Localization property for the convolution of generalized periodic functions

Abstract

The well-known Riemann localization principle for the Fourier series of summable functions is reformulated for the convolution of generalized periodic functions with families of functions, which usually coincide with kernels of certain linear methods of summation of Fourier series (for example, summation methods such as the Gauss—Weierstrass one). We call the families of functions, for which the Riemann localization holds, the families of functions of a class L(X) . The necessary and sufficient conditions of belonging the family of functions to the class L(X) are found in the case where X is a sufficiently broad non-quasi-analytic class of periodic functions or X is a class of analytic periodic functions (in particular, X =G{β} for β > 1 and X =G{β} if 0 < β1). The definition of “analytic functional equal to zero on an open set” is also substantiated; a specific example of analytic functional is given, which is 0 on (a, b)⊂[0, 2π] . The use of the obtained result in partial differential equation theory allows us to obtain a new property (localization property, the property of local convergence improvement) of many problems of mathematical physics, since such solutions are often depicted as a convolution of some family of basic functions from the space X with a function F defined at the boundary of the domain, F may be a generalized function from a space X′ .