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Korovkin tests, approximation, and ergodic theory

Authors:	Stefano Serra Capizzano

Institution:	Dipartimento di Energetica, Via Lombroso 6/17, 50134 Firenze, Italy; Dipartimento di Informatica, Corso Italia 40, 56100 Pisa, Italy

Abstract:	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 -variate $s\times t$ matrix-valued function , and, for any , we suppose that $A_n(\cdot)$ is a linear and positive operator. For every fixed we approximate the matrix 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 when 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 , 2. the sequence $\{A_n(f)-\mathcal{P}_{k(n)}(A_n(f))\}$ is distributed as the constant function (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.

Keywords:	Distributions and ergodic theory Toeplitz matrices Korovkin theorem circulants and $\tau$ matrices discrete transforms

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