闭逐块光滑流形上奇异积分的Poincaré-Bertrand公式

本文把复变函数论中著名的Poincare-Bertrand公式拓广到闭逐块C（1）光滑流形上Bochner-Martinelli型奇异积分的含有点ζ的立体角系数α（ζ）的更一般的置换公式.     

闭光滑流形上的奇异积分方程

<正> §1.前言 自Giraud G.以来,已有不少关于闭光滑流形上的奇异积分和奇异积分方程的研究(见[10]),但利用多复变函数的Cauchy型积分作为工具者,至今不多(如[3—8]),而在一维奇异积分方程论中,复变函数的Cauchy型积分起着基本的作用.本文试以定理在多复变函数论中的拓广为基础,讨论闭光滑流形上奇异积分的合成和奇异积分方程的求解,其方法和结论,都是与Giraud G.等人的工作全然不同的.     

实Clifford分析中六类拟Bochner-Martinelli型高阶奇异积分的几个问题

本文借助于Ｈａｄａｍａｒｄ关于高阶奇异积分有限部分的思想,研究关于实 Ｃｌｉｆｆｏｒｄ分析中六个类型(含一个奇点或二个奇点的)拟Ｂｏｃｈｎｅｒ－Ｍａｒｔｉｎｅｌｌｉ型高阶奇异积分的归纳定义、Ｈａｄａｍａｒｄ主值的存在性、递推公式、计算公式、微分公式、Ｐｏｉｎｃａｒｅ－Ｂｅｒｔｒａｎｄ置换公式以及拟Ｂ－Ｍ型高阶奇异积分的Ｈｏｌｄｅｒ连续性等问题．这些问题是研究单、多元复分析的学者们在研究奇异积分时,通常要涉及到的几个问题．     

The Poincaré-Bertrand Formula for the Bochner-Martinelli Integral

The Poincaré-Bertrand formula and the composition formula for the Bochner-Martinelli integral on piecewise smooth manifolds are obtained. As an application, the regularization problem for linear singular integral equation with Bochner-Martinelli kernel and variable coefficients is discussed.     

A Cauchy's Integral Formula for Functions with Values in a Universal Clifford Algebra and its Applications

In this paper, we obtain Cauchy's integral formula on certain distinguished boundary for functions with values in a universal Clifford algebra, which is similar to the classical Cauchy's integral formula on the distinguished boundary of polycylinder for several complex variables. By using it, both the mean value theorem and the maximum modulus theorem are given.     

A theorem concerning a product of a general class of polynomials and theH-function of several complex variables

A theorem concerning a product of a general class of polynomials and theH-function of several complex variables is given. Using this theorem certain integrals and expansion formula have been obtained. This general theorem is capable of giving a number of new, interesting and useful integrals, expansion formulae as its special cases.     

A Cauchy-Pompeiu formula in super Dunkl-Clifford analysis

Using a distributional approach to integration in superspace, we investigate a Cauchy-Pompeiu integral formula in super Dunkl-Clifford analysis and several related results, such as Stokes formula, Morera’s theorem and Painlevé theorem for super Dunklmonogenic functions. These results are nice generalizations of well-known facts in complex analysis.     

THE PERMUTATION FORMULA OF SINGULAR INTEGRALS WITH BOCHNER-MARTINELLI KERNEL ON STEIN MANIFOLDS

Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C(1) smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner-Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.     

Cm 中高阶偏微分方程的代数体解

该文利用多复变函数值分布理论和技巧, 研究了C^m中高阶偏微分方程的代数体函数解的存在性问题, 建立了C^m中高阶偏微分方程的Malmquist型定理.     

The several transformation formula in several complex variables and its applications

In this paper, by the method of global analysis, the authors give a new global integral transformation formula and obtain the Plemelj formula with Hadamard principal value of higher-order partial derivatives for the integral of Bochner-Martinelli type on a closed piecewise smooth orientable manifold C ⁿ . Moreover, the authors obtain the composition formula, Poincaré-Bertrand extended formula of the corresponding singular integral. As the application of some results, the authors also study a higher-order Cauchy boundary problem and a regularization problem of higher-order linear complex differential singular integral equation with variable coefficients.     

On Paravector Valued Homogeneous Monogenic Polynomials with Binomial Expansion

The aim of this note is to study a set of paravector valued homogeneous monogenic polynomials that can be used for a construction of sequences of generalized Appell polynomials in the context of Clifford analysis. Therefore, we admit a general form of the vector part of the first degree polynomial in the Appell sequence. This approach is different from the one presented in recent papers on this subject. We show that in the case of paravector valued polynomials of three real variables, there exist essentially two different types of such polynomials together with two other trivial types of polynomials. The proof indicates a way of obtaining analogous results in the case of polynomials of more than three variables.     

