A chain of controllable partitions of unity on the cube and the approximation of Hölder continuous functions

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.