Fourier级数两个收敛定理的关系及反例

Fourier级数的收敛问题一直是很多数学研究者关注的问题,不同的数学分析教材和高等数学教材对收敛定理的表述各不相同。本文通过实例说明这些收敛定理之间不存在包含关系,并进一步说明这些收敛定理都是判别Fourier级数收敛的充分条件,而不是充要条件。     

关于广义Fourier级数的若干结论及其应用

讨论Hilbert空间广义Fourier级数收敛的充分和必要条件,并将相关结果应用于数学分析中具体的Fourier级数上.     

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。     

单位圆周上广义Fourier积分的收敛性

本文研究了复平面单位圆上的广义Fourier积分.利用经典的Fourier分析的结果和Carleson定理,以及复平面上解析函数在高阶导数下直角坐标和极坐标之间的关系,我们得到了前面定义的广义Fourier积分的一个收敛定理,从而推广了直线上经典Fourier积分的收敛结果.     

周期函数Fourier级数展开式的唯一性

以 2 l为周期的函数 f(x)也可看作周期为 2 kl(k=1 ,2 ,3 ,… ) .设 f(x)满足 Dirichlet充分条件 ,[2 ]证明了按 [1 ]方法展开的以 2 l为周期的 Fourier级数和以 4l为周期的 Fourier级数对应的不同表达形式是一致的 .本文则在 [2 ]的基础上 ,进一步证明了按 [1 ]方法展开的以 2 l为周期的 Fourier级数和以 2 kl(k=1 ,2 ,3 ,… )为周期的 Fourier级数对应的表达式的一致性 ,从而得出结论 :任一周期函数 f(x)按 [1 ]方法展开的Fourier级数是唯一的 .     

一类含参数的交错巴塞尔级数的和

利用Fourier级数及其和函数给出了含参数的一类交错巴塞尔级数的和，作为应用得到了一系列的特殊级数的和，最后验证了含参数的交错巴塞尔级数的和即是交错巴塞尔级数的和的推广.     

圆形薄膜振动方程基于Hamilton体系的分离变量法

对极坐标系下的振动方程,首先引入合适的对偶变量将其化为Hamilton系统,再结合Bessel函数及双Fourier级数的性质证明了导出的Hamilton算子矩阵的本征函数系的完备性,最后利用展开定理给出了Hamilton系统的解.     

广义有界变差函数及其对Fourier级数的应用

本文提出了新的广义有界变差函数类φΛBMV及фBMV,фBV分别包含了[1]—[5]各类广义有界变差函数的概念,并用于Fourier级数的理论,讨论了在新的函数类中有关Fourier级数的绝对收敛性与Fourier系数的阶的估计及其按阶的精度改进了[1]、[4]、[5]的有关结果.     

周期函数Fourier级数展开式的唯一性

以2τ为周期的函数f（x）也可看作周期为2kτ（k=1,2,3…）。设f(x)满足Dirichlet充分条件,[2]证明了按[1]方法展开的以2τ为周期的Fourier级数和以4τ为周期的Fourier级数对应的不同表达形式是一致的。本则在[2]的基础上,进一步证明了按[1]方法展开的以2τ为周期的Fourier级数和以2kτ（k=1,2,3,…）为周期的Fourier级数对应的表达式的一致性,从而得出结论：任一周期函数f(x）按[1]方法展开的Fourier级数是唯一的。     

等周问题求解新探索

将实型Fourier级数延拓成复型Fourier级数,利用复型的Fourier级数和数学分析中的格林公式以及参数方程,借助Parseval等式,对等周问题进行求解.     

Complementary Spaces and Multipliers of Double Fourier Series for Functions of Bounded Variation

Complementary spaces for Fourier series were introduced by G. Goes and generalized by M. Tynnov. In this paper we investigate a notion of complementary space for double Fourier series of functions of bounded variation. Various applications are given.     

