Normal kernel of the source-to-far-field operator ($K(\mathbf{r},{\mathbf{r}}^{\prime })=\frac{1}{4\pi }\frac{\sin (k|\mathbf{r}-{\mathbf{r}}^{\prime }|)}{k|\mathbf{r}-{\mathbf{r}}^{\prime }|}$)

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