Difference between revisions of "Localization principle"
From Encyclopedia of Mathematics
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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 | + | 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, | 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 [[#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
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