= Normal kernel of the source-to-far-field operator
{title2=$K(\mathbf r,\mathbf r')=\frac1{4\pi}\frac{\sin(k|\mathbf r-\mathbf r'|)}{k|\mathbf r-\mathbf r'|}$}
The <adjoint source-to-far-field operator> composed with the <source-to-far-field operator at fixed frequency> has volume kernel $K(\mathbf r,\mathbf r')=(4\pi)^{-1}\operatorname{sinc}(k|\mathbf r-\mathbf r'|)$, where $\operatorname{sinc}(s)=\sin(s)/s$ and $\operatorname{sinc}(0)=1$. This follows from $\int_{S_1}e^{ik\boldsymbol\theta\cdot\mathbf d}dS=4\pi\operatorname{sinc}(k|\mathbf d|)$. Its smoothing action helps explain the unstable <normal equation for a linear inverse problem>.
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