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Zbl 0993.41009
Marano Calzolari, M.; Quesada Teruel, J.M.; Navas Ureña, J.
Rate of convergence of the Pólya algorithm on convex sets. (Spanish)
[A] Cruz Lopez de Silanes, Maria (ed.) et al., Actes des 6èmes journées Zaragoza-Pau de mathématiques appliquées et de statistiques. Pau: Publications de Université de Pau et Pays de l'Adour, PUP. 417-424 (2001). ISBN 2-908930-75-7

Let $l^n_p$ be the space $\bbfR^n$ with the $p$-norm, $1\le p \le\infty$. For $K$ being a convex and closed subset of $\bbfR^n$ and $h\in \bbfR^n\setminus K$, let denote by $h_p$ a best approximation to $h$ in $l^n_p$. The purpose of this article is to study the order of uniform convergence of $h_p$ to $h^*_\infty$ as $p\to\infty$. The limit $h^*_\infty$ is called strict approximant (of $h)$, andthe procedure of constructing $h^*_\infty$ by means of $\{h^*\}$ is known as Pólya algorithm. The following definition is a modification of the one of Newman and Shapiro (1962) for continuous functions. Definition: Let $h^*$ be a best uniform approximant of $0\in\bbfR^n$. The vector $h^*$ is strongly unique if there exists $\gamma>0$ such that $\|f-h^* \|_\infty \le\gamma (\|f\|_\infty- \|h^*\|_\infty)$, $\forall f\in K$. A characterization of the above strong unicity condition for $h^*$ is given in terms of the existence of a hyperplane $\pi:\sum a_ix_i=1$ such that 1) $0\le a_i\le 1,2) \sum a_i=1$, and 3) $\sum a_ih_i\ge 1$, $\forall h\in K$. As for the order of convergence the following result is proved. Theorem: Let $h_2\ne h^*_\infty$ and suppose that there exists a hyperplane $\pi$ which satisfies conditions $1-3$. Then $\|h_p-h^*_\infty \|\asymp 1/p$. An example shows that if any hyperplane $\pi$ of the above mentioned type cannot be found, the rate of convergence can be arbitrarily slow.
[Jesus Illán González (Vigo)]

MSC 2000:: *41A25 Degree of approximation, etc.

Keywords: strict approximant

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