Spherical polyharmonics and Poisson kernels for polyharmonic functions

作     者：Grzebula, Hubert Michalik, Slawomir 

作者机构：Cardinal Stefan Wyszynski Univ Coll Sci Fac Math & Nat Sci Warsaw Poland 

出 版 物：《COMPLEX VARIABLES AND ELLIPTIC EQUATIONS》 (复变函数与椭圆型方程)

年 卷 期：2019年第64卷第3期

页      面：420-442页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Spherical polyharmonics zonal polyharmonics polyharmonic functions Poisson kernel Gegenbauer polynomials Cauchy-Hua kernel 

摘      要：We introduce and develop the notion of spherical polyharmonics, which are a natural generalisation of spherical harmonics. In particular we study the theory of zonal polyharmonics, which allows us, analogously to zonal harmonics, to construct Poisson kernels for polyharmonic functions on the union of rotated balls. We find the representation of Poisson kernels and zonal polyharmonics in terms of the Gegenbauer polynomials. We show the connection between the classical Poisson kernel for harmonic functions on the ball, Poisson kernels for polyharmonic functions on the union of rotated balls, and the Cauchy-Hua kernel for holomorphic functions on the Lie ball.