Spaces of Functions of Generalized Bounded Variation. II: Questions of Uniform Convergence of Fourier Series

Tests are given for uniform convergence of Fourier series for spaces of functions of generalized bounded variation; along with the well-known tests (of Salem–Oskolkov–Young, Chanturiya, and Waterman) we suggest new tests. We show that the Waterman test for uniform convergence of Fourier series is strongest and unimprovable. We present a theorem on exact estimates for the Fourier coefficients for spaces of functions of bounded variation which contains classical results, improves several well-known results, and gives some new results.     

Convergence of multiple fourier series for functions of bounded variation

For functions of bounded variation in the sense of Hardy, we consider the pointwise convergence of the partial sums of Fourier series over a given sequence of bounded sets in the space of harmonics. We obtain sufficient conditions for convergence; necessary and sufficient conditions are obtained for the case in which these sets are convex with respect to each coordinate direction. The Pringsheim convergence of Fourier series in this problem was established by Hardy. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 583–595, April, 1997. Translated by S. A. Telyakovskii and V. N. Temlyakov     

k解析函数的Fourier级数

讨论了复平面上k解析函数的性质,并利用k解析函数的泰勒展开定理研究了k解析函数的Fourier级数,推广了经典的解析函数的Fourier级数理论.     

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.     

Nonlinear inversion of a band-limited Fourier transform

We consider the problem of reconstructing a compactly supported function with singularities either from values of its Fourier transform available only in a bounded interval or from a limited number of its Fourier coefficients. Our results are based on several observations and algorithms in [G. Beylkin, L. Monzón, On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal. 19 (1) (2005) 17–48]. We avoid both the Gibbs phenomenon and the use of windows or filtering by constructing approximations to the available Fourier data via a short sum of decaying exponentials. Using these exponentials, we extrapolate the Fourier data to the whole real line and, on taking the inverse Fourier transform, obtain an efficient rational representation in the spatial domain. An important feature of this rational representation is that the positions of its poles indicate location of singularities of the function. We consider these representations in the absence of noise and discuss the impact of adding white noise to the Fourier data. We also compare our results with those obtained by other techniques. As an example of application, we consider our approach in the context of the kernel polynomial method for estimating density of states (eigenvalues) of Hermitian operators. We briefly consider the related problem of approximation by rational functions and provide numerical examples using our approach.     

A Fourier method for computing the perturbation to a two‐dimensional electric potential by a conductor of arbitrary shape

Introducing an electric conductor into a region pervaded by an initial electric potential perturbs that potential by inducing a charge distribution on the conductor's surface, necessary to guarantee that the surface is an equipotential of the total potential. Some numerical method is required to compute the perturbation potential, when the conductor's shape does not admit a standard analytic solution. For two‐dimensional situations, a method is proposed for solving for the perturbation potential that involves expansion of the boundary perturbation potential and its normal derivative as truncated Fourier series. This boundary potential is known to within an additive constant from the requirement that its sum with the initial potential must be a constant. The standard representation theorem for the Dirichlet problem gives a consistency relation between the boundary function and its normal derivative, which here becomes a set of linear algebraic relations between Fourier series coefficients, with matrix entries found by appropriate applications of the fast Fourier transform. These are solved for the boundary derivative coefficients; at any point exterior to the conductor, the perturbation potential can then be evaluated from the two sets of Fourier coefficients, using further application of the fast Fourier transform. Examples are shown for two conductor shapes, with several initial potentials. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 673–683, 2001     

A quantitative version of the Dirichlet-Jordan test for double Fourier series

For classes of functions with convergent Fourier series, the problem of estimating the rate of convergence has always been of interest. The classical theorem of Dirichlet and Jordan for functions of bounded variation assures the convergence of their Fourier series, but gives no estimate of the rate of convergence. Such an estimate was first provided by Bojani . Here we consider this problem in the case of functions of two variables that are of bounded variation in the sense of Hardy and Krause. The Dirichlet-Jordan test was first extended by Hardy from single to double Fourier series. Now, we provide a quantitative version of it. We prove our estimate in a greater generality, by introducing the so-called rectangular oscillation of a function of two variables over a rectangle.     

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.