Smoothness, modulus of
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 .
|||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|
|||A. Marchaud, "Sur les dérivées et sur les différences des fonctions de variables réelles" J. Math. Pures Appl. , 6 (1927) pp. 337–425|
|||A. Zygmund, "Smooth functions" Duke Math. J. , 12 (1945) pp. 47–76|
|||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)|
Smoothness, modulus of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smoothness,_modulus_of&oldid=12777