Entropy, approximation and the geometry of the cube

by    C. Richter

Preprint series: 99-42, Reports on Analysis

41A30 Approximation by other special function classes
41A17 Inequalities in approximation (Bernstein, Jackson, Nikolskiui type inequalities)
52C17 Packing and covering in $n$ dimensions, See also {05B40,

Abstract: Controllable coverings and controllable partitions of compact
metric spaces are the main tool for two approximation theories
for real functions being closely related to the entropy numbers
of the space. We consider the geometric structure of these
arrangements on the $m$-dimensional cube. The optimality of two
estimates of Jackson type and an inverse Jackson type estimate
are proved.

Keywords: entropy numbers, lattices, cube, open coverings and partitions of unity, partitions and step functions, Jackson type and Bernstein type inequalities

Upload: 1999-07-26

Update: 1999-07-27

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