Axisymmetric harmonic interpolation between concentric spheres ($\Phi (r,\theta )={\sum }_{n\ge 0}{R}_n(r)P_n(\cos \theta )$)

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 $H_n(r)=r^n-b^{2n+1}{r}^{-n-1}$. ThenBoth radial terms solve $(r^2{R}_n^{\prime }{)}^{\prime }=n(n+1)R_n$, 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.