 Spherical far-field range condition

For an outgoing field outside a sphere of radius $R$, write $u_s=k{\sum }_{n,m}{i}^{n+1}{a}_n^m{h}_n^{(1)}(kr)Y_n^m$. Its far-field pattern is $\sum a_n^m{Y}_n^m$. If its trace on that sphere lies in L2 space, Parseval's identity gives $k^2\sum |h_n^{(1)}(kR){|}^2|a_n^m{|}^2<\infty$. The fixed-argument growth of spherical Hankel functions yields the displayed equivalent range condition, with the low-order weights defined finitely. Square summability of the far-field coefficients alone does not ensure continuation to the chosen sphere.