View source for Localization principle

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

<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>

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>

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>