Asymptotically Almost Periodic Generalized Functions

Abstract

The aim of this paper is to introduce and to study an algebra of almost periodic generalized functions containing the classical Bohr almost periodic functions as well as almost periodic Schwartz distributions.

Asymptotic almost automorphy for algebras of generalized functions

“…We have G aap G aaa , where G aap is the algebra of asymptotically almost periodic generalized functions of [8].…”

“…As a by pass result, we give a Seeley type result on extention of functions in the context of the introduced generalized functions, this is needed in the proof of a fundamental result on the uniqueness of decomposition of an asymptotically almost automorphic generalized function. The papers [8] and [9] can be considered as consequences of this work. The paper aims also, as in [16], to lift a Frechet existence result of asymptotically almost automorphic solutions of differential equations to the level of neutral difference differential systems in the framework of G aaa .…”

“…In view of the result [23] on the impossibility of the multiplication of distributions, see [21] for more details, algebras of generalized functions containing spaces of Sobolev-Schwartz type distributions have been studied, see [14], [15], [1] and [20]. The concepts of almost periodicity and asymptotic almost periodicity as well as almost automorphy in the context of such algebras of generalized functions are introduced, studied and applied in the papers [6], [7] [8], [9] and [10]. So, the paper first introduces and studies a class of asymptotically almost automorphic generalized functions, denoted by G aaa .…”

Pseudo almost periodic generalized functions

“…Finally, another result is Proposition 7, in which we give an extension of the classical Bohl-Bohr's Theorem. We refer the reader to [3], [4] and [5] from which the results of this paper were inspired. In this paper we consider functions and distributions defined on R. Recall (C b , L ∞ ) the Banach algebra of bounded and continuous complex valued functions on R endowed with the norm L ∞ of uniform convergence on R. The space C ap of almost periodic functions on R, which was introduced by H. Bohr, is the closed subalgebra of (C b , L ∞ ) that contains all the functions f, satisfying: for any ε > 0, the set…”