View source for Zygmund class of functions

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z0993401.png" /> be a positive real number. The Zygmund class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z0993402.png" /> is the class of continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z0993403.png" />-periodic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z0993404.png" /> with the property that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z0993405.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z0993406.png" /> the inequality

<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/z/z099/z099340/z0993407.png" /></td> </tr></table>

holds. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z0993408.png" /> was introduced by A. Zygmund [[#References|[1]]]. In terms of this class one can obtain a conclusive solution to the Jackson–Bernstein problem on direct and inverse theorems in the theory of approximation of functions (cf. [[Bernstein theorem|Bernstein theorem]]; [[Jackson theorem|Jackson theorem]]). For example: A continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z0993409.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934010.png" /> belongs to the Zygmund class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934011.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934012.png" /> if and only if its best uniform approximation error <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934013.png" /> by trigonometric polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934014.png" /> satisfies the inequality

<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/z/z099/z099340/z09934015.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934016.png" /> is a constant. The modulus of continuity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934017.png" /> of any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934018.png" /> admits the estimate

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in which the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934020.png" /> cannot be improved on for the entire class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934021.png" /> [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund,   "Smooth functions"  ''Duke Math. J.'' , '''12''' :  1  (1945)  pp. 47–76  ((Also: Selected papers of Antoni Zygmund, Vol. 2, Kluwer, 1989, pp. 184–213.))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,   "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.V. Efimov,   "Estimation of the modules of continuity of functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934022.png" />"  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''21''' :  2  (1957)  pp. 283–288  (In Russian)</TD></TR></table>



====Comments====
The quantity

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for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934024.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934025.png" />, is its Zygmund modulus. In terms of this, Zygmund's theorem above can also be stated as: A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934026.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934027.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934028.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934029.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934030.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934031.png" />) for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099340/z09934032.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,   "Introduction to approximation theory" , Chelsea, reprint  (1982)  pp. 203ff</TD></TR></table>