典型群上Fourier级数的球平均求和

本文讨论了典型群上Fourier级数的球平均求和。首先给出了球部分和的Dirichlet核以及 Lebesgue常数,同时通过计算给了 Lebesgue常数一个上界估计。其次证明了 Fourier级数球平均求和的一个收敛定理。对δ次Bochner-Riesz平均作了较详细的讨论,给出了一些收敛的判别定理。     

无穷求和的计算Ⅰ

应用积分变换法和参数求导法,得到了一些无穷求和的积分表示.给出了这些积分表示的封闭形式或者数值解.     

Nassrallah-Rahman积分的一个新证明

应用关于ｑ－微分算子的Ｌｅｉｂｎｉｚ公式证明了关于ｑ－微分算子的两个恒等式．利用这些恒等式及ｑ－级数的一些求和公式给出了Ｎａｓｒａｌａｈ－Ｒａｈｍａｎ积分的一个新证明，进而给出了关于ｑ－级数８Φ７的积分表示的一个简易推导     

i~k求和的一个递推公式

本文给出并证明了求和的一个递推公式，最后给出了应用递推公式的一个例子．     

旋转群上的调和分析(Ⅲ)——Fourier级数的球求和

本文讨论了SO(n)上的Fourier级数的球求和,主要结果是:(1)给出了S.Bochner型球求和的一般积分表达式;(2)证明了Riesz型球求和收敛性定理;(3)给出了S.Bochner型球求和的一条一般性收敛定理。     

p—级数与几个广义积分

本文给出了ｐ—级数与广义积分∫１０ｌｎｋ－１ｘ１－ｘｄｘ，∫１０ｌｎｋ－１ｘ１＋ｘｄｘ，∫１０ｌｎｋ－１ｘ１－ｘ２ｄｘ，∫１０ｌｎｋ－１ｘ１＋ｘ２ｄｘ之间的关系．并通过一些ｐ—级数的求和，给出了上述广义积分中某些积分的积分值．     

关于包含奇-偶下标第二类契贝谢夫多项式的恒等式

给出了一类包含偶下标第二类契贝谢夫多项式和一类包含奇下标第二类契贝谢夫多项式求和的递推公式，得到了一些包含偶下标第二类契贝谢夫多项式和奇下标第二类契贝谢夫多项式的恒等式．     

两类幂级数求和函数的一般方法

本文探讨了高等数学教材中的两类幂级数求和问题,并给出这两类幂级数求和函数的一般方法,同时进行了实例分析.     

经由Faa di Bruno公式寻找各种奇异恒等式

本文给出寻找一批奇异恒等式的一般方法.这些在分划集上求和的恒等式包含一些著名的特殊数列及特殊多项式作为被加项因子.     

酉群上Fourier级数的临界指标以下的Cesàro求和

本文解决了酉群上Ｆｏｕｒｉｅｒ级数的Ｃｅｓａｒｏ求和的收敛问题。首先给出了时Ｃｅｓａｒｏ核的Ｌｅｂｅｓｇｕｅ常数的精确估计，然后得出酉群上Ｆｏｕｒｉｅｒ级数Ｃｅｓａｒｏ求和收敛的一般性结果．     

On hyperbolic summation and hyperbolic moduli of smoothness

This paper deals with the method of hyperbolic summation of tensor product orthogonal spline functions onI ^d. The spaces, defined in terms of the order of the best approximation by the elements of the space spanned by the tensor product functions with indices from a given hyperbolic set, are described both in terms of the coefficients in some basis and as interpolation spaces. Moreover, the hyperbolic modulus of smoothness is studied, and some relations between hyperbolic summation and hyperbolic modulus of smoothness are established.     

Bailey type summation formulas associated with the root system $G_{2}^{\vee}$

We present three types of summation formulas for the root system G₂^úG_{2}^{\vee}, which are generalized from Bailey’s summation formula for a very-well-poised balanced ₆ ψ ₆ basic hypergeometric series.     

Weighted L<Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-inequalities for multi-parameter Riesz type potentials and strong fractional maximal operators

We prove weighted L ^p -inequalities for multi-parameter Riesz type potentials, strong fractional maximal operators and their dyadic counterparts. Our proofs avoid the Good-λ inequalities used earlier in the R ^m -case and are based on our integrated multi-parameter summation by parts lemma, that might be of independent interest.     

Cesàro Summation for Random Fields

Various methods of summation for divergent series of real numbers have been generalized to analogous results for sums of i.i.d. random variables. The natural extension of results corresponding to Cesàro summation amounts to proving almost sure convergence of the Cesàro means. In the present paper we extend such results as well as weak laws and results on complete convergence to random fields, more specifically to random variables indexed by ℤ₊², the positive two-dimensional integer lattice points.     

Jackson-Type Inequalities in the Space S p

In the case of approximation of periodic functions in the space S ^p, we determine the exact constants in Jackson-type inequalities for the Zygmund, Rogosinski, and de la Valleé Poussin linear summation methods.     

On Twisted Fourier Analysis and Convergence of Fourier Series on Discrete Groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group C ^*-algebras of discrete groups. For amenable groups, Følner nets give the key to Fejér summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.     

A note on spherical summation multipliers

We give a new proof of a theorem of L. Carleson and P. Sjölin onL ^p -boundedness of spherical summation operators in two variables.     

The disteribution of a moment estimator for a parameter of the generalized poision distribution

The ratio of the sample variance to the sample mean estimates a simple function of the parameter which measures the departure of the Poisson-Poisson from the Poisson distribution. Moment series to order n²⁴ are given for related estimators. In one case, exact integral formulations are given for the first two moments, enabling a comparison to be made between their asymptotic developments and a computer-oriented extended Taylor series (COETS) algorithm. The integral approach using generating functions is sketched out for the third and fourth moments. Levin's summation algorithm is used on the divergent series and comparative simulation assessments are given.     

Ramanujan summation and the exponential generating function \sum_{k=0}^{\infty}\frac{z^{k}}{k!}\zeta^{\prime}(-k)

In the sixth chapter of his notebooks, Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions ∑ _n≥1log?n e ^nz and $\sum_{n\geq 1}H_{n}^{(j)}~e^{-nz}$ where $H_{n}^{(j)}=\sum_{m=1}^{n}\frac{1}{m^{j}}$ . We find a surprising relation between the two sums when j=1 from which follows a formula that connects the derivatives of the Riemann zeta-function at the negative integers to the Ramanujan sum of the divergent Euler sums ∑ _n≥1 n ^k H _n , k ≥ 0, where $H_{n}=H_{n}^{(1)}$ . Further, we express our results on the Ramanujan summation in terms of the classical summation process called the Borel sum.