In the case of stepanov functions we need a natural restriction on the size of the flux, while for besicovitch solutions certain limitations on the. Strict convexity, besicovitchorlicz space, almost periodic functions 1. Scattering from wm metallic profiles is presented in a study by savaidis et al. Composition theorems of stepanov almost periodic functions. Norm inequalities for the fourier coefficients of some. On the strict convexity of the besicovitchorlicz space of. It provides the essential foundations for the theory as well as the basic facts relating to almost periodicity. The existence and uniqueness of a generalized entropy solution in the class of besicovitch almost periodic functions is proved for the cauchy problem for a multidimensional inhomogeneous quasilinear equation of the first order. Besicovitch remarked that generalization to a more or less general class of almostperiodic functions seemed to be extremely unlikely, and adduced as evidence for this assertion the fact that smoothness of almostperiodic functions did not in general seem to enhance convergence. Besicovitch developed his theory in, rather than in. In addition, we apply our composition theorem to study the existence and uniqueness of pseudo almost periodic solutions to a class of abstract semilinear evolution equation in a banach space. If one quotients out a subspace of null functions, it can be identified with the space of l p functions on the bohr compactification of the reals. Then you can start reading kindle books on your smartphone, tablet, or computer no kindle device required. Almostperiodic function encyclopedia of mathematics.
On generalized almost periodic functions besicovitch. Once bohr established his fundamental theorem, he was able to show that any continuous almost periodic function is the limit of a uniformly convergent sequence of trigonometric polynomials. By this method, the authors establish the existence of generalized solutions and in the quasi periodic case, they prove that these solutions are classical. The fundamental tool for the proof of the main theorem is the hausdor. Bohrs equivalence relation in the space of besicovitch. Feffermans embedding of a charge space in a measure space allows us to apply standard interpolation theorems to prove norm inequalities for besicovitch almost periodic functions. As is implicit in the article, for each there is a class of almost periodic functions, denoted by. This yields an analogue of paleys inequality for the fourier coefficients of periodic functions. We deal with the quasiperiodic solutions of the following secondorder hamiltonian systems, where, and we present a new approach via variational methods and minmax method to obtain the existence of quasiperiodic solutions to the above equation.
Oct 28, 2003 the schrodinger equation appropriate to a potential described by an almost periodic function in the sense of representation theory is examined. Spaces of holomorphic almost periodic functions on a strip. On generalized almost periodic functions besicovitch 1926. Bumps of potentials and almost periodic oscillations blot, j. Limit theorems about almost periodic functions 3 our approach is di. Variational methods for almost periodic solutions of a class. Norm inequalities for the fourier coefficients of some almost. Here, too, are a number of monographs on the subject, most notably by l. In this paper, we consider the quasiperiodic solutions of the following secondorder hamiltonian system.
Structure theorem for level sets of multiplicative. Entire functions of exponential type, almost periodic in besicovitchs sense on the real hyperplane authors. Jul 11, 2018 based on bohrs equivalence relation which was established for general dirichlet series, in this paper we introduce a new equivalence relation on the space of almost periodic functions in the sense of besicovitch, \b\mathbb r,\mathbb c\, defined in terms of polynomial approximations. The cauchy problem for a firstorder quasilinear equation.
On the fractional fourier and continuous fractional wave. The concept was first studied by harald bohr and later generalized by vyacheslav stepanov, hermann weyl and abram samoilovitch besicovitch, amongst others. Fractals and useful information to identify their almostperiodic behavior can be found in works by mandelbrot 1983, falconer 1990, voss 1985, and berry and lewis 1980. Function spaces with bounded l p means and their continuous functionals picardello, massimo a. Recently, in 1, 2, diagana introduced the concept of stepanovlike pseudoalmost periodicity, which is a generalization of the classical notion of pseudoalmost periodicity, and established some properties for stepanovlike pseudoalmost periodic functions.
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, welldistributed almost periods. Robust almost periodic dynamics for interval neural networks with mixed timevarying delays and discontinuous. Besicovitch considered this class in the context of the lebesgue lpspaces. Fractional hankel and bessel wavelet transforms of almost. Besicovitch remarked that generalization to a more or less general class of almost periodic functions seemed to be extremely unlikely, and adduced as evidence for this assertion the fact that smoothness of almost periodic functions did not in general seem to enhance convergence. For further knowledge on almost periodic functions we refer the readers to the books 5,4,9. The authors use a variational method on a hilbert space of besicovitch almost periodic functions which looks like a sobolev space. Besicovitch 1932 describes almost periodic functions. Almost periodic functions paperback january 1, 1954 by a. Distinguished as a pure mathematician, particularly for his researches in the theory of functions of a real variable, the theory of analytic functions, and the theory of almost periodic functions.
