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 [4].


[1] S.N. Bernshtein, "Sur l'ordre de la meilleure approximation des fonctions continués par de polynomes de degré donné" Mem. Publ. Classe Sci. Acad. Belgique (2) , 4 (1912) pp. 1–103
[2] A. Marchaud, "Sur les dérivées et sur les différences des fonctions de variables réelles" J. Math. Pures Appl. , 6 (1927) pp. 337–425
[3] A. Zygmund, "Smooth functions" Duke Math. J. , 12 (1945) pp. 47–76
[4] A.V. Efimov, "Estimate of the modules of continuity of a function in the class " Izv. Akad. Nauk SSSR Ser Mat. , 21 (1957) pp. 283–288 (In Russian)


The modulus of smoothness is also written in terms of symmetric differences, as , where


This gives a recurrent procedure for computing (approximations of) it.

To overcome certain shortcomings of this (classical) modulus of smoothness (especially its ability to characterize the order of the best polynomial approximation to a function ), new moduli of smoothness have been introduced. They use so-called step-weight functions , and are defined by

The function is chosen for the problem at hand. Note that here the increment varies with . A basic result is that if and only if . (Here , , , , and approximation is in .) For more on such moduli, their use in approximation problems and in the interpolation of spaces, see [a1].


[a1] Z. Ditzian, V. Totik, "Moduli of smoothness" , Springer (1987)
[a2] G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966)
How to Cite This Entry:
Smoothness, modulus of. Encyclopedia of Mathematics. URL:,_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