Classical expansions and their relation to conjugate harmonic functions
Classical expansions and their relation to conjugate harmonic functions
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DOI：
10.1090/s0002-9947-1965-0199636-9
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发表时间：
1965-06
影响因子：
1.3
通讯作者：
B. Muckenhoupt;E. Stein
中科院分区：
文献类型：
--
作者：
B. Muckenhoupt;E. Stein
统一了这里得到的大多数技术和结果的共同思想集中在研究多元调和函数及其共轭及其变种。普通的傅里叶级数和积分与解析函数有着密切的联系，而它们所享有的这种关系长期以来一直是他们研究的基本工具。因此，我们的主要目标之一是为所讨论的其他展开式开发一个类似的工具。各种展开式与广义调和函数和共轭调和函数之间的联系似乎具有基本的重要性。在形式上，这种联系植根于古典文学。然而，最近，在轴对称势理论(见Weinstein[24])以及Bers[2]、[3]和Vekua[20]的拟解析函数的研究中采用了一些想法。我们现在更详细地描述我们的结果的背景和声明(‘)。我们首先回顾超球展开和傅立叶-贝塞尔变换的一些基本性质。固定A>；0，并考虑由生成关系定义的n次超球面多项式PR(T)：
The common idea which unifies most of the techniques and results obtained here centers about the study of harmonic functions and their conjugates in several variables, and their variants. Ordinary Fourier series and integrals have an intimate connection with analytic functions, and this relation which they enjoy has long provided a fundamental tool in their study. Thus one of our main ob-jectives is the development of an analogous tool for the other expansions in question.The connection that is used between various expansions and generalized har-monic and conjugate harmonic functions seems to be of basic importance. In its formal aspects this connection has its roots in the classical literature. More recently, however, some of the ideas have been taken up in the theory of axially symmetric potential theory (see Weinstein [24]) and in the study of pseudoanalytic functions of Bers [2],[3], and Vekua [20]. We now describe in more detail the background and statements of our results ('). We begin by recalling some of the basic properties of the ultraspherical expansions and Fourier-Bessel transforms. Fix A> 0, and consider the ultraspherical polynomials of degree n, PR (t), defined by the generating relation: