Generalized Hörmander's conditions, commutators and weights

We present a general framework to deal with commutators of singular integral operators with BMO functions. Hörmander type conditions associated with Young functions are assumed on the kernels. Coifman type estimates, weighted norm inequalities and two-weight estimates are considered. We give applications to homogeneous singular integrals, Fourier multipliers and one-sided operators.     

Spectral Analysis for Linear Pencils N — λP of Ordinary Differential Operators

Boundary eigenvalue problems for linear pencils N — λ of two ordinary differential operators are studied where P is of lower order than N. In a suitable scale of subspaces of Sobolev spaces and spaces of continuously differentiable functions results on minimality and basis properties of the eigenfunctions and associated functions are proved, including explicit formulas for the Fourier coefficients. As an application the Orr - Sommerfeld equation is considered.     

Decomposition theorems and approximation by a ``floating" system of exponentials

The main problem considered in this paper is the approximation of a trigonometric polynomial by a trigonometric polynomial with a prescribed number of harmonics. The method proposed here gives an opportunity to consider approximation in different spaces, among them the space of continuous functions, the space of functions with uniformly convergent Fourier series, and the space of continuous analytic functions. Applications are given to approximation of the Sobolev classes by trigonometric polynomials with prescribed number of harmonics, and to the widths of the Sobolev classes. This work supplements investigations by Maiorov, Makovoz and the author where similar results were given in the integral metric.

     

Fourier�CHaar coefficients of continuous functions

Properties of Fourier?CHaar coefficients of continuous functions are studied. It is established that Fourier?CHaar coefficients of continuous functions are monotonic in a certain sense for convex functions. Questions of quasivariation of Fourier?CHaar coefficients of continuous functions are also considered.     

Nonharmonic Fourier series and its applications

Nonharmonic Fourier series with coefficients in certain spaces are considered. When we expand functions as nonharmonic Fourier series, we give a relationship between the spaces of coefficients and those of functions. The problem of controllability for a one-dimensional vibrating system is considered as an application.     

<Emphasis Type="Italic">p</Emphasis>-Adic Mikusinski calculus and its applications to the fourier and the mahler expansions of locally constant functions

We consider the p-adic counterpart of Mikusinski’s operational calculus based on the algebra C(ℤ _p ) of continuous functions on ℤ _p taking values in ℂ _p and equipped with the discrete Laplace convolution. Elements of the field (hyperfunctions) corresponding to shift operators, difference operators, and the indefinite sum operator are considered. A notion of p-adic exponent is generalized. Applications to the Fourier and the Mahler expansions of the indicator function of a ball and the convolution of two indicator functions are provided. Two ways of applying the p-adic analog of Mikusinski’s operational calculus lead us to the Fourier expansion for the fractional part of a p-adic number.     

Hecke-Siegel's pull-back formula for the Epstein zeta function with a harmonic polynomial

In this paper, we discuss the generalization of the Hecke's integration formula for the Epstein zeta functions. We treat the Epstein zeta function as an Eisenstein series come from a degenerate principal series. For the Epstein zeta function of degree two, Siegel considered the Hecke's formula as the constant term of a certain Fourier expansion of the Epstein zeta function and obtained the other Fourier coefficients as the Dedekind zeta functions with Grössencharacters of a real quadratic field. We generalize this Siegel's Fourier expansion to more general Eisenstein series with harmonic polynomials. Then we obtain the Dedekind zeta functions with Grössencharacters for arbitrary number fields.     

Fourier Restriction of H<Superscript>1</Superscript> Functions on Polynomial Surfaces

We prove a Fourier restriction theorem for H¹ functions, which is a generalization of a classical inequality of Hardy. As a consequence of our result, we explicitly obtain a class of BMO functions.     

О сходимости в метрике L четных тригонометрических интерполяционных полиномов по равноотстояшим уэлам

Even, 2π-periodic, continuous for all x ≠ 2πn, n = 0, 1, …, functions, represented by Fourier series are considered. The question of convergence in the metric L of the trigonometric interpolation cosine polinomials of such functions with convex, quasiconvex, monotone and quasimonotone Fourier coefficients is investigated.     

