Measure-valued almost periodic functions

We consider Stepanov almost periodic functions μ ∈ ranging in the metric space of Borel probability measures on a complete separable metric space is equipped with the Prokhorov metric). The main result is as follows: a function , belongs to if and only if for each bounded continuous function , the function is Stepanov almost periodic (of order 1) and

Vector-valued pseudo almost periodic functions

Vector-valued pseudo almost periodic functions are defined and their properties are investigated. The vector-valued functions contain many known functions as special cases. A unique decomposition theorem is given to show that a vector-valued pseudo almost periodic function is a sum of an almost periodic function and an ergodic perturbation.     

Approximation of almost periodic functions by periodic ones

It is not the purpose of this paper to construct approximations but to establish a class of almost periodic functions which can be approximated, with an arbitrarily prescribed accuracy, by continuous periodic functions uniformly on =(+).     

Carathéodory-Toeplitz and Nehari problems for matrix valued almost periodic functions

In this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated. We present necessary and sufficient conditions for the existence of extensions in terms of Toeplitz and Hankel operators on Besicovitch spaces and Lebesgue spaces. Furthermore, when a solution exists a special extension (the band extension) is constructed which enjoys a maximum entropy property. A linear fractional parameterization of the set of all extensions is also provided. The techniques used in the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method.

     

Gabor systems and almost periodic functions

Inspired by results of Kim and Ron, given a Gabor frame in $L^2(\mathbb{R})$, we determine a non-countable generalized frame for the non-separable space ${\mathit{AP}}_2(\mathbb{R})$ of the Besicovic almost periodic functions. Gabor type frames for suitable separable subspaces of ${\mathit{AP}}_2(\mathbb{R})$ are constructed. We show furthermore that Bessel-type estimates hold for the AP norm with respect to a countable Gabor system using suitable almost periodic norms of sequences.     

Unbounded functions with almost periodic differences

The structure of continuous real-valued functions F(t) on the real line, such that for any fixed y the difference F(t+y)–F(t) is an almost periodic Bohr function, is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 10, pp. 1409–1413, October, 1991.     

Weakly almost periodic functions on semisimple Lie groups

IfG is a semisimple analytic group with finite center, it is proved thatG admits only the “obvious” weakly almost periodic functions. The analysis yields also an intrinsic proof of Moore's ergodicity theorem [7].     

Weighted pseudo almost periodic functions and applications

This Note introduces a new class of functions called weighted pseudo almost periodic functions, which generalize in a natural fashion the classical pseudo almost periodic functions due to C. Zhang. Properties of those weighted pseudo almost periodic functions are discussed including a composition result of weighted pseudo almost periodic functions, which plays a crucial role for the solvability of some weighted pseudo almost periodic semilinear differential and partial differential equations. To cite this article: T. Diagana, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Limits of integrals involving almost periodic functions

Let Sp ? R⁺ be a discrete countable set, let {α_λ}_λ∈Sp be a sequence in l¹(Sp) and f(x) ? ∑_λ∈Spα_λsin(λx). f is an almost periodic odd function with {λ: ±λ ∈ Sp} as spectrum. We give some conditions about the set S so that $\int _1^{+\infty}\ f(x){\rm sin}(Rx){dx\over x}\rightarrow 0$ whenever R → +∞, R ∈ S.     

The almost periodic type difference equations

In this paper, we study the existence of almost periodic type solutions of difference equations by means of exponential dichotomy and trichotomy of linear difference equations.     

On the corona theorem for almost periodic functions

LetAP ⁺ (R ⁿ ) denote the Banach algebra of all continuous almost periodic functions onR ⁿ whose Bohr-Fourier spectrum is contained in an additive semi-group [0, ) ⁿ . We show that the maximal ideal space ofAP ⁺ (R ⁿ ) may have a nonempty corona and we characterize all for which the corona is empty. Analogous results are established for algebras of almost periodic functions with absolutely convergent Fourier series.     

Sets of values of equivalent almost periodic functions

The Ramanujan Journal - This paper presents a full generalization of Bohr’s equivalence theorem for the case of almost periodic functions, which improves a recent result that was uniquely...     

Approximations of almost periodic functions by entire ones

In this paper we propose a new proof of the well-known theorem by S. N. Bernstein, according to which among entire functions which give on (−∞,∞) the best uniform approximation of order σ of periodic functions there exists a trigonometric polynomial whose order does not exceed σ. We also prove an analog of this Bernstein theorem and an analog of the Jackson theorem for uniform almost periodic functions with an arbitrary spectrum.