Localization principle

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)