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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
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<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>
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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
+
Let  $  M $
 +
be a positive real number. The Zygmund class $  Z _ {M} $
 +
is the class of continuous $  2 \pi $-periodic functions  $  f $
 +
with the property that for all  $  x $
 +
and all  $  h > 0 $
 +
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>
+
$$
 +
| f ( x + h ) - 2f ( x) + f ( x - h ) |  \leq  M h
 +
$$
  
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
+
holds. The class  $  Z _ {M} $
 +
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  $  2 \pi $-
 +
periodic function  $  f $
 +
belongs to the Zygmund class  $  Z _ {M} $
 +
for some  $  M > 0 $
 +
if and only if its best uniform approximation error  $  E _ {n} ( f  ) $
 +
by trigonometric polynomials of degree  $  \leq  n $
 +
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/z09934019.png" /></td> </tr></table>
+
$$
 +
E _ {n} ( f  )  \leq 
 +
\frac{A}{n}
 +
,
 +
$$
  
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]]].
+
where  $  A > 0 $
 +
is a constant. The modulus of continuity  $  \omega ( \delta , f  ) $
 +
of any function  $  f \in Z _ {M} $
 +
admits the estimate
 +
 
 +
$$
 +
\omega ( \delta , f  )  \leq 
 +
\frac{M}{2  \mathop{\rm ln}  \sqrt {2 } + 1 }
 +
 
 +
\delta  \mathop{\rm ln} 
 +
\frac \pi  \delta
 +
+ O ( \delta )
 +
$$
 +
 
 +
in which the constant $  M / 2  \mathop{\rm ln}  ( \sqrt {2 } + 1 ) $
 +
cannot be improved on for the entire class $  Z _ {M} $[[#References|[3]]].
  
 
====References====
 
====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>
 
<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====
 
====Comments====
 
The quantity
 
The quantity
  
<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/z09934023.png" /></td> </tr></table>
+
$$
 +
\omega _ {f}  ^ {*} ( h)  = \sup _ { x }
 +
\sup _ {| \delta | \leq  n } \
 +
| f( x+ \delta ) - 2f( x) + f( x- \delta ) | ,
 +
$$
  
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" />.
+
for a $  2 \pi $-periodic function $  f $,  
 +
is its Zygmund modulus. In terms of this, Zygmund's theorem above can also be stated as: A $  2 \pi $-periodic function $  f $
 +
satisfies $  E _ {n} ( f  ) \leq  n  ^ {- 1} A $
 +
for some $  A $
 +
if and only if $  \omega _ {f}  ^ {*} ( h ) \leq  Bh $ ($  h > 0 $)  
 +
for some $  B $.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 04:01, 15 June 2022


Let $ M $ be a positive real number. The Zygmund class $ Z _ {M} $ is the class of continuous $ 2 \pi $-periodic functions $ f $ with the property that for all $ x $ and all $ h > 0 $ the inequality

$$ | f ( x + h ) - 2f ( x) + f ( x - h ) | \leq M h $$

holds. The class $ Z _ {M} $ was introduced by A. Zygmund [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; Jackson theorem). For example: A continuous $ 2 \pi $- periodic function $ f $ belongs to the Zygmund class $ Z _ {M} $ for some $ M > 0 $ if and only if its best uniform approximation error $ E _ {n} ( f ) $ by trigonometric polynomials of degree $ \leq n $ satisfies the inequality

$$ E _ {n} ( f ) \leq \frac{A}{n} , $$

where $ A > 0 $ is a constant. The modulus of continuity $ \omega ( \delta , f ) $ of any function $ f \in Z _ {M} $ admits the estimate

$$ \omega ( \delta , f ) \leq \frac{M}{2 \mathop{\rm ln} \sqrt {2 } + 1 } \delta \mathop{\rm ln} \frac \pi \delta + O ( \delta ) $$

in which the constant $ M / 2 \mathop{\rm ln} ( \sqrt {2 } + 1 ) $ cannot be improved on for the entire class $ Z _ {M} $[3].

References

[1] 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.))
[2] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[3] A.V. Efimov, "Estimation of the modules of continuity of functions of class " Izv. Akad. Nauk. SSSR Ser. Mat. , 21 : 2 (1957) pp. 283–288 (In Russian)

Comments

The quantity

$$ \omega _ {f} ^ {*} ( h) = \sup _ { x } \sup _ {| \delta | \leq n } \ | f( x+ \delta ) - 2f( x) + f( x- \delta ) | , $$

for a $ 2 \pi $-periodic function $ f $, is its Zygmund modulus. In terms of this, Zygmund's theorem above can also be stated as: A $ 2 \pi $-periodic function $ f $ satisfies $ E _ {n} ( f ) \leq n ^ {- 1} A $ for some $ A $ if and only if $ \omega _ {f} ^ {*} ( h ) \leq Bh $ ($ h > 0 $) for some $ B $.

References

[a1] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff
How to Cite This Entry:
Zygmund class of functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zygmund_class_of_functions&oldid=14439
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article