A chain of controllable partitions of unity on the cube and the approximation of Hölder continuous functions (Q1282990)

The paper concerns so-called controllable partitions of unity \(\Phi= \{\varphi_1, \varphi_2,\dots, \varphi_k\} \subseteq C(X)\) of compact metric spaces \((X,d)\), which are defined by the uniformity condition \(\varepsilon_1 (\text{supp} (\varphi_i))< \varepsilon_{k-1}(X)\) for \(1\leq i\leq k\) in terms of the metric entropy (due to I. Stephani). The author constructs a chain \((\Phi_n)_{n=1}^\infty\) of controllable partitions of unity on the \(m\)-dimensional cube \(([0,2]^m, d_{\max})\) such that \(\text{span} (\Phi_1)\subset \text{span} (\Phi_2)\subset \text{span} (\Phi_3)\subset\dots\) with \(\dim(\text{span} (\Phi_n))= k_n\) and \(k_1< k_2< k_3<\dots\;\). By inserting suitable intermediate peaked partition, a related chain \((\widetilde{\Phi}_n)_{n=1}^\infty\) is defined, which fails the sharp controllability condition but has the property \(\text{card} (\widetilde{\Phi}_n)=n\). The corresponding approximation quantities, obtained by approximating functions \(f\in C([0,2]^m)\) by elements from \(\text{span} (\Phi_n)\) (or \(\text{span} (\widetilde{\Phi}_n)\)), give rise to error estimates of Jackson type as well as to Bernstein type inequalities so far as \(f\) is Hölder continuous. The results are generalized to \(C([0,2]^m)\)-valued bounded linear operators, and, partly, to spaces \(([0,2]^m,d)\) with an equivalent metric \(d\) instead of \(d_{\max}\).