For formal series identifying linear continuous functionals given on the space of trigonometric polynomials and summarized by Gauss-Weierstrass methods, we prove an analog of the Riemann localization principle: if $\{f_1,f_2\} ⊂ L_1 [0,2π]$ are converge at the interval $(a, b) ⊂ [0, 2π] $ then at each segment $[a + ε, b − ε] ⊂ (a, b)$ their difference of Fourier series uniformly converges to zero.
Generally speaking, the principle of localization for Fourier series of $2π$-periodic generalized functions is not fulfilled. When studying various problems of mathematical physics and analysis, it is often used not the Fourier series itself, but the series summarized by one or another regular method, so it is natural to fulfill the principle of localization for such series. For example, the solution of the Dirichlet problem for the Laplace equation in a unit circle is represented by the Fourier series of the boundary function summarized by the Abel-Poisson method; the solution of the Cauchy periodic problem for the equation of thermal conductivity and the initial condition in the space of generalized periodic functions is treated as a formal Fourier series of the initial function summarized by the Gauss-Weierstrass method. 

The paper investigates multiple Fourier series of periodic hyperfunctions and ultradistributions.

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