Rate of Convergence of the Discrete Polya Algorithm from Convex Sets. A Particular Case

Volume 3, Issue 4, 2002

Article 50

RATE OF CONVERGENCE OF THE DISCRETE POLYA ALGORITHM FROM CONVEX SETS. A PARTICULAR CASE

DEPARTAMENTO DE MATEMÁTICAS
UNIVERSIDAD DE JA\'EN
PARAJE LAS LAGUNILLAS
CAMPUS UNIVERSITARIO
23071 JAÉN, SPAIN.
E-Mail: mmarano@ujaen.es

E-Mail: jnavas@ujaen.es

E-Mail: jquesada@ujaen.es

Received 4 December, 2001; Accepted 28 May, 2002.
Communicated by: A. Babenko

ABSTRACT. In this work we deal with best approximation in $\ell_{p}^n$ , $1 <p \leq \infty$ , $n\geq 2$ . For $1<p<\infty$ , let $h_{p}$ denote the best $\ell_{p}^n$ -approximation to $f\in \mathbb{R}^n$ from a closed, convex subset

of $\mathbb{R}^n$ , $f \not \in K$ , and let

be a best uniform approximation to

from

. In case that

$=(\rho_1,\rho_2,\cdots,\rho_n)$ , $\vert\rho_j\vert=\rho\;$ for $j=1,2,\cdots,n$ , we show that the behavior of $\Vert h_{p}-h^*\Vert$ as $p \to \infty$ depends on a property of separation of the set

from the $\ell^n_{\infty}$ -ball $\{x\in\mathbb{R}^n:\Vert x-f\Vert\leq\rho\}$ at

Key words:
Best uniform approximation, Rate of convergence, Polya Algorithm, Strong uniqueness.

2000 Mathematics Subject Classification:
26D15.

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