Introduction

  • Z. Ditzian
  • V. Totik
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 9)

Abstract

The aim of this book is to introduce and apply a new “natural” modulus of smoothness. This will be a measure of smoothness that will provide us with a better tool to deal with the rate of best approximation, inverse theorems and imbedding theorems. The classical modulus of smoothness
$$\omega ^r (f,t) = \mathop {\sup }\limits_{0 < h \leqslant t} \parallel \Delta _h^r f\parallel $$
has proved to be very useful for problems of the above type. However, in recent years some shortcomings of the classical modulus and the need for a new modulus for measuring smoothness have become evident. In answer to this we suggest our modulus given by
$$\omega _\phi ^r (f,t)_p \equiv \mathop {\sup }\limits_{0 < h \leqslant t} \parallel \Delta _{h\phi }^r f\parallel _{L_p } ,$$
(1)
where the function ϕ(x) and the interval in question will be related to the problem at hand. We should point out that a vital feature of (1) is that the increment (x) varies with x. For ϕ(x)≡ 1, (1) is reduced to the classical modulus.

Keywords

Sten 

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Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Z. Ditzian
    • 1
  • V. Totik
    • 2
  1. 1.Department of MathematicsUniversity of AlbertaEdmontonCanada
  2. 2.Bolyai InstituteAttila Jozsef UniversitySzegedHungary

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