Bochner-Type Property on Spaces of Generalized Almost Periodic Functions

Abstract

Our paper is focused on spaces of generalized almost periodic functions which, as in classical Fourier analysis, are associated with a Fourier series with real frequencies. In fact, based on a pertinent equivalence relation defined on the spaces of almost periodic functions in Bohr, Stepanov, Weyl and Besicovitch's sense, we refine the Bochner-type property by showing that the condition of almost periodicity of a function in any of these generalized spaces can be interpreted in the way that, with respect to the topology of each space, the closure of its set of translates coincides with its corresponding equivalence class.

"On the real projections of zeros of analytic almost periodic functions"

“…This new equivalence relation, which is widely used in this paper, coincides with that of Bohr for the particular case of general Dirichlet series whose sets of exponents have an integral basis (see [22,Proposition 1]). Other important results derived from this equivalence relation can be seen in [19,20,21,22,23].…”

A note on spaces of almost periodic functions with values in Banach spaces

“…exists uniformly with respect to a ∈ R, and, at most, a countable set of values of In this paper, we consider an equivalence relation on the functions with values in a Banach space X which can be represented with a Fourier-like series (see the comments before Definition 3.2), and which satisfies the important property consisting of that an equivalence class is completely contained in the space AP(R, X) when at least one of its functions is almost periodic (see Corollary 3.3). In this way, by analogy with the recent developments which we made for the space AP(R, C), the Besicovitch almost periodic functions, and other spaces of generalized almost periodic functions (see [16,17,19], respectively), this equivalence relation leads to refine the Bochnertype property or normality in the sense that the condition of almost periodicity in AP(R, X) implies that every sequence of translates has a subsequence that converges, with respect to the topology of AP(R, X), to an equivalent function (see Theorem 3.7 and Corollary 3.8). This extends our previous work on the case of numerical almost periodic functions (see [16,17], but also [18,20]).…”

The class of c-almost periodic functions defined on vertical strips in the complex plane

“…It is known that Bohr's notion of almost periodicity of a function defined on R is equivalent to the relative compactness of the set of its translates with respect to the topology of the uniform convergence (see e.g. [3,16,17]). In the same terms, the property of relative compactness of the vertical translates, with respect to the topology of the uniform convergence on reduced strips, identifies the class of almost periodic functions defined on vertical strips of the complex plane.…”