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Vysotskaya I. A., Strukova I. I. The research of some classes of almost periodic at infinity functions. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2021, vol. 21, iss. 1, pp. 4-14. DOI: 10.18500/1816-9791-2021-21-1-4-14

# The research of some classes of almost periodic at infinity functions

The article under consideration is devoted to continuous almost periodic at infinity functions defined on the whole real axis and with their values in a complex Banach space. We consider different subspaces of functions vanishing at infinity, not necessarily tending to zero at infinity. We introduce the notions of slowly varying and almost periodic at infinity functions with respect to these subspaces. For almost periodic at infinity functions (with respect to a subspace) we give four different definitions. The first definition (approximating) is based on the approximation theorem. In the classical version, for almost periodic functions, they are represented as uniform closures of trigonometric polynomials. In our case, the Fourier coefficients are slowly varying at infinity functions. The second definition, which is an analogue of G. Bohr’s definition of an almost periodic function, is based on the concept of an $\varepsilon$-period. The third definition meets S. Bochner’s criterion for the almost periodicity of functions. The fourth definition is given in terms of factor space. With the help of the results of the theory of almost periodic vectors in Banach modules those four definitions are proved to be equivalent. In addition, it was proved that the introduced spaces of slowly varying and almost periodic at infinity functions with respect to different subspaces of functions vanishing at infinity coincide with the spaces of ordinary slowly varying and almost periodic at infinity functions, respectively. The feasibility of consideration of these functions is due to the fact that the solutions of some important classes of differential and difference equations are almost periodic at infinity. We consider differential equations whose right-hand side is a function vanishing at infinity and obtain necessary and sufficient conditions for their bounded solutions to be almost periodic at infinity functions. We also study an asymptotic representation of the solutions.

- Gelfand I. M., Raikov D. A., Shilov G. E. Commutative normed rings. Uspehi Matem. Nauk (N. S.), 1946, vol. 1, no. 2 (12), pp. 48–146 (in Russian).
- Baskakov A. G., Strukova I. I., Trishina I. A. Solutions Almost Periodic at Infinity to Differential Equations With Unbounded Operator Coefficients. Siberian Mathematical Journal, 2018, vol. 59, iss. 2, pp. 231–242. https://doi.org/10.1134/S0037446618020052
- Trishina I. A. Almost Periodic at Infinity Functions Relative to the Subspace of Functions Integrally Decrease at Infinity. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 2017, vol. 17, iss. 4, pp. 402–418 (in Russian). https://doi.org/10.18500/1816-9791-2017-17-3-402-418
- Strukova I. I. On Wiener’s Theorem for functions periodic at infinity. Siberian Mathematical Journal, 2016, vol. 57, no. 1, pp. 145–154. https://doi.org/10.1134/S0037446616010146
- Baskakov A., Strukova I. Harmonic analysis of functions periodic at infinity. Eurasian Mathematical Journal, 2016, vol. 7, no. 4, pp. 9–29.
- Baskakov A. G. Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations. Russian Mathematical Surveys, 2013, vol. 68, no. 1, pp. 69–116. https://doi.org/10.1070/RM2013v068n01ABEH004822
- Baskakov A. G. Harmonic and spectral analysis of power bounded operators and bounded semigroups of operators on Banach spaces. Mathematical Notes, 2015, vol. 97, no. 2, pp. 164–178. https://doi.org/10.1134/S0001434615010198
- Baskakov A. G., Kaluzhina N. S. Beurling’s theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations. Mathematical Notes, 2012, vol. 92, no. 5, pp. 587–605. https://doi.org/10.1134/S0001434612110016
- Strukova I. I. About harmonic analysis of periodic at infinity functions. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 2014, vol. 14, iss. 1, pp. 28–38 (in Russian). https://doi.org/10.18500/1816-9791-2014-14-1-28-38.
- Trishina I. A. Functions slowly varying at infinity. Proceeding of Voronezh State University. Series: Physics. Mathematics, 2017, vol. 4, pp. 134–144 (in Russian).
- Baskakov A. G., Krishtal I. A. Harmonic analysis of causal operators and their spectral properties. Izvestiya: Mathematics, 2005, vol. 69, no. 3, pp. 439–486. http://doi.org/10.1070/IM2005v069n03ABEH000535
- Baskakov A. G. Representation theory for Banach algebras, Abelian groups, and semigroups in the spectral analysis of linear operators. Journal of Mathematical Sciences, 2006, vol. 137, no. 4, pp. 4885–5036. https://doi.org/10.1007%2Fs10958-006-0286-4
- Baskakov A. G., Krishtal I. A. Spectral analysis of abstract parabolic operators in homogeneous function spaces. Mediterranean Journal of Mathematics, 2016, vol. 13, no. 5, pp. 2443–2462. https://doi.org/10.1007/s00009-015-0633-0

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