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

$$\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}{a}_n\cos nx+b_n\sin nx$$

is the result of integrating it twice, that is,

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

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

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