ALMOST PERIODIC FUNCTIONS AND PARTIAL DIFFERENTIAL OPERATORS
Corollary: There exists uniformly almost-periodic functions with an arbitrary countable set of Fourier exponents. If particular, the Fourier exponents of a uniformly almost-periodic function may have finite limit points or may even be everywhere dense. Other definitions of almost-periodic functions of the above classes rely on the concept of an almost-period and generalizations thereof.
Besides the concept of closure or that of an almost-period, the concept of a translation can also be used for the definition of almost-periodic functions. Thus, a function is uniformly almost-periodic if and only if every infinite sequence of functions where the translation numbers are arbitrary real numbers, contains a uniformly convergent subsequence. This definition serves as a starting point in considering almost-periodic functions on groups.
The main results in the theory of almost-periodic functions remain valid if one considers the concept of a generalized translation. Other generalizations are possible and useful: almost-periodic functions with values in an -dimensional space or in a Banach or metric space, and analytic or harmonic almost-periodic functions.
The theory of almost-periodic functions was initiated by H.
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Bohr, who developed the notion of a uniformly almost-periodic function in his study of Dirichlet series. This original definition was the one using -almost-periods and the approximation theorem was one of the main achievements of the theory. For a modern approach see [a4] , Sect. As to the equivalence of the approach starting from a certain structural property that is a generalization of pure periodicity, and on the other hand, the approach starting from approximation by trigonometric polynomials also in the definition of the various classes of generalized almost-periodic functions , see  , Chapt.
An interesting generalization not mentioned above are the Levitan almost-periodic functions see [a5] , where they are called -almost periodic functions. A new approach to the theory of uniformly almost-periodic functions is given in [a1] ; this has lead to the study of so-called almost-automorphic functions [a9] , a class of functions closely related to the Levitan almost-periodic functions mentioned above [a7]. Outside the realm of harmonic analysis an important application of uniformly almost-periodic functions lies in the theory of differential equations; see e.
In this context the theory of dynamical systems cf. Dynamical system is relevant and, in particular, the study of various types of almost-periodic and or recurrent motions, see [a2] , [a3] and [a6] beware of conflicting terminology in the literature.
As a replacement for  , [a11] may be used, which is of the same flavour as . Log in. Namespaces Page Discussion. Views View View source History.
Almost Periodic Functions: Second Edition
Jump to: navigation , search. Here is the set of continuous bounded functions on the real line with the metric and and for are the sets of functions that are measurable and whose -th powers are integrable on every finite interval of the real line, the metrics being Let be the set of trigonometric polynomials where the are arbitrary real numbers and the are complex coefficients, and let the symbol denote the closure of in. Then If and , then For every -a.
With a function in any of the classes defined above one can associate its Fourier series For -a. There is also a uniqueness theorem: If two functions and have the same Fourier series, then In particular, for uniformly almost-periodic functions the uniqueness theorem states that for Stepanov almost-periodic periodic functions: almost-everywhere. A consequence of the definition of the classes of almost-periodic functions through the concept of closure is the approximation theorem: For every almost-periodic function from or or and every there is a finite trigonometric polynomial in , satisfying the inequality The approximation theorem may serve as a starting point of the definition of various classes of almost-periodic functions.
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References  H. Besicovitch, "Almost periodic functions" , Cambridge Univ. Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. American Institute of Mathematical Sciences. Previous Article Condensing operators and periodic solutions of infinite delay impulsive evolution equations. A periodic and diffusive predator-prey model with disease in the prey. Keywords: Neutral functional dynamic equation , Stepanov-almost periodic function , the contraction mapping principle , time scales.
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Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales. References:  B. Google Scholar  L. Google Scholar  J. Google Scholar  J. Google Scholar  S. Google Scholar  S. Google Scholar  S. Google Scholar  M.
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