On the absolute summability of Fourier series of almost-periodic besicovitch functions

For almost-periodic Besicovitch functions whose spectrum has a limit point only at infinity, we establish criteria for the absolute Cesàro summability of their Fourier series of order greater than –1.     

双尺度收敛和拟线性抛物方程的均匀化

利用Ｆｏｕｒｉｅｒ分析的某些技巧，得到殆周期国数的一些新性质，在此基础上将均匀化理论中的双尺度收敛方法推广到殆周期振荡的情形．然后利用一些新技巧讨论了殆周期振荡的拟线性方程的均匀化问题，得到了相应的弱收敛和纠正子结果．     

Summability of Fourier series for almost-periodic functions on locally compact Abelian groups

Some results concerning the summability of Fourier series of continuous 2π-periodic functions are generalized for the case of almost-periodic functions defined on locally compact Abelian groups.     

Two-scale Convergence and Homogenization for a Class of Quasilinear Elliptic Equations

By use of Fourier analysis techniques, we obtain some new properties of the almost-periodic functions and extend the two-scale convergence method in the homogenization theory to the case of almost-periodic oscillations. Then, we use some new techniques to study the homogenization for quasilinear elliptic equations with almostperiodic coefficients: div a(x,x/ε, u, Du) = f(x) in Ω and obtain the weak convergence and corrector result.     

Об абсолютной суммируемости рядов Фурье почти-периодических функций

In the article we establish some criteria of absolute summability of Fourier series of almost periodic functions in the Besicovitch space. We consider the case where the Fourier exponents have a limit point at infinity.     

Spaces of piecewise-continuous almost-periodic functions and of almost-periodic sets on the line. I

In connection with the study of piecewise-continuous almost-periodic functions we introduce the notion of a countable almost-periodic number set. We investigate various properties of it: In particular, we prove that the space of almost-periodic sets is closed with respect to the operation of free union.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1613–1619, December, 1991.     

Fourier Integrals, Special Functions, and the Semicontinuity Phenomenon

For a real weighted homogeneous hypersurface germ, we consider elliptic deformations and related special functions. Singularities of these special functions are characterized by some rational numbers called energy exponents. We apply the residue mapping to the corresponding Fourier integrals and give a geometric interpretation of the energy exponents in the terms of the volume of the associated Lagrangian manifold. The energy exponents are calculated for a series of examples. Two conjectures concerning the energy exponents are discussed.     

Approximation of the functions in weighted Lebesgue spaces with variable exponent

In the present work, we investigate the approximation problems of the functions by Fejér, and Zygmund means of Fourier trigonometric series in weighted Lebesgue spaces with variable exponents and of the functions by Fejér and Abel–Poisson sums of Faber series in weighted Smirnov classes with variable exponents defined on simply connected domains with a Dini-smooth boundary of the complex plane.     

Abstract differential equations with almost-periodic solutions

In this paper we present some quite simple results concerning almost-periodic solutions of abstract differential equations. We start with a general proposition about linear equations, then we establish some new facts about bounded or relatively compact solutions which become almost-periodic; finally, we study “separation from 0 properties” of non-trivial almost-periodic solutions for equations with bounded or unbounded operator coefficients.     

Holomorphic Almost-Periodic Functions

The present paper gives a survey of up-to-date results of holomorphic almost-periodic functions and mappings in one and several complex variables.     

Fourier coefficients of functions with a given modulus of continuity

In this paper, we study the role of the convexity condition for the modulus of continuity in the problem of finding an upper bound for the Fourier coefficients taken over the class of functions with a given modulus of continuity. Also, we solve the problem of the Fourier coefficients for the Rademacher system.     

Finite-automaton transformations of strictly almost-periodic sequences

Different versions of the notion of almost-periodicity are natural generalizations of the notion of periodicity. The notion of strict almost-periodicity appeared in symbolic dynamics, but later proved to be fruitful in mathematical logic and the theory of algorithms as well. In the paper, a class of essentially almost-periodic sequences (i.e., strictly almost-periodic sequences with an arbitrary prefix added at the beginning) is considered. It is proved that the property of essential almost-periodicity is preserved under finite-automaton transformations, as well as under the action of finite transducers. The class of essentially almost-periodic sequences is contained in the class of almost-periodic sequences. It is proved that this inclusion is strict.     

Global attractivity of almost-periodic solution in a model of hematopoiesis with feedback control

By means of properties of almost-periodic system, sufficient conditions are obtained for the existence and global attractivity of a positive almost-periodic solution for an almost-periodic model of hematopoiesis with feedback control.     

Some remarks concerning the Fourier transform in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> (ℝ)

Two useful estimates are proved for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus.     

On differential operators with Bohr-Neugebauer type property

Let B be a bounded linear operator of a Banach space X into itself. If the differential operator $\text{(}\text{d}\text{dt}\text{) ? B}$ has a property more general than Bohr-Neugebauer property for Bochner almost-periodic functions, then any Stepanov-bounded solution of the differential equation $\text{(}\text{d}\text{dt}\text{) u(t) ? Bu(t) = g(t)}$ is also almost-periodic, with g(t) being continuous and Stepanov almost-periodic.     

Proof of the favard theorem on the existence of almost-periodic solution for an arbitrary Banach space

The well-known Favard-Amerio theorem on the existence of an almost-periodic solution of a linear equation is based on the geometry of a uniformly convex space, since the almost-periodic solution is found by the minimax condition. In the present note an essentially different method for finding the almost-periodic solution is developed, which enables us to prove the Favard-Amerio theorem for an arbitrary Banach space.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 121–126, January, 1978.     

On some estimates for best approximations of bivariate functions by Fourier–Jacobi sums in the mean

Some problems in computational mathematics and mathematical physics lead to Fourier series expansions of functions (solutions) in terms of special functions, i.e., to approximate representations of functions (solutions) by partial sums of corresponding expansions. However, the errors of these approximations are rarely estimated or minimized in certain classes of functions. In this paper, the convergence rate (of best approximations) of a Fourier series in terms of Jacobi polynomials is estimated in classes of bivariate functions characterized by a generalized modulus of continuity. An approximation method based on “spherical” partial sums of series is substantiated, and the introduction of a corresponding class of functions is justified. A two-sided estimate of the Kolmogorov N-width for bivariate functions is given.     

On Some Almost-Periodicity Problems in Various Metrics

The Bohr-type and the Bochner-type definitions for almost periodic functions are examined in various metrics (Stepanov, Weyl and Besicovitch). The correct definitions of Besicovitch-like multifunctions are given. Weak almost-periodic solutions are proved for differential equations and inclusions. This problem is also discussed as a fixed-point problem in function spaces.     

Convergence rate estimates for “spherical” partial sums of double Fourier series

The convergence of Fourier double series of 2π-periodic functions from the space $\mathbb{L}_2$ is analyzed. The convergence rate of spherical partial sums of a double Fourier series is estimated for some classes of functions characterized by a generalized modulus of continuity.     

Congruences for coefficients of level 2 modular functions with poles at 0

We give congruences modulo powers of 2 for the Fourier coefficients of certain level 2 modular functions with poles only at 0, answering a question posed by Andersen and the first author. The congruences involve a modulus that depends on the binary expansion of the modular form’s order of vanishing at \(\infty \).