= Source-to-far-field operator at fixed frequency
{title2=$T_k:L^2(A)\to L^2(S_1)$}
For $(\Delta+k^2)\psi=Q$, the <outgoing Green function for the three-dimensional Helmholtz equation> yields the <far-field pattern>
$$
(T_kQ)(\boldsymbol\theta)=-\frac1{4\pi}\int_A e^{-ik\boldsymbol\theta\cdot\mathbf r}Q(\mathbf r)\,d^3\mathbf r,\qquad |\boldsymbol\theta|=1.
$$
The sign corresponds to the Green function satisfying $(\Delta+k^2)G=-\delta$. This <Hilbert-Schmidt operator> samples the source <Fourier transform> on the sphere $|\boldsymbol\xi|=k$; for a ball of radius $r_0$, its squared Hilbert-Schmidt norm is $r_0^3/3$.
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