Korovkin tests, approximation, and ergodic theory

We consider sequences of $s\cdot k(n)\times t\cdot k(n)$ matrices $\{A_n(f)\}$ with a block structure spectrally distributed as an $L_1$ $p$-variate $s\times t$ matrix-valued function $f$, and, for any $n$, we suppose that $A_n(\cdot)$ is a linear and positive operator. For every fixed $n$ we approximate the matrix $A_n(f)$ in a suitable linear space $\mathcal{M}_n$ of $s\cdot k(n)\times t\cdot k(n)$ matrices by minimizing the Frobenius norm of $A_n(f)-X_n$ when $X_n$ ranges over $\mathcal{M}_n$. The minimizer $\hat{X}_n$ is denoted by $\mathcal{P}_{k(n)}(A_n(f))$. 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 $\{\mathcal{P}_{k(n)}(A_n(f))\}$ is distributed as $f$,

2.

the sequence $\{A_n(f)-\mathcal{P}_{k(n)}(A_n(f))\}$ 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.