均值有界变差条件及其在Fourier分析中的应用

本文宣布了关于Fourier (三角)级数的系数数列单调性条件的一个最终推广.我们已经证明了该条件对正弦级数保持一致收敛性方面已不可作任何推广,一些Fourier分析中重要和有趣的经典定理也已经在此最终条件下重新建立.本文因而对长达90年的研究历史作了一个简要综述.这一系列论文中的第一篇原始论文公布在arXiv:math.CA/0611805 v1, November 27, 2006.     

全变差有界函数列的一致(R)可积性

给出了一致有界单调函数列一致可积性定理 ,由此得出全变差序列有界的收敛函数列的一致可积性 .说明了该结论可判断一些非一致收敛函数列的逐项积分性质 .     

三角级数一致收敛性问题在复空间的完整推广

本文研究了三角级数在复空间中的一致收敛性问题.利用Abel变换等方法,获得了三角级数在复空间中满足上确界有界变差(SBVS)条件下一致收敛的充分必要条件,推广了Korus在实空间中的结论.     

关于分组有界变差数列的几个等价定理

在分析中,人们从各种不同的角度推广了分组有界变差数列的概念,并给出了许多有趣的性质.本文对分组有界变差数列做了进一步的研究,根据Leindler近期的研究,对周颂平等建立的渐近关系的结果做一些补充和推广.     

借助于复变数解析函数求一些三角级数的和

把求三角级数的和转化为求复数域内相应幂级数的和,所要求的余弦级数的和与正弦级数的和分别等于幂级数和的实部与虚部系数.     

基于分段均值有界变差条件下正弦与余弦积分的收敛性

本文介绍了分段均值有界变差函数(PMBVF)的定义,分别给出了正弦与余弦积分一致收敛性的充分必要条件.     

SSB算子和SSK算子对有界变差函数的点态逼近度

关于用线性算子逼近有界变差函数,到目前为止已经有一些杰出的工作,其中绝大多数都是沿着Bojanic引进的方法对不同的算子进行的.在这里引进两种算子: 称L_n为Stancu—Sikkcma—Bernstcin算子,L_n称为Stancu—Sikkema—Kantoro vich算子,简称为SSB算子和SSK算子. 我们研究了L_n(f,x)和L_n(f,x)对[0,1]上的有界变差函数的点态逼近度,主要结果是定理1 对于任意的x∈(0,1),当n充分大时,有     

一类三角级数的收敛性

利用Euler公式求三角级∞∑m=1 sinmx/n与∞∑m=1 cosmx/m的和函数并讨论其一致收敛性.     

Feller-Trotter概率型算子对有界变差函数的收敛速度

本文运用概率方法研究了Feller-Trotter概率型算子对有界变差函数的收敛速度。由于该算子包括许多常见的算子,从而由关于该算子的一般结论可导出许多常见算子对有界变差。函数的收敛速度。作为一般结论的应用,本文列举了Baskakov算子、Szasz-Mirakjan算子、Gauss-Weierstrass算子、Gramma算子、Post-Gamma算子对有界变差函数的收敛速度。其中,关于Szasz-Mirakjan算子的结论推广并改进了Fuhua Cheng的结论,其它结论是作者首次得到。     

Uniform Convergence of Trigonometric Series with Rarely Changing Coefficients

We consider the series and whose coefficients satisfy the condition for , where the sequence can be expressed as the union of a finite number of lacunary sequences. The following results are obtained. If as , then the series is uniformly convergent. If for all , then the sequence of partial sums of this series is uniformly bounded. If the series is convergent for and as , then this series is uniformly convergent. If the sequence of partial sums of the series for is bounded and for all , then the sequence of partial sums of this series is uniformly bounded. In these assertions, conditions on the rates of decrease of the coefficients of the series are also necessary if the sequence is lacunary. In the general case, they are not necessary.     

On the Rate of Convergence of Fourier Series of Functions of Bounded Variation

Mathematical Notes -     

On Pointwise Divergence of Trigonometric Series of General Type

Mathematical Notes -     

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.     

A Condition for Convergence in Mean of Trigonometric Series

We construct an example of a trigonometric series showing that a certain condition closely related to convergence in mean of trigonometric series cannot be sharpened.     

Uniform Convergence of Fourier–Jacobi Series

Necessary and sufficient conditions which imply the uniform convergence of the Fourier–Jacobi series of a continuous function are obtained under an assumption that the Fourier–Jacobi series is convergent at the end points of the segment of orthogonality [−1,1]. The conditions are in terms of the modulus of continuity, Λ-variation, and the modulus of variation of a function.