Canonical Sets of BestL1-Approximation

Abstract

In mathematics, the termapproximationusually 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 calledcanonical sets of best approximation. The present paper summarizes results on canonical sets of bestL1-approximation with emphasis on multivariate interpolation and bestL1-approximation by blending functions. The bestL1-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 multivariateL1-approximation by sums of univariate functions. Explicit constructions of best one-sidedL1-approximants give rise to well-known and new inequalities.