Almost automorphic generalized functions

Abstract

The paper deals with a new algebra of generalized functions. This algebra contains Bochner almost automorphic functions and almost automorphic distributions. Properties of this algebra are studied.

Almost periodic generalized ultradistributions

“…Denoting by E the space of infinitely derivable functions, we recall the space of almost periodic infinitely derivable functions and the space of infinitely derivable bounded functions, see [13] and [3], are denoted and defined respectively by…”

“…Remark 7.4. The generalized spectra of an almost periodic generalized ultradistribution Λ u is not necessarily countable, see [3].…”

“…In view of the problem of multiplication of distributions, algebras of generalized functions containing different classes of distributions have been introduced and developed, see [11]. Consequently, an algebra of almost periodic generalized functions containing almost periodic functions and almost periodic distributions has been introduced and studied in [3], and an application to systems of ordinary differential equations is given in [4].…”

“…So we have a new reservoir of mathematical objects that could be useful for studying problems with nonlinear operations on ultradistributions. The paper aims to introduce an algebra of almost periodic generalized ultradistributions containing almost periodic Beurling ultradistributions of [6], as well as the algebra of almost periodic generalized functions of [3], and then it studies fundamental properties of this algebra.…”

Vector-Valued Almost Periodic Ultradistributions and Their Generalizations

“…The concept of almost automorphic ultradistributions will be introduced and analyzed in our follow-up research with S. Pilipović and D. Velinov [24] (cf. C. Bouzar, M. T. Khalladi, F. Z. Tchouar [5] for the notion of an almost automorphic Colombeau generalized function, C. Bouzar, Z. Tchouar [6] for the notion of an almost automorphic distribution, C. Bouzar, M. T. Khalladi [7] for the notion of an almost periodic Colombeau generalized function, and M. F. Hasler [19] for the notion of a Bloch-periodic Colombeau generalized function).…”