Smoothness, modulus of
The modulus of continuity of the derivative of order of a function
defined on a Banach space
, i.e. the expression
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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:
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does not decrease together with
;
if
is an integer, then
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for any ,
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if , then
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if , then
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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
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The modulus of continuity of such functions satisfies the condition
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, and the constant
cannot be improved [4].
References
[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 ![]() |
Comments
The modulus of smoothness is also written in terms of symmetric differences, as
, where
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and
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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
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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].
References
[a1] | Z. Ditzian, V. Totik, "Moduli of smoothness" , Springer (1987) |
[a2] | G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966) |
Smoothness, modulus of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smoothness,_modulus_of&oldid=48741