Entropy and the approximation of continuous functions
The author develops a quantitative approximation theory for continuous functions on an arbitrary compact metric space (X,d) taking Jackson's theorem (1) \(E_ n(f)\leq c\cdot \omega (f;(b-a)/2n)\) from the theory of polynomial approximation as a model. Of course, the notion of a polynomial does not make sense on an arbitrary compact metric space (X,d). Linear combinations of so-called controllable partitions of unity on X are used as a substitute for the polynomials. They give rise to a sequence of approximation numbers \(a_ n(f)\) for any \(f\in C(X)\) which corresponds to the sequence of optimal errors \(E_ n(f)\) in the theory of polynomial approximation. The final result is (2) \(a_ n(f)\leq \omega (f;\epsilon_ n(X))\) for \(n=1,2,3,...\), where \(\omega\) (f;\(\delta\)) is the modulus of continuity of the function \(f\in C(X)\) and where \(\epsilon_ n(X)\) denotes the nth entropy number of X. Since X is compact \(\epsilon_ n(X)\) is the smallest radius \(\epsilon\geq 0\) such that n closed balls of radius \(\epsilon\) cover X. Hence in the case of an interval \(X=[a,b]\) we have \(\epsilon_ n([a,b])=(b-a)/2n.\) Accordingly, the argument \(\delta =b-a/2n\) under the modules of continuity \(\omega\) (f;\(\delta\)) in Jackson's inequality (1) is nothing other but the nth entropy number of the interval [a,b]. This remark discloses the analogy between the result (2) of the present paper and Jackson's result (1). In any case the approximability of a function \(f\in C(X)\) is determined by the degree of continuity of f in terms of its modulus of continuity \(\omega\) (f;\(\delta\)) and by the degree of compactness of (X,d) in terms of the sequence of entropy numbers \(\epsilon_ n(X)\).
scientific article; zbMATH DE number 4192338