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Mathematics of Computation

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Korovkin tests, approximation, and ergodic theory


Author: Stefano Serra Capizzano
Journal: Math. Comp. 69 (2000), 1533-1558
MSC (1991): Primary 65F10, 65D15, 15A60, 47B65, 28Dxx
Published electronically: March 6, 2000
MathSciNet review: 1697646
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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 $L_1$ $p$-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.


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Additional Information

Stefano Serra Capizzano
Affiliation: Dipartimento di Energetica, Via Lombroso 6/17, 50134 Firenze, Italy; Dipartimento di Informatica, Corso Italia 40, 56100 Pisa, Italy
Email: serra@mail.dm.unipi.it

DOI: http://dx.doi.org/10.1090/S0025-5718-00-01217-5
Keywords: Distributions and ergodic theory, Toeplitz matrices, Korovkin theorem, circulants and $\tau$ matrices, discrete transforms
Received by editor(s): February 2, 1998
Received by editor(s) in revised form: November 20, 1998
Published electronically: March 6, 2000
Article copyright: © Copyright 2000 American Mathematical Society