Almost Everywhere Convergence of Multiple Trigonometric Fourier Series of Functions from Sobolev Classes

Mathematical Notes - In this paper, we study the almost everywhere convergence of spherical partial sums of multiple Fourier series of functions from Sobolev classes. It is proved that almost...     

Convergence almost Everywhere of Certain Partial Sums of Fourier Integrals

Suppose that R goes to infinity through a second-order lacunaryset. Let S_R denote the Rth spherical partial inverse Fourierintegral on R^d. Then S_Rf converges almost everywhere to f, providedthat f satisfies . 2000 Mathematics SubjectClassification 42B15, 42B25, 42B08.     

一类非负函数无穷积分收敛性的刻画

针对无穷积分教学的需要,在利用随机理论的重要概念——失效函数来刻画出非负函数的递减趋于坐标横轴程度的基础上,运用一般推理方法及无穷积分敛散性的比较判别法,得到了一类非负函数的无穷积分敛散性的较为简便而有效的判别方法.     

Distributional Point Values and Convergence of Fourier Series and Integrals

In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier transforms in the following way: If and , and is locally integrable, then distributionally if and only if there exists k such that , for each a > 0, and similarly in the case when is a general distribution. Here means in the Cesaro sense. This result generalizes the characterization of Fourier series of distributions with a distributional point value given in [5] by . We also show that under some extra conditions, as if the sequence belongs to the space for some and the tails satisfy the estimate ,\ as , the asymmetric partial sums\ converge to . We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We apply these results to lacunary Fourier series of distributions.     

Divergence of Spherical General Terms of Double Fourier Series

We prove the following theorem: For arbitrary there exists a nonnegative function such that and

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.     

Numerical Evaluation of a Class of Double Integrals of Oscillatory Functions

A general method is described for the numerical evaluation ofintegrals of the form where, in a typical case, (x, y) is a lengthy polynomial ofdegree 10 in (x, y) and A is the region common to three overlappingcircles, although the method is in no way restricted to suchcases. Illustrative numerical examples are given.     

Approximation of Classes of Analytic Functions by Fourier Sums in Uniform Metric

We find asymptotic equalities for upper bounds of approximations by partial Fourier sums in the uniform metric on classes of Poisson integrals of periodic functions belonging to the unit balls in the spaces L _p , 1 ≤ p ≤ ∞. We generalize the results obtained to the classes of (ψ, )-differentiable (in the sense of Stepanets) functions that admit an analytic extension to a fixed strip of the complex plane. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1079 – 1096, August, 2005.     

Convergence of Series of Fourier Coefficients of p-Absolutely Continuous Functions

In the paper, we generalize the well-known criteria of Bernstein and Stechkin on the absolute convergence in terms of best approximations and moduli of smoothness of continuous functions. We give conditions for the convergence of the series of Fourier coefficients raised to the power in terms of best approximations in the space of p-absolutely continuous functions and in terms of fractional moduli of continuity with respect to this space. We also prove the sharpness of our conditions for 0 < 1 with no restriction and for 1 < 2 under some restrictions.     

Convergence of Double Fourier Series after a Change of Variable

In this paper, we prove that for any compact set there exists a homeomorphism of the closed interval such that for an arbitrary function f the Fourier series of the function F(x,y) = f((x),(y)) converges uniformly on simultaneously over rectangles, over spheres, and over triangles.     

Unconditional Convergence of Fourier Series for Functions of Bounded Variation

This article concerns the unconditional convergence a.e. of Fourier series with respect to general orthonormal systems. We find certain conditions to be satisfied by the functions in the orthonormal system so that the Fourier series of each function of finite variation unconditionally converge a.e. The results are best possible.     

Sets of Uniform Convergence of Fourier Expansions of Piecewise Smooth Functions

For Fourier expansions of piecewise smooth functions associated with an elliptic operator of order m in Rⁿ, the sets of uniform convergence are described.     

Numerical Evaluation of Fourier Integrals

A method is developed for evaluating Fourier integrals of theform A() = ¹_–1f(x) e^fax dx, 0. The method consists of expanding the function f in a seriesof Chebyshev polynomials and expressing the integral A() asa series of the Bessel functionsJ_r+(), r= 0, 1, 2,.... A partialsum A_N() of the series provides an approximant to A(). The principalfeature of the method is that one set of N+1 evaluations off(x) suffices for the calculation of A_N() for all , and alsothe truncation error A()–A_N() is essentially independentof . Numerical tests show that the method is accurate, economicaland reliable. An application to the inversion of Fourier andLaplace transforms is briefly described.     

稳定积分的弱收敛

令{X;X_n≥1}是一列严平稳的随机变量,且其分布F在一个α-稳定分布的吸引场,这里0α1.本文考虑∑_(i=1)~n f_n(β,i/n)(X_i)/(a_n)的弱收敛性.不同于经典意义下的随机过程弱收敛,本文将∑_(i=1)~n∫_n(β,in/)(X_i)/(a_n)看作β变化的随机元,利用点过程收敛方法得到了其弱收敛性.     

Waterman classes and triangular sums of double Fourier series

Let ^a ={nln^a (n+1)}, where a R. The following results are established: For every &fnof ^a BV ((- ]²), the triangular partial sums of its Fourier series are uniformly bounded if a = -1, and converge everywhere if a < -1.For every a>0, there exists &fnof ^a BV ((- ]²) such that the triangular partial sums of its Fourier series are unbounded at the point (0;0).     

Pointwise Behavior of Double Fourier Series of Functions of Bounded Variation

We extend some recent results of S. A. Telyakovskii on the uniform boundedness of the partial sums of Fourier series of functions of bounded variation to periodic functions of two variables, which are of bounded variation in the sense of Hardy. As corollaries, we obtain the classical Parseval formula, the convergence theorem of the series involving the sine Fourier coefficients, and a lower estimate of the best approximation by trigonometric polynomials in the metric of L in a sharpened version. This research was supported by the Hungarian National Foundation for Scientific Research under Grants TS 044 782 and T 046 192.     

Pointwise Behavior of Double Fourier Series of Functions of Bounded Variation

We extend some recent results of S. A. Telyakovskii on the uniform boundedness of the partial sums of Fourier series of functions of bounded variation to periodic functions of two variables, which are of bounded variation in the sense of Hardy. As corollaries, we obtain the classical Parseval formula, the convergence theorem of the series involving the sine Fourier coefficients, and a lower estimate of the best approximation by trigonometric polynomials in the metric of L in a sharpened version.     

Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. II

In normed spaces of functions analytic in the Jordan domain , we establish exact order estimates for the Kolmogorov widths of classes of functions that can be represented in by Cauchy-type integrals along = with densities f(·) such that . Here, is a conformal mapping of onto {w: |w| > 1}, and is a certain subset of infinitely differentiable functions on T = {w: |w| = 1}.     

Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. I

In the Banach space of functions analytic in a Jordan domain , we establish order estimates for the Kolmogorov widths of certain classes of functions that can be represented in by Cauchy-type integrals along the rectifiable curve = and can be analytically continued to or to .