Bochner-Type Property on Spaces of Generalized Almost Periodic Functions
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Bochner-Type Property on Spaces of Generalized Almost Periodic Functions J. M. Sepulcre
and T. Vidal
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 Bochnertype 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. Mathematics Subject Classification. 42A75, 42A16, 42B05, 46Axx, 42Axx. Keywords. Almost periodic functions, Besicovitch almost periodic functions, Stepanov almost periodic functions, Weyl almost periodic functions, Bochner’s theorem, approximation by trigonometric polynomials, exponential sums, Fourier series, Bohr’s equivalence relation.
1. Introduction The theory of almost periodic functions was mainly created during the 1920s by the Danish mathematician H. Bohr (1887–1951). The definition given by Bohr of an almost periodic function is based upon two properly generalized concepts: the periodicity to the so-called almost periodicity, and the periodic distribution of periods to the so-called relative density of almost periods. Specifically, let f (t) be a real or complex function of an unrestricted real variable t, the notion above of almost periodicity involves the fact that f (t) must be continuous, and for every ε > 0, there corresponds a number l = l(ε) > 0 such that each interval of length l contains a number τ satisfying |f (t + τ ) − f (t)| < ε for all t. We will denote as AP (R, C) the space of almost periodic functions in the sense of this definition (Bohr’s condition). The theory of almost periodic functions opened a way to study a wide class of trigonometric series of the general type and even exponential series 0123456789().: V,-vol
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J. M. Sepulcre and T. Vidal
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(in this context, we can cite among others the papers [3–6,8,11]). Furthermore, it has many important applications in problems of ordinary differential equations, dynamical systems, stability theory and partial differential equations (see, for example, recent developments in [10,12,13]). A very important result of this theory is the approximation theorem according to which the class of almost periodic functions AP (R, C) coincides with the class of limit functions of uniformly convergent sequences of trigonometric polynomials of the type a1 eiλ1 t + · · · + an eiλn t
(1)
with arbitrary real exponents λj and arbitrary complex coefficients aj . Moreover, an equivalent definition for AP (R, C), called normality, was provided by S. Bochner (in fact, sometimes it is called the Bochner-type definition) and it makes use of the generalization of the relative compactness, in the sense of uniform convergence,
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