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Rate of Convergence of the Discrete Polya Algorithm from Convex Sets. A Particular Case  
 
  Authors: M. Marano, J. Navas, J.M. Quesada,  
  Keywords: Best uniform approximation, Rate of convergence, Polya Algorithm, Strong uniqueness.  
  Date Received: 04/12/01  
  Date Accepted: 28/05/02  
  Subject Codes:

26D15

  
  Editors: Alexander G. Babenko,  
 
  Abstract:

In this work we deal with best approximation in $ ell_{p}^n$, $ 1<p leq infty$, $ ngeq 2$. For $ 1<p<infty$, let $ h_{p}$ denote the best $ ell_{p}^n$-approximation to $ fin mathbb{R}^n$ from a closed, convex subset $ K$ of $ mathbb{R}^n$, $ f not in K$, and let $ h^*$ be a best uniform approximation to $ f$ from $ K$. In case that $ h^*-f$ $ =(rho_1,rho_2,cdots,rho_n)$, $ vertrho_jvert=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 $ K$ from the $ ell^n_{infty}$-ball $ {xinmathbb{R}^n:Vert x-fVertleqrho}$ at $ h^*-f$.;


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