View source for Smoothness, modulus of

The modulus of continuity of the derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859201.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859202.png" /> defined on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859203.png" />, i.e. the expression

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859206.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859207.png" />, the modulus of smoothness is the ordinary modulus of continuity (cf. [[Continuity, modulus of|Continuity, modulus of]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859208.png" />. Basic properties of the modulus of smoothness (in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859209.png" /> and the space of continuous functions) are:

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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592011.png" /> does not decrease together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592012.png" />;

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592013.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592014.png" /> is an integer, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592015.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592016.png" />,

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if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592018.png" />, then

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if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592020.png" />, then

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592023.png" /> are constants independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592024.png" />.

Certain problems in the theory of approximation of functions can ultimately be solved only in terms of a modulus of smoothness of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592025.png" />. In the theory of approximations of functions an important class is the class of periodic continuous functions with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592026.png" /> 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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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592029.png" />, and the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592030.png" /> cannot be improved [[#References|[4]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Marchaud,   "Sur les dérivées et sur les différences des fonctions de variables réelles"  ''J. Math. Pures Appl.'' , '''6'''  (1927)  pp. 337–425</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Zygmund,   "Smooth functions"  ''Duke Math. J.'' , '''12'''  (1945)  pp. 47–76</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Efimov,   "Estimate of the modules of continuity of a function in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592031.png" />"  ''Izv. Akad. Nauk SSSR Ser Mat.'' , '''21'''  (1957)  pp. 283–288  (In Russian)</TD></TR></table>



====Comments====
The modulus of smoothness <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592032.png" /> is also written in terms of symmetric differences, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592033.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592036.png" /> to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592037.png" />), new moduli of smoothness have been introduced. They use so-called step-weight functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592038.png" />, and are defined by

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The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592040.png" /> is chosen for the problem at hand. Note that here the increment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592041.png" /> varies with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592042.png" />. A basic result is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592043.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592044.png" />. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592048.png" />, and approximation is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592049.png" />.) For more on such moduli, their use in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592050.png" /> approximation problems and in the interpolation of spaces, see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Z. Ditzian,   V. Totik,   "Moduli of smoothness" , Springer  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.G. Lorentz,   "Approximation of functions" , Holt, Rinehart &amp; Winston  (1966)</TD></TR></table>