x t (.) k(n) matrices by minimizing the Frobenius norm of A(n)(f) - X-n when X-n ranges over M-n. The minimize r (X) over cap(n) is denoted by P-k(n)(A(n)(f)). We show that only a simple Korovkin test over a finite number of polynomial test functions has to be performed in order to prove the following general facts: 1. the sequence {P-k(n)(A(n)(f))} is distributed as f, 2. the sequence {A(n)(f) - P-k(n) (A(n)(f))} is distributed as the constant function 0 (i.e, is spectrally clustered at zero). The first result is an ergodic one which can be used for solving numerical approximation theory problems. The second has a natural interpretation in t he theory of the preconditioning associated to cg-like algorithms.