Localization property for the convolution of generalized periodic functions

TitleLocalization property for the convolution of generalized periodic functions
Publication TypeJournal Article
Year of Publication2020
AuthorsHorodets’kyi, VV, Martynyuk, OV
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
Date Published4/2020

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′ .

Keywordsconvolution, Fourier series, generalized function, localization property

1. Riemann, B. (1948). Collected Works. Moscow: Gostehizdat (in Russian).
2. Gorbachuk, V. I. & Gorbachuk, M. L. (1981). Trigonometric series and generalized periodic functions. Dokl. AN SSSR, 257, No. 4, pp. 799-803 (in Russian).
3. Gorbachuk, V. I. (1982). On the Fourier series of periodic ultra-distributions. Ukr. Math. J., 34, No. 2, pp. 118-123 (in Russian). Doi: https://doi.org/10.1007/BF01091513
4. Izvekov, I. G. (1986). The Riemann localization principle for Fourier series in spaces of generalized functions. Dokl. AN SSSR. Ser. A, No. 2, pp. 5-8 (in Russian).
5. Tillmann, H. G. (1953). Die Randverteilungen analytischer Funktionen und Distributionen. Math. Z., 59, pp. 61-83. Doi: https://doi.org/10.1007/BF01180242
6. Sato, M. (1959). Theory of hyperfunctions, I. J. Fac. Sci. Univ. Tokyo, Sect. I, 8, pp. 139-193.
7. Sato, M. (1960). Theory of hyperfunctions, II. J. Fac. Sci. Univ. Tokyo, Sect. II, 8, pp. 387-437.
8. Sato, M. (1958). On a generalization of the concept of functions. Proc. Japan. Acad., 34, pp. 126-130. Doi: https://doi.org/10.3792/pja/1195524746
9. Bremerman, G. (1968). Distributions, complex variables and Fourier transforms. Moscow: Mir (in Russian).
10. Gotynchan, T. I. (1998). On analytical image of ultra-distributions of S′ type. Bulletin of the University of Kiev, Ser. Phys.-Math. Sci., Iss. 1, pp. 37-41 (in Ukrainian).
11. Gorodetskii, V. V. & Gotynchan, T. I. (1998). On zero sets of generalized functions from space 1 α ′ (S1/ ), Boundary value problems for differential equations: Collection of scientific works, Iss. 1 (pp. 79-89). Kiev: Institute of Mathematics of the NAS of Ukraine (in Ukrainian).
12. Gorodetskii, V. V. & Martynyuk, O. V. (2016). Evolutionary pseudodifferential equations in numerically normalized spaces. Chernivtsi: Tehnodruk (in Ukrainian).