Best approximation by linear combinations of characteristic functions of half-spaces

Authors:
Paul C. Kainen;Věra Kůrková;Andrew Vogt
Affiliations:
Department of Mathematics, Georgetown University, Box 571233, 37th and O Streets N. W., Washington, DC;Institute of Computer Science, Academy of Sciences of the Czech Republic, P.O. Box 5, 182 07 Prague 8, Czech Republic;Department of Mathematics, Georgetown University, Box 571233, 37th and O Streets N. W., Washington, DC
Venue:
Journal of Approximation Theory
Year:
2003

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Abstract

It is shown that for any positive integer n and any function f in Lp([0,1]d) with p ∈ [1,∞) there exist n half-spaces such that f has a best approximation by a linear combination of their characteristic functions. Further, any sequence of linear combinations of n half-space characteristic functions converging in distance to the best approximation distance has a subsequence converging to a best approximation, i.e., the set of such n-fold linear combinations is an approximatively compact set.