# Smoothness, modulus of

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The modulus of continuity of the derivative of order of a function defined on a Banach space , i.e. the expression  where . If , the modulus of smoothness is the ordinary modulus of continuity (cf. Continuity, modulus of) of . Basic properties of the modulus of smoothness (in the case and the space of continuous functions) are:  does not decrease together with ;

if  is an integer, then for any , if , then if , then where and are constants independent of .

Certain problems in the theory of approximation of functions can ultimately be solved only in terms of a modulus of smoothness of order . In the theory of approximations of functions an important class is the class of periodic continuous functions with period and with second-order modulus of smoothness satisfying the condition The modulus of continuity of such functions satisfies the condition  , and the constant cannot be improved .

How to Cite This Entry:
Smoothness, modulus of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smoothness,_modulus_of&oldid=12777
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article