**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*

**Upload:** 1999-07-26

**Update:** 1999-07-27

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