The Fourier series is known to be a very powerful tool in connection with various problems involving partial differential equations.Ī graph of periodic function f(x) that has a period equal to L exhibits the same pattern for every L unit along the x-axis so that f(x + L) is equal to f(x) for each value of x. The Fourier series can be defined as a way of representing a periodic function (possibly infinite) as a sum of sine functions and cosine functions. So, let’s begin with what the Fourier series is. In this article, we will discuss the Fourier series, formulas, and uses and applications of the Fourier series. Examples of the Fourier series are trigonometric functions like sin x and cos x with period \ and tan x with period \. It can be done by using a process called Fourier analysis. These periodic functions could be analysed into their constituent components (fundamentals and harmonics). For instance, current and voltage in an alternating current circuit. ![]() For instance, the function \(x \mapsto \cos x + \cos (5x)\) is periodic, and this remains true when 5 is replaced by any other rational number.Most of the phenomena that are studied in Engineering and Science are periodic in nature. In fact, the first motivation for the study of almost periodic functions is the set of various ways to combine periodic functions with different periods. One can remark that speaking about Stepanov, Weyl or Besicovitch metrics implicitly means dealing with the related quotient spaces, because otherwise we should rather speak about Stepanov, Weyl or Besicovitch. Some extensions of Bohr’s concept have been introduced, most notably by Besicovitch, Stepanov, Weyl and Eberlein. He showed that these functions include certain earlier generalizations of the notion of almost periodic functions. ![]() In 1962, Bochner defined and studied the almost periodic functions with values in Banach spaces. with respect to this matter, we cite and the references therein. Afterwards, the theory of almost periodic functions was continuously getting established by several mathematicians like Amerio and Prouse, Levitan, Besicovitch, Bochner, von Neumann, Fréchet, Pontryagin, Lusternik, Stepanov, Weyl, etc. The first of these papers dealt with the almost periodic functions of a real variable, while the third one took up the case of a complex variable. ![]() The theory of almost periodic functions was developed in its main features by Bohr as a generalization of pure periodicity in three rather long papers, under the common title ‘ Zur Theorie der fastperiodischen Funktionen’ in 19. Indeed, the prehistory of almost periodicity begins with Esclangon and Bohl. The theory of almost periodic functions has gradually been increased to a comprehensive and extensive theory by the contributions of numerous mathematicians. It is worth noting that the topics dealt with in this paper seem to be of an intrinsic connection with the problem of existence and uniqueness of solutions of differential and difference equations, in both determinist and stochastic cases. In order to make a picture as complete and clear as possible, several illustrating examples and counter-examples are given. We actually introduce eight new classes of asymptotically almost periodic functions and analyze relations between them. The class of asymptotically Weyl-almost periodic functions, introduced in this work, seems to be not considered elsewhere even in the scalar-valued case. We also recollect some basic results regarding equi-Weyl-almost periodic functions and Weyl-almost periodic functions. Special accent is put on the Stepanov generalizations of almost periodic functions and asymptotically almost periodic functions. ![]() In this paper, we review indispensable properties and characterizations of almost periodic functions and asymptotically almost periodic functions in Banach spaces.
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