Harmonic functions on the unit disc I
Harmonic functions on the unit disc I
复制标题
单位圆盘 I 上的谐波函数
DOI：
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发表时间：
1968
期刊：
影响因子：
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通讯作者：
G. Johnson
中科院分区：
文献类型：
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作者：
G. Johnson
2andin(1.2)P(Ot)([(e-}Re)/(e Re‘)]是单位圆盘的Poisson核。P)是Pr的n阶导数。文献[5]中报道了这一定理。它已被Douglas[2]用作函数在有限点集上逼近的圆盘上调和连续的全局约束。Sayler是Douglas的学生，他将这些结果推广到R中区域上解析系数为紧解析边界的线性椭圆型二阶偏微分方程解的情形[10]。从Fatou的工作开始，关于Poisson积分和Poisson-Stieltjes积分有丰富的边值理论。在[4]中可以很好地处理这一理论中的新旧结果。鉴于上面的结果，自然会问是否存在一个边界函数，在某种广义意义上，通过泊松表示与任意调和函数相关联。事实上，表示fr(O)f(r，O)并使用(1.2)，形式为fr E：-op(*g，E：-opr*g P，*E：=0g()=P*g。
2andin (1.2) P(O t) ([(e -}rei)/(e re’)] is the Poisson kernel for the unit disc. P) is the nth derivative of Pr. This theorem was reported in [5]. It has been used by Douglas [2] as global constraint for harmonic continuation in the disc of a function which is approximated at a finite set of points. Saylor, a student of Douglas, has extended these results to the case of solutions of a linear elliptic second order partial differential equation with analytic coefficients on domain in R bounded by a compact analytic boundary [10]. There is a rich boundary-value theory concerning the Poisson and the Poisson-Stieltjes integrals beginning with the work of Fatou. A nice treatment of old and new results in this theory may be found in [4]. In view of the above result it is natural to ask whether there is a boundary function, in some generalized sense, associated with an arbitrary harmonic function by means of a Poisson representation. In fact, denoting fr(O) f(r, O) and using (1.2) we have, formally fr E:-oP( *g, E:--oPr*g P,* E:=0g()= P*g.