## Axisymmetric harmonic interpolation polynomials in $\textbf {R}^{N}$

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- by Morris Marden
- Trans. Amer. Math. Soc.
**196**(1974), 385-402 - DOI: https://doi.org/10.1090/S0002-9947-1974-0348130-6
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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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## Bibliographic Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**196**(1974), 385-402 - MSC: Primary 31B99
- DOI: https://doi.org/10.1090/S0002-9947-1974-0348130-6
- MathSciNet review: 0348130