关于正弦级数$\bm{L^{1}}$-收敛性研究中对单调递减条件的确切推广

本文在处理$L^1$-收敛性问题中给出了一个确切的条件和一种更直接的方式.     

关于具有拟单调系数的三角级数L1-收敛性的注记

假设三角级数的系数具有拟单调性，给出了级数按L^1[0，2π]中的范数收敛于其和函数的一个判别条件，推广了文献中的有关结果．     

关于正弦级数L~1-收敛性研究中对单调递减条件的确切推广

本文在处理L1-收敛性问题中给出了一个确切的条件和一种更直接的方式.     

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.     

具有分段有界变差系数的三角级数

通过引入所称谓的分段有界变差数列(PBVS),将关于具有非负系数的三角级数的收敛性和可积性的经典结果推广到具有可变系数(允许变号)的三角级数.     

一类三角级数的最佳逼近度

本文在复值连续函数空间中给出一类具有O-正则变化拟单调系数之三角级数的最佳逼近度的一个完备结果.特别是第3节的定理3.2,即使在实连续函数空间中亦是一个新的结论,而且也是Xie与Zhou[6]的定理2在一个方向上的本质推广.最后,在第4节给出一个应用.     

一类三角级数的最佳逼近度

本文在复值连续函数空间中给出一类具有Ｏ一正则变化拟单调系数之三角组数的最佳逼近度的一个完备结果．特别是第 ３节的定理 ３．２,即使在实连续函数空间中亦是一个新的结论,而且也是 Ｘｉｅ与Ｚｈｏｌｌ［６］的定理 ２在一个方向上的本质推广．最后,在第４节给出一个应用．     

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.     

交错级数收敛性的几个结果及其应用

莱布尼兹判别法只是一个充分条件。有大量交错级数虽然不满足其条件,但却是收敛的．对于无法用莱布尼兹判别法判定的三类交错级数,利用常数项级数收敛的定义及相关结果,可以证明在一定条件下它们都是收敛的．并通过实例说明所得结果的应用价值．     

关于SL(2,R)上主级数表示、离散级数表示的矩阵系数的平方可积性

人们知道SU（２）的不可约酉表示的矩阵元是相互正交且平方可积的（Ｐｅｔｅｒ Ｗｅｙｌ定理）． 对于SU（２，R）的主级数表示和离散级数表示的矩阵系数是否有类似的结果？在该文中，作者部分给出了这个问题的肯定回答，即关于主级数表示的矩阵系数是准平方可积的，关于离散级数表示 的矩阵系数是平方可积的． 此外，他们还得到了离散级数表示（除狀＝±１外）在子空间′狀上的矩阵系数是绝对可积的．     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> integrability of trigonometric series with special varying coefficients

Very recently, Yu, Le and Zhou introduced the so called △B1^＊ and △B2^＊ conditions, which are generalizations of the monotone condition. By applying these two new conditions, the author essentially generalizes the classical results of Chen on the necessary and sufficient conditions of the Lp integrability of trigonometric series. In fact, the present paper gives the first result on the necessary and sufficient conditions of the Lp integrability of trigonometric series, where coefficients may have different signs.     

A remark on ``two-sided' monotonicity condition: an application to L p convergence

Summary To verify the universal validity of the ``two-sided' monotonicity condition introduced in [4], we will apply it to include more classical examples. The present paper selects the L^p convergence case for this purpose. Furthermore, Theorem 3 shows that our improvements are not trivial.     

Points of monotonicity in Orlicz-Lorentz function spaces

In Orlicz-Lorentz function space ?_M,ω[0,γ), points of upper monotonicity, lower monotonicity, upper local uniform monotonicity (γ<∞) and lower local uniform monotonicity (γ<∞) are characterized. As the corollary, we can easily obtain the criteria for strict monotonicity, upper local uniform monotonicity (γ<∞) and lower local uniform monotonicity (γ<∞) of ?_M,ω[0,γ).     

A generalization of the monotonicity condition and applications

We introduce a new class of sequences called NBVS to generalize GBVS, essentially extending monotonicity from “one sided” to “two sided”, while some important classical results keep true. Supported in part by Natural Science Foundation of China under grant number 10471130.     

An application of MVBV condition in real sense for L 1-convergence of trigonometric series

We continue to investigate the application of the Mean Value Bounded Variation (MVBV) concept in real sense, and prove the necessary and sufficient condition for L ¹-convergence of trigonometric series.     

三角级数在Burgers-KdV混合型方程中的应用

利用三角级数法将Burgers-KdV混合型方程转化为一组非线性代数方程,进而用待定系数法求解方程组,最后求出了Burgers-KdV混合型方程的精确解.     

关于MVBV条件下Fourier级数的L~1收敛性的注记(英文)

本注记给出了关于在Fourier级数L~1收敛中MVBV条件不可减弱的一个猜想的肯定回答。     

Application of the absolute Euler method to some series related to Fourier series and its conjugate series

We study the absolute Euler summability problem of some series associated with Fourier series and its conjugate series generalizing some known results in the literature. Also, it is shown that absolute Euler summability of rth derived Fourier series and rth derived conjugate series can be ensured under local conditions.     

Comparison of two orthogonal series methods of estimating a density and its derivatives on an interval

We compare the merits of two orthogonal series methods of estimating a density and its derivatives on a compact interval—those based on Legendre polynomials, and on trigonometric functions. By examining the rates of convergence of their mean square errors we show that the Legendre polynomial estimators are superior in many respects. However, Legendre polynomial series can be more difficult to construct than trigonometric series, and to overcome this difficulty we show how to modify trigonometric series estimators to make them more competitive.     

Trigonometric series of classesL^p (\mathbb{T}), p \in ]1;\infty [ and their conservative meansand their conservative means

Suppose that a lower triangular matrix μ:[μ _m ⁽ⁿ⁾ ] defines a conservative summation method for series, i.e.,