Absolute convergence of fourier series of almost-periodic functions

We present necessary and sufficient conditions for the absolute convergence of the Fourier series of almost-periodic (in the sense of Besicovitch) functions when the Fourier exponents have limit points at infinity or at zero. The structural properties of the functions are described by the modulus of continuity or the modulus of averaging of high order, depending on the behavior of the Fourier exponents. The case of uniform almost-periodic functions of bounded variation is considered.     

Sharp estimates for the convergence rate of Fourier series of complex variable functions in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(D, <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">z</Emphasis>))

Sharp estimates are derived for the convergence rate of Fourier series in terms of orthogonal systems of functions for certain classes of complex variable functions, and the Kolmogorov N-widths of these classes are determined. These issues find applications in numerical analysis methods.     

Inversion formulas for the short-time Fourier transform

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L₂. In the present article the so-called θ-summability (with a function parameter θ) is considered which induces norm convergence for a large class of function spaces. Under some conditions on θ we prove that the summation of the short-time Fourier transform of ƒ converges to ƒ in Wiener amalgam norms, hence also in the L_p sense for L_p functions, and pointwise almost everywhere.     

Inversion of the Short-Time Fourier Transform Using Riemannian Sums

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L₂ and modulation spaces. In the present note the Riemannian sums of the inverse short-time Fourier transform are investigated. Under some conditions on the window functions we prove that the Riemannian sums converge to f in the modulation spaces and inWiener amalgam norms, hence also in the L_p sense.     

Approximation of periodic functions by Steklov averages in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

In the space L₂ of periodic functions, we establish exact (in the sense of constants) estimates from below for the deviation of the Steklov functions of the first and second order in terms of the modulus of continuity of the second order. Similar results are also established for even continuous periodic functions with nonnegative Fourier coefficients in the space C. Bibliography: 5 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 79–90     

Uniform convergence of expansions into a multiple trigonometric Fourier series or a Fourier integral

Questions of convergence almost everywhere of expansions into a multiple trigonometric Fourier series or a Fourier integral are studied for functions from L_p, p≥1, with summation over rectangles. Moreover, a “generalized localization principle,” understood in the sense of convergence almost everywhere, is considered in the paper.     

Continuous functions with universally divergent Fourier series on small subsets of the circle

It is shown that quasi all continuous functions on the unit circle have the property that, for many small subsets E of the circle, the partial sums of their Fourier series considered as functions restricted to E exhibit certain universality properties.     

Absolute convergence of multiple Fourier series revisited

In a recent paper [4], Gogoladze and Meskhia generalized the classical results of Bernstein, Szász, Zygmund and others related to absolute convergence of single trigonometric Fourier series. Our aim is to extend these results from single to multiple Fourier series. To this effect, we introduce the notions of multiplicative moduli of continuity and that of smoothness. Multiplicative Lipschitz classes of functions in several variables, and functions of bounded s-variation in the sense of Vitali are also considered.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for bi-parameter and bilinear Fourier integral operators

Fourier integral operators play an important role in Fourier analysis and partial differential equations. In this paper, we deal with the boundedness of the bilinear and bi-parameter Fourier integral operators, which are motivated by the study of one-parameter FIOs and bilinear and bi-parameter Fourier multipliers and pseudo-differential operators. We consider such FIOs when they have compact support in spatial variables. If they contain a real-valued phase φ(x, ξ, η) which is jointly homogeneous in the frequency variables ξ, η, and amplitudes of order zero supported away from the axes and the antidiagonal, we can show that the boundedness holds in the local-L² case. Some stronger boundedness results are also obtained under more restricted conditions on the phase functions. Thus our results extend the boundedness results for bilinear and one-parameter FIOs and bilinear and bi-parameter pseudo-differential operators to the case of bilinear and bi-parameter FIOs.     

On a high order numerical method for functions with singularities

By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various types of singularities of importance in applications are considered. Relations between the discrete and the continuous Fourier series for the singular functions are established. Of particular interest are piecewise smooth functions, for which various important applications are indicated, and for which numerous numerical results are presented.

     

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.