#### For citation:

Vysotskaya I. A. Almost Periodic at Infinity Functions Relative to the Subspace of Functions Integrally Decrease at Infinity. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2017, vol. 17, iss. 4, pp. 402-418. DOI: 10.18500/1816-9791-2017-17-4-402-418

# Almost Periodic at Infinity Functions Relative to the Subspace of Functions Integrally Decrease at Infinity

In the paper we introduce and study a new class of almost periodic at infinity functions, which is defined by means of a subspace of integrally decreasing at infinity functions. It is wider than the class of almost periodic at infinity functions introduced in the papers of A.G.Baskakov (with respect to the subspace of functions vanishing at infinity). It suffices to turn to the approximation theory for a new class of functions, where the Fourier coefficients are slowly varying at infinity functions with respect to the subspace of functions that decrease integrally at infinity. Three equivalent definitions of functions almost periodic at infinity with respect to integrally decreasing functions at infinity are formulated. For their investigation, the theory of Banach modules over the algebra L 1(R) of summable functions is applied. Almost periodic functions at infinity appear naturally as a solution of differential equations. Criteria for the almost periodicity at infinity of bounded solutions of ordinary differential equations of the form x(t) = Ax (t) + z (t), t ∈ J are formulated, where A is a linear operator and z is an integrally decreasing function at infinity, defined on infinite interval J that coincides with one of the sets R or R+.

- Bor G. Pochti periodicheskie funkcii [Almost periodic functions]. Moscow, OGIZ, 1934. 126 p. (in Russian).
- Восhner S. Uber gewisse Differential und algemeinere Gleichungen deren Losungen fast- periodisch sin. Math. Ann., 1930, vol. 103, pp. 588–597.
- Веsiсоvitсh A. S. On generalist almost periodic functions. Рrос. London Math. Soc., 1926, vol. 25, pp. 495–512.
- Favard J. On the convergence of the Sturm–Liouville Series. Ann. Math., 1923, vol. 24, no. 2, pp. 109–120.
- Levitan B. M., Stepanov V. V. O pochti periodicheskih funkciyah, prinadlezhashchih v sobstvennom smysle klassu W [On almost periodic functions belonging in the proper sense to the class W]. Dokl. AN SSSR [Reports of the Academy of Sciences of the USSR], 1939, vol. 22, pp. 229–232 (in Russian).
- Shtern A. I. Almost periodic functions and representations in locally convex spaces. Russian Math. Surveys, 2005, vol. 60, iss. 3, pp. 489–557. DOI: https://doi.org/10.1070/RM2005v060n03ABEH000849.
- Baskakov A. G. Representation theory for Banach algebras, Abelian groups, and semi- groups in the spectral analysis of linear operators. J. Math. Sci., 2006, vol. 137, no. 4, pp. 4885–5036. DOI: https://doi.org/10.1007/s10958-006-0286-4.
- Baskakov A. G., Kaluzhina N. S. Beurling’s theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations. Math. Notes, 2012, vol. 92 no. 5, pp. 587–605. DOI: https://doi.org/10.1134/S0001434612110016.
- Baskakov A. G., Kaluzhina N. S., Polyakov D. M. Slowly varying at infinity operator semigroups. Russian Math. (Iz. VUZ), 2014, vol. 58, no. 7, pp. 1–10. DOI: https://doi.org/10.3103/S1066369X14070019.
- Ryzhkova A. A., Trishina I. A. About periodic functions at infinity. Belgorod State University Scientific Bulletin. Mathematics & Physics, 2014, vol. 36, iss. 19(190), pp. 71–75 (in Russian).
- Trishina I. A. Algebraicheskie svojstva pochti periodicheskih na beskonechnosti funkcij [Algebraic properties of almost periodic functions at infinity]. Vestnik fakul’teta prikladnoj matematiki, informatiki i mekhaniki [Vestnik of the Department of Applied Mathematics, Informatics and Mechanics], 2016, no. 12, pp. 223–227 (in Russian).
- Ryzhkova A. A., Trishina I. A. Almost periodic at infinity solutions of difference equations. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2015, vol. 15, iss. 1, pp. 45–49 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2015-15-1-45-49.
- Levitan B. M., Zhikov V. V. Pochti-periodicheskie funkcii i differencial’nye uravneniya [Almost-periodic functions and differential equations]. Moscow, Moscow Univ. Press, 1978. 205 p. (in Russian).
- Baskakov A. G., Krishtal I. A. Harmonic analysis of causal operators and their spectral properties. Izv. Math., 2005, vol. 69, no. 3, pp. 439–486. DOI: https://doi.org/10.1070/IM2005v069n03ABEH000535.
- Baskakov A. G. Harmonic and spectral analysis of power bounded operators and bounded semigroups of operators on Banach spaces. Math. Notes, 2015, vol. 97, no. 2, pp. 164–178. DOI: https://doi.org/10.1134/S0001434615010198.
- Daleckij U. L., Krejn M. G. Ustojchivost reshenij differencialnyx uravnenij v banakhovom prostranstve [Stability of solutions of differential equations in a Banach space]. Moscow, Nauka, 1970. 534 p. (in Russian).

- 614 reads