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.
Citation
Christian Richter. "A chain of controllable partitions of unity on the cube and the approximation of Hölder continuous functions." Illinois J. Math. 43 (1) 159 - 191, Spring 1999. https://doi.org/10.1215/ijm/1255985343
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