一类高阶偏微分方程的亚纯解

利用多复变值分布理论,我们将Steinmetz的代数微分方程的Malmqiust型定理推广到复偏微分方程中．     

多复变函数的Picard定理

本文给出了多复变函数的Ｋ－拟亚纯函数的定义,并且得到了一个关于多复变函数的Ｋ－拟亚纯函数的正规定则,从这个正规定则,我们证明了多复变函数的全纯函数Ｐｉｃａｒｄ定理。     

THE GROWTH THEOREM FOR STARLIKE MAPPINGS ON BOUNDED STARLIKE CIRCULAR DOMAINS

1.IntroductionOnthegeometricfunctiontheoryofonecomplexvariable,thefollowinggrowthandi-coveringtheoremiswellknown(see[2]).TheoremA.Foreachno~alizedunivalentjunctionfontheunitdiscDCC,ESPecially,theleft-handsideOftheaboveinequalityimpliesf(D)2ID.ForeachzED,z/0,equalityoccursintheaboveinequalityifandonlyiffisKoe6efunctionK(z)=theoritsrotatione--"K(e"z).Itisnaturaltoextendthisandotherresultsonthegeometricfunctiontheoryofonevariablestoseveralvariables.Butasearlyasfiftyyearsago,H.Cartanpointe…     

SHANNON SAMPLING AND ESTIMATION OF BAND-LIMITED FUNCTIONS IN THE SEVERAL COMPLEX VARIABLES SETTING

1IntroductionSystematic studies on sinc function of one complex variable may be traced to the work ofStenger[10].Numerical methods based on the sinc function were subsequently studied in[12].The results arising from the numerical methods were further appl…     

A Martinelli-Bochner formula on fractal domains

A new perspective on a Cauchy integral formula for Clifford algebras valued functions on domains with quite smooth boundaries was discussed in [5]. On the other hand, the Cauchy transform associated to Clifford analysis has been involved recently with fractional metric dimensions and fractals, see [1, 2, 3]. In this paper we consider the question of possible generalizations of the Cauchy integral formula to domains with fractal boundary. As an application, we prove a Martinelli-Bochner type formula for several complex variables on such pathological domains. The proof makes heavy use of the isotonic approach of the monogenic functions theory. Received: 8 October 2008     

The Sampling Theorem and Several Equivalent Results in Analysis

First we show that several fundamental results on functions from theBernstein spaces (such as Bernstein's inequality andthe reproducing formula) can be deduced from a weak form of the classicalsampling theorem. In §3 we discuss the mutual equivalence of thesampling theorem, the derivative sampling theorem and a harmonic functionsampling theorem. In §§4–6 we discuss connections between thesampling theorem and various important results in complex analysis andFourier analysis. Our considerations include Cauchy's integral formula,Poisson's summation formula, a Gaussian integral, certain properties ofweighted Hermite polynomials, Plancherel's theorem, the maximum modulusprinciple, and the Phragmén–Lindelöf principle.     

THE DIFFERENTIAL INTEGRAL EQUATIONS ON SMOOTH CLOSED ORIENTABLE MANIFOLDS

1 IntroductionSillce tl1e limit value fOrlnula, viz. tl1e Plemelj fOrn1ula, of the Cauthe type integraJ withBochner-Martinelli kernel was proved in 1957[1], it has beell successfully used to the study Ofsingular i1ltegral equatious, solvi11g the 0b--equation, holomorphic extension, 0--closed exten-sion and C-R 111al1ifolds[2-51. Evideutly, the researcl1 of higher order singular integrals withBochuer-Martinelli kerllel itself also l1as important significallce. In 1952, J. Hadanmrd firstde…     

One-dimensional maps via complex analysis in several variables

In this article we use elements from the theory of several complex variables to establish dynamical properties of certain one-dimensional maps (e.g. decay of correlations, central limit theorems, etc.). I am grateful to David Ruelle for introducing me to the Bochner theorem in September 1984 and for making a copy of his article [15] available to me prior to publication. I would also like to thank S. Krantz, E. Bedford and H. Weiss for advice on the complex analysis and G. Keller for comments on interval maps. The author currently holds a Royal Society University Research Fellowship.     

实Clifford分析中三类高阶奇异积分及其非线性微分积分方程

本文第一部分借助于高阶异积分的Ｈａｄａｍａｒｄ主值的思想以及归纳法的思想,在证明了６个引理的基础上讨论实Ｃｌｉｆｆｏｒｄ分析中三类高阶异积分的归纳定义,Ｈａｄａｍａｒｄ主值的存在性,递推公式,计算公式以及高阶奇异积分在Ｈａｄａｍａｒｄ主值意义下的１２个微分公式,受多复变中解析函数积分表示式多样笥的,本文采用的算子的积分表达式就与个公式和微分公式都十分乘法本文第二部分在引进并证明了Ｈｉｌｅ引理型的基