Almost Automorphic Distributions

Vector-valued almost automorphic distributions and vector-valued almost automorphic ultradistributions

“…Using the fact that the first derivative of a differentiable almost automorphic function is almost automorphic iff it is uniformly continuous [7], it can be easily verified that we have E AA (X) = E(X) ∩ AA(R : X); furthermore, E AA (X) * L 1 (R) ⊆ E AA (X) and E AA (X) is a closed subspace of D L ∞ (X) (see [4,Proposition 5]). We have, actually, that E AA (X) is the space of those…”

“…It can be simply verified that a regular distribution (ultradistribution of * -class) determined by an almost automorphic vector-valued function that is not almost periodic is an almost automorphic vector-valued distribution (ultradistribution of * -class) that cannot be almost periodic (cf. [4,Example 2]). Now we would like to state the following result: (ii) For every real sequence (b n ), there exist a subsequence (a n ) of (b n ) and a vector-valued ultradistribution S ∈ D ′ * (X) such that lim n→∞ T an , ϕ = S, ϕ , ϕ ∈ D * and lim n→∞ S −an , ϕ = T, ϕ , ϕ ∈ D * .…”

“…In Subsection 1.1, we remind ourselves of the elementary facts about Komatsu's approach to vector-valued ultradistributions. Section 2 is written in an expository manner, and its aim is to transfer the results of C. Bouzar and Z. Tchouar [4] to vector-valued case. In Section 3, we introduce the notion of a vector-valued almost automorphic ultradistribution and further analyze this concept.…”

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).…”

Almost automorphic generalized functions

“…Schwartz introduced and studied in [6] almost periodic distributions. The study of almost automorphic Schwartz distributions is done in the work [4].…”