= Axisymmetric harmonic interpolation between concentric spheres
{title2=$\Phi(r,\theta)=\sum_{n\geq0}R_n(r)P_n(\cos\theta)$}
= Spherical-shell harmonic interpolation
{synonym}
Suppose the inner and outer boundary values on spheres of radii $0<a<b$ have convergent <Legendre polynomial> expansions with coefficients $u_n,v_n$. For each degree set $F_n(r)=r^n-a^{2n+1}r^{-n-1}$ and $H_n(r)=r^n-b^{2n+1}r^{-n-1}$. Then
$$
R_n(r)=u_n\frac{H_n(r)}{H_n(a)}+v_n\frac{F_n(r)}{F_n(b)}.
$$
Both radial terms solve $(r^2R_n')'=n(n+1)R_n$, so the displayed angular expansion is <harmonic> wherever the series and required derivatives converge. The two radial factors take endpoint values $(1,0)$ and $(0,1)$, respectively, proving the boundary matching. For continuous data with the usual harmonic-extension convergence, the <maximum principle for harmonic functions> gives uniqueness. The construction separates the angular <orthogonality> from the two-boundary radial interpolation.
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