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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