Lemma: Retarded convolution and fixed Fourier pole

The causal convolution is defined and

For $t>0$ away from the vertex,

With temporal Fourier convention

and $\rho =\text{abs}\xi >0$, the retarded multiplier is

On ${\lambda }_s=\omega -s\rho$, $\text{abs}{\lambda }_s<c\rho$, $0<c<1$,

The singular coefficient is fixed by retarded support.

Proof. For fixed output in a compact set, intermediate points in the convolution lie in a bounded causal diamond; thus causal convolution is defined. The spatial Fourier transform of $R_{\mathrm{ret}}$ is $\Theta (t)\sin (\rho t)/\rho$. Its temporal self-convolution is

This equals the spatial Fourier transform of the displayed $B_{\mathrm{ret}}$, obtained by integrating over the ball $\text{abs}x\le t$. Differentiation proves the separated formula.

The identity

follows by positive damping. Partial fractions reconstruct $\Theta (t)\sin (\rho t)/\rho$. On one sheet the opposite denominator is smooth, giving (eq:pole). Changing its delta coefficient changes the existing retarded kernel.