Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Axisymmetric harmonic interpolation polynomials in $ {\bf R}\sp{N}$

Author: Morris Marden
Journal: Trans. Amer. Math. Soc. 196 (1974), 385-402
MSC: Primary 31B99
MathSciNet review: 0348130
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Abstract: Corresponding to a given function $ F(x,\rho )$ which is axisymnetric harmonic in an axisymmetric region $ \Omega \subset {{\text{R}}^3}$ and to a set of $ n + 1$ circles $ {C_n}$ in an axisymmetric subregion $ A \subset \Omega $, an axisymmetric harmonic polynomial $ {\Lambda _n}(x,\rho ;{C_n})$ is found which on the $ {C_n}$ interpolates to $ F(x,\rho )$ or to its partial derivatives with respect to x. An axisymmetric subregion $ B \subset \Omega $ is found such that $ {\Lambda _n}(x,\rho ;{C_n})$ converges uniformly to $ F(x,\rho )$ on the closure of B. Also a $ {\Lambda _n}(x,\rho ;{x_0},{\rho _0})$ is determined which, together with its first n partial derivatives with respect to x, coincides with $ F(x,\rho )$ on a single circle $ ({x_0},{\rho _0})$ in $ \Omega $ and converges uniformly to $ F(x,\rho )$ in a closed torus with $ ({x_0},{\rho _0})$ as central circle.

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Keywords: Axisymmetric harmonic polynomial, axisymmetric harmonic function, harmonic interpolation polynomial, Bergman operator method
Article copyright: © Copyright 1974 American Mathematical Society