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Difference between revisions of "Localization principle"

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For any [[Trigonometric series|trigonometric series]] with coefficients tending to zero, the convergence or divergence of the series at some point depends on the behaviour of the so-called Riemann function in a neighbourhood of this point.
 
For any [[Trigonometric series|trigonometric series]] with coefficients tending to zero, the convergence or divergence of the series at some point depends on the behaviour of the so-called Riemann function in a neighbourhood of this point.
  
The Riemann function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060300/l0603001.png" /> of a given trigonometric series
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The Riemann function $F$ of a given trigonometric series
  
<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/l/l060/l060300/l0603002.png" /></td> </tr></table>
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$$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx$$
  
 
is the result of integrating it twice, that is,
 
is the result of integrating it twice, that is,
  
<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/l/l060/l060300/l0603003.png" /></td> </tr></table>
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$$F(x)=\frac{a_0}{4}x^2+Cx+D-\sum_{n=1}^\infty\frac{a_n\cos nx+b_n\sin nx}{n^2}.$$
  
 
There is a generalization of the localization principle for series with coefficients that do not tend to zero (see [[#References|[2]]]).
 
There is a generalization of the localization principle for series with coefficients that do not tend to zero (see [[#References|[2]]]).

Latest revision as of 16:43, 19 September 2014

For any trigonometric series with coefficients tending to zero, the convergence or divergence of the series at some point depends on the behaviour of the so-called Riemann function in a neighbourhood of this point.

The Riemann function $F$ of a given trigonometric series

$$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx$$

is the result of integrating it twice, that is,

$$F(x)=\frac{a_0}{4}x^2+Cx+D-\sum_{n=1}^\infty\frac{a_n\cos nx+b_n\sin nx}{n^2}.$$

There is a generalization of the localization principle for series with coefficients that do not tend to zero (see [2]).

References

[1] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[2] A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)
How to Cite This Entry:
Localization principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Localization_principle&oldid=17475
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article