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Journal of Function Spaces and Applications
Volume 2012, Article ID 435945, 38 pages
http://dx.doi.org/10.1155/2012/435945
Review Article

Canonical Sets of Best -Approximation

1Department of Mathematics and Statistics, Concordia University, Montreal, QC, Canada H3G 1M8
2Numerical Modeling Department, Leibniz Institute for Crystal Growth, Max-Born-Street 2, D-12489 Berlin, Germany
3Department of Numerical Methods and Algorithms, Faculty of Mathematics and Informatics, Sofia University, 5 James Bourchier blvd., 1164 Sofia, Bulgaria

Received 23 February 2012; Accepted 7 May 2012

Academic Editor: Henryk Hudzik

Copyright © 2012 Dimiter Dryanov and Petar Petrov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best -approximation with emphasis on multivariate interpolation and best -approximation by blending functions. The best -approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate -approximation by sums of univariate functions. Explicit constructions of best one-sided -approximants give rise to well-known and new inequalities.