by **
C. Richter**

**Preprint series:**
99-50, Reports on Analysis

**The paper is published:**
Math. Nachr. (to appear)

**MSC:**- 41A30 Approximation by other special function classes
- 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
- 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.), {For properties determined by Fourier coefficients, See 42A16; for those determined by approximation properties, See 41A25, 41A27}
- 26A30 Singular functions, Cantor functions, functions with other special properties

**Abstract:** Controllable step functions on a compact metric space $(X,d)$

are defined on partitions of $X$ into subsets whose sizes fulfil

a particular uniformity condition in terms of entropy numbers.

The paper deals with the class $A(X)$ of all bounded real-valued

functions $f \in M(X)$ which can be approximated uniformly by

controllable step functions. We show that every function

$f \in A(X)$ is a controllable step function itself or possesses

a successive procedure of approximation by a sequence of

controllable step functions on an ascending chain of

controllable partitions. It turns out that $A(X)$ coincides with

$M(X)$ if and only if $X$ is finite. Furthermore, we prove a

discretized formula for computing corresponding approximation

quantities and obtain results concerning the nonlinearity of

$A(X)$ and continuity properties of functions $f \in A(X)$.

Applications concern the Riemann integrability of approximable

functions on cubes $([-1,1]^m,d_\infty)$ and the approximation

of so-called regulated functions on compact intervals. The

approximation on the cube leads to Riemann integrable

quasi-continuous functions, which are of particular interest in

global optimization.

**Keywords:** *compact metric space, entropy numbers, nonlinear approximation, n-term approximation, approximation by step functions, regulated functions, quasi-continuous functions, Riemann integrable functions*

**Upload:** 1999-12-09

**Update:** 2001-08-31

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