View source for 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.

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 [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR></table>