Korovkin tests, approximation, and ergodic theory

Abstract

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-valued function f, and, for any n, we suppose that A n (.) is a linear and positive 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 minimizer X 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 the theory of the preconditioning associated to cg-like algorithms.