Korovkin tests, approximation, and ergodic theory

We consider sequences of s · k(n)× t · k(n) matrices {An(f)} with a block structure spectrally distributed as an L1 p-variate s× t matrix-valued function f , and, for any n, we suppose that An(·) is a linear and positive operator. For every fixed n we approximate the matrix An(f) in a suitable linear space Mn of s · k(n) × t · k(n) matrices by minimizing the Frobenius norm of An(f)−Xn when Xn ranges overMn. The minimizer X̂n is denoted by Pk(n)(An(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 {Pk(n)(An(f))} is distributed as f , 2. the sequence {An(f)−Pk(n)(An(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.