by **
C. Richter**

**Preprint series:**
99-41, Reports on Analysis

**The paper is published:**
Math. Nachr. 198 (1999), 179-188

**MSC:**- 41A30 Approximation by other special function classes
- 41A17 Inequalities in approximation (Bernstein, Jackson, Nikolskiui type inequalities)
- 41A25 Rate of convergence, degree of approximation
- 47A58 Operator approximation theory
- 52C17 Packing and covering in $n$ dimensions, See also {05B40,

**Abstract:** The paper deals with the approximation of bounded real functions

$f$ on a compact metric space $(X,d)$ by so-called controllable

step functions in continuation of a common paper with I. Stephani.

These step functions are connected with controllable coverings,

that are finite coverings of compact metric spaces by subsets

whose sizes fulfil a uniformity condition depending on the entropy

numbers $\varepsilon_n(X)$ of the space $X$. We show that a strong

form of local finiteness holds for these coverings on compact

metric subspaces of $R^m$ and $S^m$. This leads to a Bernstein

type theorem if the space is of finite convex deformation. In this

case the corresponding approximation numbers $\hat{a}_n(f)$ have

the same asymptotics as $\omega(f,\eps_n(X))$ for $f \in C(X)$.

Finally, the results concerning functions $f \in M(X)$ and

$f \in C(X)$ are transferred to operators with values in $M(X)$

and $C(X)$, respectively.

**Keywords:** *compact metric space, entropy number, approximation by step functions, approximation of operators, inequalities of Jackson and Bernstein type, coefficient of convex deformation*

**Upload:** 1999-07-26

**Update:** 1999-07-27

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