The besicovitch space of almost periodic functions, b2r is the closure of trigonometric polynomials of the form n a s n. If the coefficients are replaced by constants, our main result concerning the conditional oscillation reduces to the classical one. 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. Because of this approach some measure theoretical problems arose whose solution seems to be hard. We construct frame decompositions for almost periodic functions using these two transforms. The moving wall represents the time period between the last issue available in jstor and the most recently published issue of a journal. A scale of almost periodic functions spaces corduneanut, c. On lerays problem for almost periodic flows internet archive. Representations of almost periodic pseudodifferential. Moreover, diagana studied the existence of pseudoalmost periodic solutions to the abstract semilinear evolution equation. Besicovitch almost periodic solutions for a class of second. Also a norm equality of this signal is given using the continuous fractional wave packet. This yields an analogue of paleys inequality for the fourier coefficients of periodic functions. The first part of the article deals with, the rest is more general.
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. 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. Almostperiodic multipliers almostperiodic multipliers bruno, giordano. Some qualitative aspects of band structure are discussed.
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. We denote by bp the space of generalized pbesicovitch almost periodic functions or sequences, i. The cauchy problem for a firstorder quasilinear equation in. Spaces of holomorphic almost periodic functions on a strip. Besicovitch almost periodic solutions for a class of. Robust almost periodic dynamics for interval neural networks with mixed timevarying delays and discontinuous. Almost periodic functions and the theory of disordered systems.
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. Scattering from wm metallic profiles is presented in. For further knowledge on almost periodic functions we refer the readers to the books 5,4,9. Strict convexity, besicovitchorlicz space, almost periodic functions 1. Finally, observe that f, as a product of two besicovitch rationally almost periodic functions, is itself besicovitch rationally almost periodic. Entire functions of exponential type, almost periodic in besicovitchs sense on the real hyperplane authors. 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. Limit theorems about almost periodic functions 3 our approach is di.
B p the space of besicovitch almost periodic functions on g see also 11 for. We study three representations of such algebras, one of which was introduced by coburn, moyer and singer and the other two inspired by results in probability theory by gladyshev. Structure theorem for level sets of multiplicative. For these equations, we explicitly find an oscillation constant. Then you can start reading kindle books on your smartphone, tablet, or computer no kindle device required. 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. Besicovitch s candidacy for the royal society reads.
In this paper, we consider the quasiperiodic solutions of the following secondorder hamiltonian system. Almost periodic functions hardcover january 1, 1954 by abram samoilovitch besicovitch author. Structure theorem for level sets of multiplicative functions. By this method, the authors establish the existence of generalized solutions and in the quasi periodic case, they prove that these solutions are classical. Almostperiodic multipliers, acta applicandae mathematicae. Almost periodic functions hardcover january 1, 1954. Variational methods for almost periodic solutions of 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. Norm inequalities for the fourier coefficients of some almost. Fractional hankel and bessel wavelet transforms of almost.
Pdf almost periodic functions, bohr compactification. Danilov, measurevalued almost periodic functions and almost periodic selections of multivalued maps. Oct 28, 2003 the schrodinger equation appropriate to a potential described by an almost periodic function in the sense of representation theory is examined. It provides the essential foundations for the theory as well as the basic facts relating to almost periodicity. Besicovitchs candidacy for the royal society reads. 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. General references may be found under almost periodic function. 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. Besicovitch considered this class in the context of the lebesgue lpspaces.
In previous papers the proofs were based on the ergod theorem. Besicovitch 1932 describes almost periodic functions. Function spaces with bounded l p means and their continuous functionals picardello, massimo a. 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. The space b p of besicovitch almost periodic functions for p. Scattering from wm metallic profiles is presented in a study by savaidis et al. In six structured and selfcontained chapters, the author unifies the treatment of various classes of almost periodic functions, while uniquely addressing oscillations and waves in the almost periodic case. 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. 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. On generalized almost periodic functions besicovitch. Bohrs equivalence relation in the space of besicovitch. 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.
Strict convexity, besicovitch orlicz space, almost periodic functions 1. In six structured and selfcontained chapters, the author unifies the treatment of various classes of almost periodic functions, while uniquely addressing oscillations and waves in. Variational methods for almost periodic solutions of a class. 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. In rare instances, a publisher has elected to have a zero moving wall, so their current issues are available. On generalized almost periodic functions besicovitch 1926. Pdf almost periodic functions, bohr compactification, and. We extend shu and xu 2006 variational setting for periodic solutions of nonlinear neutral delay equation to the almost periodic settings. Entire functions of exponential type, almost periodic in.
Introduction the class of bohrs almost periodic functions denoted by u. 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. 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. The authors use a variational method on a hilbert space of besicovitch almost periodic functions which looks like a sobolev space. The asteroid 16953 besicovitch is named in his honour. As is implicit in the article, for each there is a class of almost periodic functions, denoted by. Here, too, are a number of monographs on the subject, most notably by l.
Almost periodic functions paperback january 1, 1954 by a. Applications to partial differential equations are also given. The cauchy problem for a firstorder quasilinear equation. Floquet theorem is generalized to almost periodic functions and some properties of the density of states and spatial localization of the electrons are obtained. A lmost periodic functions stepanov, weyl and besicov.
Norm inequalities for the fourier coefficients of some. Bohrs theory of analytic almost periodic functions of a complex variable, their dirichlet series and their behaviour in and on the boundary of a. 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. We denote by apr the set of all almost periodic functions in the sense of bohr on r.
Besicovitch almostperiodic functions encyclopedia of. We construct a generalized frame and write new relations and inequalities using almost periodic functions, strong limit. We provide new variational settings to study the a. General references may be found under almostperiodic function. Almostperiodic function encyclopedia of mathematics. As is implicit in the article, for each there is a class of almostperiodic functions, denoted by. We investigate secondorder halflinear differential equations with asymptotically almost periodic coefficients. 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. 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. On the strict convexity of the besicovitchorlicz space of. Also a norm equality of this signal is given using the continuous fractional wave packet transform. Besicovitch developed his theory in, rather than in. 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. Oscillation of halflinear differential equations with.
The main objective of this paper is to study the hankel, fractional hankel, and bessel wavelet transforms using the parseval relation. The class of boars i, 2, 3 almost periodic functions may be considered from two different points of view. Bumps of potentials and almost periodic oscillations blot, j. Enter your mobile number or email address below and well send you a link to download the free kindle app. By continuing to use this site you agree to our use of cookies. From this, we show that in an important subspace \b2\mathbb r,\mathbb c\subset b\mathbb r. Robust almost periodic dynamics for interval neural networks with mixed timevarying delays and. 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. 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 fundamental tool for the proof of the main theorem is the hausdor. Urbanik 19 has shown for r that if a hartman almost periodic function f is in mp for some p 1, then. The notions of almost periodicity in the sense of weyl and besicovitch of the order p are extended to holomorphic functions on a strip.
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