Title

The Polya Algorithm on Cylindrical Sets

Document Type

Article

Publication Date

6-1988

Publication Source

Journal of Approximation Theory

Volume

53

Issue

3

Inclusive pages

335-349

DOI

10.1016/0021-9045(88)90027-5

Abstract

We define the property “E-cylindrical,” which relates to a subset of m certain directed cylinders. We investigate some of the consequences of this definition, showing, for example, that polyhedral convex sets and smooth, rotund convex bodies are E-cylindrical. Suppose X is a finite set, F is the set of all real-valued functions on X, fϵF, andKF is closed, convex, and E-cylindrical. For 1 < p < ∞, let fp be the best lp-approximation to f by elements of K. We show that limp → ∞fp exists. We give an example to show that {fp} may fail to converge if X is countably infinite. We discuss the relationship between discrete (lp) and continuous (Lp) approximation.

Disciplines

Mathematics

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