【摘要】 <正> 设u(x,y)是上半平面内的调和函数且对任何y>0一致地有 integral from n=-∞ to ∞ |u(x,y)|dx≤A, (A是常数) (1) 则存在在(-∞,∞)上的有界变差函数g(t)使 u(x,y)=1/πintegral from n=-∞ to ∞ y/((t-x)~2+y~2)dg(t),(2) 这是大家熟知的一个基本结果。但在实际问题中积分(1)往往是不存在的,例如在Titchmarsh所著按二阶微分方程特征函数展开一书中所遇到的解析函数m(λ),ψ(x,λ),φ(x,λ)的虚部,一般说来都不满足(1)。本文应用围道积分的方法在比(1)弱得多的条件下给出积分表示式(2),而且成功的将我们的结果应用到特征函数展开及解析函数角形边值问题的研究。更多还原

【Abstract】 In this paper we shall prove several main results as follows.Theorem A. Let u(x, y) be harmonic in the upper half-plane and for any T≥1 and 0<y≤T, the relationholds uniformly, where a and A are constants. Then there exists a function g(t) of bounded variation in any finite interval with the total variation satisfyingsuch that u(x, y) admits representationConversely, for any function g(t) satisfying (2), the integral on the right of formula (3) just defines a harmonic function u(x, y) satisfying (1).When a = 0, the above-mentioned and following results are well-known in the theory of Hp-spaces.Theorem B. Let s(x, y) be defined a positive, continuous subharmonic function in the upper half-plane and for any T≤1 and 0<y≤T, the relationholds uniformly. Then there exists a harmonic function m (x, y) satisfying (l) and.Theorem C. Suppose f(z) is analytic in the upper half-plane and for any T≥1 and 0<y≤T, the relationholds uniformly. Then angular boundary values of f(z) exist almost everywhere, that is,where C is a positive constant.更多还原