We provide new variational settings to study the a. Representations of almost periodic pseudodifferential. Permanence and almost periodic solutions of a discrete ratiodependent leslie system with time delays and feedback controls yu, gang and lu, hongying, abstract and applied analysis, 2012. Structure theorem for level sets of multiplicative functions. In the case of stepanov almostperiodic functions the proof is based on a detailed variational analysis of a linear inverse problem, while in the besicovitch setting the proof follows by a precise analysis in wavenumbers. The cauchy problem for a firstorder quasilinear equation in. Almost periodicity and general trigonometric series. Almost periodic functions hardcover january 1, 1954 by abram samoilovitch besicovitch author. Besicovitch almostperiodic functions encyclopedia of. Besicovitch s candidacy for the royal society reads. Floquet theorem is generalized to almost periodic functions and some properties of the density of states and spatial localization of the electrons are obtained. Urbanik 19 has shown for r that if a hartman almost periodic function f is in mp for some p 1, then. A lmost periodic functions stepanov, weyl and besicov.
The space b p of besicovitch almost periodic functions for p. Because of this approach some measure theoretical problems arose whose solution seems to be hard. For these equations, we explicitly find an oscillation constant. Moreover, diagana studied the existence of pseudoalmost periodic solutions to the abstract semilinear evolution equation. Also a norm equality of this signal is given using the continuous fractional wave packet. The asteroid 16953 besicovitch is named in his honour. Entire functions of exponential type, almost periodic in. We denote by bp the space of generalized pbesicovitch almost periodic functions or sequences, i.
We construct frame decompositions for almost periodic functions using these two transforms. The class of boars i, 2, 3 almost periodic functions may be considered from two different points of view. As is implicit in the article, for each there is a class of almostperiodic functions, denoted by. By continuing to use this site you agree to our use of cookies. We state the fractional fourier transform and the continuous fractional wave packet transform as ways for analyzing persistent signals such as almost periodic functions and strong limit power signals. Almostperiodic multipliers almostperiodic multipliers bruno, giordano. We denote by apr the set of all almost periodic functions in the sense of bohr on r. If the coefficients are replaced by constants, our main result concerning the conditional oscillation reduces to the classical one. Besicovitch almost periodic solutions for a class of second. A scale of almost periodic functions spaces corduneanut, c. General references may be found under almostperiodic function.
Almostperiodic multipliers, acta applicandae mathematicae. The moving wall represents the time period between the last issue available in jstor and the most recently published issue of a journal. Almost periodic functions hardcover january 1, 1954. Besicovitch almost periodic solutions for a class of. Next, we use our results to construct a unique almost periodic solution to the so called lerays problem concerning 3d fluid motion in two semiinfinite cylinders connected by a bounded reservoir. Finally, observe that f, as a product of two besicovitch rationally almost periodic functions, is itself besicovitch rationally almost periodic. Oscillation of halflinear differential equations with.
We obtain results on the structure of the set of the a. The first part of the article deals with, the rest is more general. Some qualitative aspects of band structure are discussed. Besicovitchs candidacy for the royal society reads.
Two of the representations are shown to be unitarily equivalent for. Bohrs theory of analytic almost periodic functions of a complex variable, their dirichlet series and their behaviour in and on the boundary of a. From this, we show that in an important subspace \b2\mathbb r,\mathbb c\subset b\mathbb r. In rare instances, a publisher has elected to have a zero moving wall, so their current issues are available. The notions of almost periodicity in the sense of weyl and besicovitch of the order p are extended to holomorphic functions on a strip. The main objective of this paper is to study the hankel, fractional hankel, and bessel wavelet transforms using the parseval relation.
Strict convexity, besicovitch orlicz space, almost periodic functions 1. We investigate secondorder halflinear differential equations with asymptotically almost periodic coefficients. Based on bohrs equivalence relation which was established for general dirichlet series, in this paper we introduce a new equivalence relation on the space of almost periodic functions in the sense of besicovitch, \b\mathbb r,\mathbb c\, defined in terms of polynomial approximations. Introduction the class of bohrs almost periodic functions denoted by u. We construct a generalized frame and write new relations and inequalities using almost periodic functions, strong limit. In previous papers the proofs were based on the ergod theorem. Enter your mobile number or email address below and well send you a link to download the free kindle app. Almost periodic functions and the theory of disordered systems. Robust almost periodic dynamics for interval neural networks with mixed timevarying delays and. Fractals and useful information to identify their almost periodic behavior can be found in works by mandelbrot 1983, falconer 1990, voss 1985, and berry and lewis 1980. Gibson, onthe almost periodicity oftrigonometric polynomials inconstructive mathematics, nederl. General references may be found under almost periodic function. Pdf almost periodic functions, bohr compactification. We extend shu and xu 2006 variational setting for periodic solutions of nonlinear neutral delay equation to the almost periodic settings.
B p the space of besicovitch almost periodic functions on g see also 11 for. Applications to partial differential equations are also given. The first half of the book lays the groundwork, noting the basic properties of almost periodic functions, while the second half of this work addresses applications whose main emphasis is on the solvability of ordinary or partial differential equations in the class of almost periodic functions. The bohr compactification is shown to be the natural setting for studying almost periodic functions. We establish a composition theorem of stepanov almost periodic functions, and, with its help, a composition theorem of stepanovlike pseudo almost periodic functions is obtained. The besicovitch space of almost periodic functions, b2r is the closure of trigonometric polynomials of the form n a s n. Pdf almost periodic functions, bohr compactification, and. Variational methods for almost periodic solutions of a. Scattering from wm metallic profiles is presented in. Danilov, measurevalued almost periodic functions and almost periodic selections of multivalued maps. Besicovitch 1932 describes almostperiodic functions.
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