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

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.