Moduli of Smoothness pp 1-4 | Cite as

# Introduction

Chapter

## 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
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
where the function ϕ(

$$\omega ^r (f,t) = \mathop {\sup }\limits_{0 < h \leqslant t} \parallel \Delta _h^r f\parallel $$

$$\omega _\phi ^r (f,t)_p \equiv \mathop {\sup }\limits_{0 < h \leqslant t} \parallel \Delta _{h\phi }^r f\parallel _{L_p } ,$$

(1)

*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*hϕ*(*x*) varies with*x*. For ϕ(*x*)≡ 1, (1) is reduced to the classical modulus.## Keywords

Polynomial Approximation Bernstein Polynomial Weighted Sobolev Space Inverse Theorem Simple Polytopes
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag New York Inc. 1987