Entropy and the approximation of bounded functions and operators (Q1816490)

The paper completes a quantitative approximation theory developed by the second author in [Arch. Math. 58, No. 3, 280-287 (1992; Zbl 0749.41022)]. This theory concerns the approximation of continuous functions on compact metric spaces by linear combinations \(\varphi= \sum^n_{j=1} \lambda_j \varphi_j\) of so-called controllable partitions of unity \(\varphi_1, \varphi_2, \dots, \varphi_n\). Among others it is shown that the Jackson type inequality \(a_n (f)\leq \omega (f; \varepsilon_n (X))\), relating the corresponding approximation numbers \(a_n (f)\) of \(f\in C(X)\) to the modulus of continuity \(\omega (f; \delta)\) at \(\delta= \varepsilon_n (X)\) (\(n\)-th entropy number of \(X\)), is optimal in some sense. After that controllable partitions of unity \(\varphi_1, \varphi_2, \dots, \varphi_n\) are replaced by the characteristic functions \(I_{P_1}, I_{P_2}, \dots, I_{P_n}\) of so-called controllable partitions \({\mathcal P}= \{P_1, P_2, \dots, P_n\}\) of \(X\), the condition of controllability saying that \(\varepsilon_1 (P_i)< \varepsilon_{n-1} (X)\) for \(1\leq i\leq n\). The approximation number \(\widehat {a}_n (f)\) defined with the approximation by controllable step functions \(\widehat {\varphi}= \sum^k_{i=1} \lambda_i I_{P_i}\), \(\{P_1, P_2, \dots, P_k\}\) a controllable partition of \(X\) with \(k\leq n\), for continuous functions \(f\) on \(X\) are subject to \(a_n (f)\leq \widehat {a}_n (f)\). Furthermore, a Jackson type inequality \(\widehat {a}_n (f)\leq \omega (f; \varepsilon_n (X))\) now turns out to be true for arbitrary bounded functions \(f\) on \(X\). This inequality in the end leads to a Jackson type inequality \(c_{n+1} (T)\leq \omega (T; \varepsilon_n (X))\) for operators \(T\in {\mathcal L} (E, C(X))\) mapping an arbitrary Banach space \(E\) into \(C(X)\), \(c_{n+1} (T)\) denoting the \((n+1)\)-st Gelfand number of \(T\). (E-mail: richter\@minet.uni-jena.de).