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Entropy, Approximation Quantities and the Asymptotics of the Modulus of Continuity

✍ Scribed by Christian Richter

Publisher
John Wiley and Sons
Year
1999
Tongue
English
Weight
380 KB
Volume
198
Category
Article
ISSN
0025-584X

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✦ Synopsis

A b s t r a c t . The paper deals with the approximation of bounded real functions f on a compact iiii*tric space (A', d) by so-called controllable step functions in continuation of (Ri/Ste]. These step liirictions are connected with controllable coverings, that are finite coverings of compact metric spaces by subset6 whose sizes fulfil a uniformity condition depending on the entropy numbers cn(X) of the tipirce X. We show that a strong form of local finiteness holds for these coverings on compact metric rliibspaces of R" and S". This leads to a Bernstein type theorem if the space is of finite convex (Idormation. In this case the corresponding approximation numbers &(f) have the 8ame asymptotics IL\ w(f,cn(X)) for j E C ( X ) . Finally, the results concerning functions f E M ( X ) and f E C ( X ) are 1i;iiisferred to operators with values in M ( X ) and C ( X ) , respectively.

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