Approximation in the Cone of Positive Harmonic Functions

In the course of studying quadrature domains Gustafsson, Sakai and Shapiro were led to the question of whether it is the case that the positive integrable harmonic functions on a bounded domain are dense among all positive harmonic functions (w.r.t. uniform convergence on compact subsets). In this article we will show how such approximation problems are related to representing measures on the Martin boundary, and then we use these results to give a counterexample to the question posed above. This research was supported by Science Foundation Ireland under Grant 06/RFP/MAT057.     

Densities with the mean value property for harmonic functions in a Lipschitz domain

Let be a bounded domain in , , and let . We consider positive functions on such that for all bounded harmonic functions on . We determine Lipschitz domains having such with .

     

Boundary Values of Harmonic Bloch Functions in Lipschitz Domains: A Martingale Approach

In this paper, the boundary behaviour of a certain class of harmonic functions in Lipschitz domains is studied. It continues the work of N. Makarov on the boudary behaviour of analytic Bloch functions in the unit disk. Certain means associated with the function allow to approximate it by a martingale, and the boudary properties of the function are transferred to the asymptotic behaviour of the martingale.     

多元函数的高阶微分中值定理

对凸区域DRn上的二次可微函数,本文采用构造“混合函数”的方法,将多元函数微分中值定理推广到了高阶的情形,并给出了应用示例.     

论中值定理类命题证明中的辅助函数构造

借助实例分析的方法,讨论在证明微分与积分相结合的中值定理类命题时,关于辅助函数的构造技巧及其变形思想.     

A Note on Boundary Unique Contination for Harmonic Functions in Non-Smooth Domains

In this note, we prove the boundary unique continuation property for harmonic function dfined on convex domain     

On a Generalized Cochrane Sum and Its Hybrid Mean Value Formula

The main purpose of this paper is to use a mean value theorem of Dirichlet L-functions to study the asymptotic property of a hybrid mean value of a generalized Cochrane sum, and give an interesting mean value formula.2000 Mathematics Subject Classification: Primary—11F20; Secondary—11F99     

Positive Harmonic Functions that Vanish on a Subset of a Cylindrical Surface

This paper investigates positive harmonic functions on domains that are complementary to a subset of a cylindrical surface. It characterizes, both in terms of harmonic measure and of a Wiener-type criterion, those domains that admit minimal harmonic functions with exponential growth. Illustrative examples are provided. Two applications are also given. The first of these concerns minimal harmonic functions associated with an irregular boundary point, and amplifies a recent construction of Gardiner and Hansen. The second concerns the possible non-approximability of positive harmonic functions by integrable positive harmonic functions. This research was supported by Science Foundation Ireland under Grant 06/RFP/MAT057, and is also part of the programme of the ESF Network “Harmonic and Complex Analysis and Applications” (HCAA).     

Tangential Boundary Behaviour of Harmonic Functions on Trees

We study the behaviour of harmonic functions on a homogeneous tree from the point of view of the tangential boundary covergence.     

关于广义函数赋值问题的完善与积分问题

本文采用非标准分析方法，首先对广义函数的赋值问题进行了完善；接着界定了广义函数的积分；最后讨论了广义函数的广义连续性．     

多元函数中值定理中介值的渐近性

讨论了n元函数中值定理中介值的渐近性质.     

Mean Values of Multiplicative Functions with Weight

We study the characteristics of the mean values of multiplicative functions, given the values of the functions on primes.     

两个微分中值定理证明中辅助函数的多种作法

在数学分析中 ,三个微分中值定理极为重要 .在证明 Lagrange中值定理和 Cauchy中值定理时 ,都少不了作辅助函数 ,各种版本的《数学分析》或《高等数学》书本中 ,都只给出了一种形式的辅助函数 .为了扩展思路 ,给出了多种形式的辅助函数 ,并得出了一般形式 .     

两个微分中值定理证明中辅助函数作法探讨

在证明拉格朗日中值定理与柯西中值定理时都要作辅助函数,这里直接依据所证明的结论,给出两种求辅助函数的方法.     

一个类似广义Dedekind和S_2(h,n,k)的二次均值

本文的主要目的是利用 Dirichlet L -函数的均值研究 n为奇数时 ,类似广义 Dedekind和 S2 ( h,n,k)的二次均值 ,得到了两个有趣的渐近公式 .     

亚纯函数及其导数在角域内的唯一性

In this paper, applying the Nevanlinna theory of meromorphic function in one angular domain, we deal with a problem of uniqueness for meromorphic functions and their derivatives sharing three finite value ignoring multiplicities in an angular domain instead of the whole complex plane. Obtained results improve a recent result of Lin Weichuan and Seiki Mori.     

C~2空间中广义解析函数的一个带位移带共轭值的非线性边值问题

讨论C2空间中广义解析函数的一个带位移带共轭的非线性边值问题.首先讨论解的形式,然后用积分方程的理论和Schauder不动点定理证明了解的存在性.     

Clifford分析中无界域上正则函数的边值问题

在引入修正Cauchy核的基础上,讨论了无界域上正则函数的带共轭值的边值问题：a(t)Φ (t) b(t)Φ (t) c(t)Φ-(t) d(t)Φ=g(t)．首先给出了无界域上正则函数的Plemelj公式,然后利用积分方程方法和压缩不动点原理证明了问题解的存在唯一性．     

Clifford分析中双曲调和函数的一个带位移的非线性边值问题

讨论了Clifford分析中双曲调和函数的一个带位移的非线性边值问题,先讨论了解析函数的一个边值问题的解的存在性,然后利用Clifford分析中双曲调和函数与解析函数的关系讨论了此边值问题的解,并给出了解的积分表达式.     

Clifforrd分析中广义双正则函数的(线性)非线性边值问题

本文研究Ｃｌｉｆｏｒｄ分析中广义双正则函数的一个非线性边值问题：Ａ（ｔ１，ｔ２）Ｗ＋＋（ｔ１，ｔ２）＋Ｂ（ｔ１，ｔ２）Ｗ＋－（ｔ１，ｔ２）＋Ｃ（ｔ１，ｔ２）Ｗ－＋（ｔ１，ｔ２）＋Ｄ（ｔ１，ｔ２）Ｗ－－（ｔ１，ｔ２）＝ｇ（ｔ１，ｔ２）ｆｔ１，ｔ２，Ｗ＋＋（ｔ１，ｔ２），Ｗ＋－（ｔ１，ｔ２），Ｗ－＋（ｔ１，ｔ２），Ｗ－－（ｔ１，ｔ２）［］．先讨论解的积分表示式，再研究几个奇异算子，最后用Ｓｃｈａｕｄｅｒ不动点原理（压缩映射定理）证明了解的存在性（唯一性）．目前还没有见到其它国内外学者研究广义双正则函数的非线性边值问题．本文推广了Ｆ．Ｂｒａｃｋｓ，Ｗ．Ｐｉｎｃｋｅｔ［１０］，ＬｅＨｕａｎｇ Ｓｏｎ［１１］，Ｒ．Ｐ．ＧｉｌｂｅｒｔａｎｄＪ．Ｌ．Ｂｕｃｈｎａｎ［１５］和黄沙［１３］的工作