A chain of controllable partitions of unity on the cube and the approximation of Hölder continuous functions

by    C. Richter

Preprint series: 99-43, Reports on Analysis

The paper is published: Illinois J. Math. 43 (1999), 159-191

MSC:
41A30 Approximation by other special function classes
41A17 Inequalities in approximation (Bernstein, Jackson, Nikolskiui type inequalities)
47A58 Operator approximation theory
41A25 Rate of convergence, degree of approximation
41A36 Approximation by positive operators

Abstract: Controllable partitions of unity in $C(X)$ are partitions of
unity whose supports fulfil a uniformity condition depending
on the entropy numbers of the compact metric space $X$. We
construct a chain of such partitions in $C([0,2]^m)$ such that
the span of any partition is a proper subspace of the span of
the following one. This chain gives rise to approximation
quantities for functions from $C([0,2]^m)$ as well as for
$C([0,2]^m)$-valued operators and to corresponding Jackson type
inequalities. Inverse inequalities are presented for Hölder
continuous functions and operators.

Keywords: cube, entropy number, chain of partitions of unity, Lebesgue singular function, approximation of functions and operators, inequalities of Jackson and Berstein type

Upload: 1999-07-26

Update: 1999-07-27


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