n-Term approximation by controllable step functions

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