Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity

We construct an example of a function from the class $H_1^{\omega ^ * } $ , where $\omega ^ * (t) = \sqrt {\log \log (t^{ - 1} )/\log (t^{ - 1} )} $ , $0 < t \leqslant t_0 $ , whose trigonometric Fourier series is divergent almost everywhere. We obtain sharp integrability conditions for the majorants of the partial sums of trigonometric Fourier series in terms of whether the functions in question belong to the classes $H_1^\omega $ .     

Continuous and Discrete Nonlinear Approximations Based on Fourier Series

Continuous and discrete nonlinear extensions of Fourier seriesare derived by the method of generating functions, and examplesof their use are presented. Gibbs' phenomenon of the continuousnonlinear approximation is investigated.     

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.     

Regularization and Fourier Series for Fredholm Integral Equations of the Second Kind with a Weakly Singular Kernel

To compute an approximate solution, of a weakly singular integral equation, we first regularize the kernel, and then truncate the associated Fourier series. An application to an astrophysical transfer equation is presented.     

Universal Fourier Series

Mathematical Notes -     

On Fourier Series on the Torus and Fourier Transforms

Mathematical Notes - The question of the representability of a continuous function on $$mathbb R^d$$ in the form of the Fourier integral of a finite Borel complex-valued measure on $$mathbb R^d$$...     

Curvelets and Fourier Integral Operators

A recent body of work introduced new tight-frames of curvelets E. Candès, D. Donoho, in: (i) Curvelets – a suprisingly effective nonadaptive representation for objects with edges (A. Cohen, C. Rabut, L. Schumaker (Eds.)), Vanderbilt University Press, Nashville, 2000, pp. 105–120; (ii) http://www.acm.caltech.edu/~emmanuel/publications.html, 2002 to address key problems in approximation theory and image processing. This paper shows that curvelets essentially provide optimally sparse representations of Fourier Integral Operators. To cite this article: E. Candès, L. Demanet, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

两种基本函数项级数系数公式注记

分析部分高等数学教材在推导幂级数和傅里叶级数的系数公式时存在的不足,并从函数逼近论的角度,就此问题提出改进思想和方法.     

傅里叶变换中一个含参变量广义积分的计算

本文利用含参变量广义积分计算中常用的求导和变量替换等技巧等给出了傅里叶变换中一个重要的含参变量广义积分的计算方法.     

Global h Fourier Integral Operators with Complex-Valued Phase Functions

The author considers globally defined h Fourier integral operators(h FIO) with complex-valued phase functions. Symbolic calculusof h FIO is considered and, using a new complex Gauss transform,the composition of h pseudodifferential operators (h PDO) andh FIO is considered. For a self-adjoint h PDO A(h) and h PDOP(h) and Q(h) with compactly supported symbols, the resultsare applied to approximate the kernel of the operator by a single, globally defined h-oscillatoryintegral. 2000 Mathematics Subject Classification 81Q20, 35S30.     

积分区域与原函数的存在性

讨论由Newton-Leibuniz公式引起的积分区域对原函数存在性的影响问题.     

Fourier Series with Linear Splines on an Interval

We construct orthonormal bases of linear splines on a finite interval [a, b] and then we study the Fourier series associated to these orthonormal bases. For continuous functions defined on [a, b], we prove that the associated Fourier series converges pointwisely on (a, b) and also uniformly on [a, b], if it convergences pointwisely at a and b.     

On Lp-boundedness of Fourier Integral Operators

Potential Analysis - In this paper, we get an Lp boundedness of Fourier integral operators with rough amplitude $a\in L^{\infty } S^{m}_{\varrho },~$ and phase $\varphi \in L^{\infty }{\Phi }^{2}$...     

关于Taylor级数与Fourier级数的几点注记

Taylor级数与Fourier级数是两类非常重要的函数项级数,二者在发展与应用背景、展开条件、收敛性和展开的唯一性等方面不尽相同,本文对此作了一些总结与探讨。     

关于Fourier级数的一点推广

以２ｌ为周期的函数ｆ（ｘ）也可看作周期为４ｌ．设ｆ（ｘ）满足 Ｄｉｒｉｃｈｌｅｔ充分条件,按［１］方法展开的以 ２ｌ为周期的 Ｆｏｕｒｉｅｒ 级数和以 ４ｌ为周期的 Ｆｏｕｒｉｅｒ 级数则对应于不同表达式．本文证明了这两种表达形式是一致的     

An Asymptotic Series for the Weber--Schafheitlin Integral

We obtain an asymptotic power series for the Weber--Schafheitlin integral whose coefficients are distributions.     

Pointwise Convergence of Fourier Series

The problem of studying the largest rearrangement invariantspace of functions with almost everywhere convergent Fourierseries is considered. A quasi-Banach space QA of functions isdefined with almost everywhere convergent Fourier series thatstrictly contains Antonov and Soria's spaces (L log L log loglog L and ).     

Almost Everywhere Convergent Fourier Series

We study some properties of the logconvex quasi-Banach space QA defined by Arias-de-Reyna and show several applications to convergence of Fourier series. In particular, we describe the Banach envelope of QA and prove that there exists a Lorentz space strictly bigger than the Antonov space in which the almost everywhere convergence of the Fourier series holds. We also give a necessary condition for a Banach rearrangement invariant space X to be contained in QA. As an application, we show that for some classes of Banach spaces there is no the largest Banach space in a given class which is contained in QA.     

On the Absolute Convergence of Fourier Series

The necessary and sufficient conditions of the absolute convergence of a trigonometric Fourier series are established for continuous 2-periodic functions which in [0, 2] have a finite number of intervals of convexity, and whose nth Fourier coefficients are O((l/n; f)/n), where ( f) is the continuity modulus of the function f.