Abstract: The paper aims to to study the concept of asymptotic almost automorphy in the context of generalized functions. We introduce an algebra of asymptotically almost automorphic generalized functions which contains the space of smooth asymptotically almost automorphic functions as a subalgebra. The fundamental importance of this algebra, is related to the impossibility of multiplication of distributions; it also contains the asymptotically almost automorphic Sobolev-Schwartz distributions as a subspace. Moreover, it is shown that the introduced algebra is stable under some nonlinear operations. As a by pass result, the paper gives a Seeley type result on extension of functions in the context of the algebra of bounded generalized functions and the algebra of bounded generalized functions vanishing at infinity, these results are used to prove the fundamental result on the uniqueness of decomposition of an asymptotically almost automorphic generalized function. As applications, neutral difference-differential systems are considered in the framework of the algebra of generalized functions.
Keywords: asymptotic almost automorphy, generalized functions, neutral difference differential equations.
For citation: Bouzar, Ch. and Slimani, M. Asymptotic Almost Automorphy for Algebras of Generalized Functions, Vladikavkaz Math. J., 2023, vol. 25, no. 2, pp. 38-55.
DOI 10.46698/j1917-4964-8877-l
1. Bochner, S. Uniform Convergence of Monotone Sequences of Functions,
Proceedings of the National Academy of Sciences USA,
1961, vol. 47, no. 4, pp. 582-585. DOI: 10.1073/pnas.47.4.582.
2. Bochner, S. Continuous Mappings of Almost Automorphic and Almost Periodic Functions,
Proceedings of the National Academy of Sciences USA,
1964, vol. 52, no. 4, pp. 907-910. DOI: 10.1073/pnas.52.4.907.
3. Bochner, S. A New Approach to Almost Periodicity,
Proceedings of the National Academy of Sciences USA,
1962, vol. 48, no. 12, p. 2039-2043.
DOI: 10.1073/pnas.48.12.2039.
4. Bohr, H. Almost Periodic Functions, Chelsea Publishing Company, 1947.
5. Stepanoff, V. V. Sur Quelques Generalisations des Fonctions Presque Periodiques,
Comptes Rendus de l'Academie des Sciences, Paris, 1925, vol. 181, pp. 90-92.
6. Levitan, B. M. Pochti-periodicheskie funktsii [Almost Periodic Functions],
Moscow, Gos. Izdat. Tekh-Teor. Lit., 1953 (in Russian)
7. Frechet, M. Les Fonctions Asymptotiquement Presque Periodiques,
Revue Sci., 1941, vol. 79, pp. 341-354.
8. N'Guerekata, G. M. Some Remarks on Asymptotically Almost Automorphic Functions,
The Mathematical Revue of the University of Parma (4),
1987, vol. 13, pp. 301-303.
9. Bouzar, C. and Tchouar, F. Z. Asymptotic Almost Automorphy Of Functions And Distributions,
Ural Mathematical Journal, 2020, vol. 6, no. 1, pp. 54-70.
DOI: 10.15826/umj.2020.1.005.
10. Sobolev, S. L. Applications of Functional Analysis In Mathematical Physics,
American Mathematical Society, 1963.
11. Schwartz, L. Theorie des Distributions, 2nd ed., Hermann, 1966.
12. Cioranescu, I. Asymptotically Almost Periodic Distributions,
Applicable Analysis, 1989, vol. 34, no. 3-4, pp. 251-259.
DOI: 10.1080/00036818908839898.
13. Bouzar, C. and Tchouar, F. Z.
Almost Automorphic Distributions,
Mediterranean Journal of Mathematics , 2017, vol. 14, no. 04, pp. 1-13.
DOI: 10.1007/s00009-017-0953-3.
14. Schwartz, L. Sur l'Impossibilite de la Multiplication des Distributions,
Comptes Rendus de l'Academie des Sciences,
1954, vol. 239, pp. 847-848.
15. Colombeau, J.-F. Elementary Introduction to New Generalized Functions,
North Holland, 1985.
16. Egorov, Yu. V. A Contribution to the Theory of Generalized Functions,
Russian Mathematical Surveys, 1990, vol. 45, no. 5, pp. 1-49.
DOI: 10.1070/RM1990v045n05ABEH002683.
17. Antonevich, A. B. and Radyno, Ya. V. On a General Method of Constructing Algebras
of New Generalized Functions, Soviet Mathematics - Doklady, 1991, vol. 43, no. 3, pp. 680-684.
18. Bouzar, C. and Khalladi, M. T. Almost Periodic Generalized Functions,
Novi Sad Journal of Mathematics, 2011, vol. 41, no. 1, pp. 33-42.
19. Bouzar, C. and Khalladi, M. T. Linear Differential Equations in the Algebra of
Almost Periodic Generalized Functions, Rendiconti del Seminario Matematico Universita
e Politecnico di Torino, 2012, vol. 70, no. 2, pp. 111-120.
20. Bouzar, C. and Khalladi, M. T. Asymptotically Almost Periodic Generalized Functions,
Operator Theory: Advances and Applications, 2013, vol. 231, pp. 261-272.
21. Bouzar, C. and Khalladi, M. T. On Asymptotically Almost Periodic Generalized
Solutions of Differential Equations, Operator Theory: Advances and Applications, 2015, vol. 245, pp. 35-43.
22. Bouzar, C., Khalladi M. T. and Tchouar, F. Z. Almost Automorphic Generalized Functions,
Novi Sad Journal of Mathematics, 2015, vol. 45, no. 1, pp. 207-214.
23. Garetto, C. Topological Structures in Colombeau Algebras,
Monatshefte fur Mathematik, 2005, vol. 146, no. 3, pp. 203-226.
DOI: 10.1007/s10440-005-6700-y.
24. Seeley, R. T. Extension of \(C^{\infty}\) Functions Defined in a Half Space,
Proceedings of the American Mathematical Society, 1964, vol. 15, no. 4, pp. 625-626.
DOI: 10.1090/s0002-9939-1964-0165392-8.
