Entropy, approximation quantities and the asymptotics of the modulus of continuity

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