Entropy and the approximation of bounded functions and operators

Preprint series: 99-39, Reports on Analysis

The paper is published: Arch. Math. 67 (1996), 478-492

MSC:
41A17 Inequalities in approximation (Bernstein, Jackson, Nikolskiui type inequalities)
41A30 Approximation by other special function classes
47B06 Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators

Abstract: The paper completes a quantitative approximation theory developed
by the second named author in Arch. Math. 58, 280-287, 1992. This
theory concerns the approximation of continuous functions on compact
metric spaces by linear combinations $\varphi= \sum\limits_{j=1}^n \lambda_j \varphi_j$ of so-called controllable partitions of unity
$\varphi_1, \varphi_2, \ldots, \varphi_n$. Among others it is shown
that the Jackson type inequality $a_n(f) \le \omega(f;\eps_n(X))$,
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
$\varphi_1, \varphi_2, \ldots, \varphi_n$ are replaced by the
characteristic functions $I_{P_1}, I_{P_2}, \ldots, I_{P_n}$ of
so-called controllable partitions ${\cal P}= \{ P_1, P_2, \ldots, P_n \}$ of $X$, the condition of controllability saying that
$\varepsilon_1(P_i) < \varepsilon_{n-1}(X)$ for $1 \le i \le n$.
The approximation numbers $\hat{a}_n(f)$ defined with the
approximation by controllable step functions $\hat{\varphi}= \sum\limits_{i=1}^k \lambda_i I_{P_i}$, $\{ P_1, P_2, \ldots, P_k \}$
a controllable partition of $X$ with $k \le n$, for continuous
functions $f$ on $X$ are subject to $a_n(f) \le \hat{a}_n(f)$.
Furthermore, a Jackson type inequality $\hat{a}_n(f) \le \omega(f;\varepsilon_n(X))$ now turns out to be true for arbitrary
bounded functions $f$ on $X$. This inequality in the end leads to
a Jackson type inequality $c_{n+1}(T) \le \omega(T;\varepsilon_n(X))$ for operators $T \in {\cal L}(E,C(X))$
mapping an arbitrary Banach space $E$ into $C(X)$, $c_{n+1}(T)$
denoting the $(n+1)$-st Gelfand number of $T$.

Keywords: approximation of functions and operators, entropy numbers, partitions of unity, step functions, Jackson type inequalities, Gelfand numbers