Sine-kernel Volterra resolvent ($x+\sin \ast x=F⟹x=F-\frac{1}{\sqrt{2}}\sin (\sqrt{2}\cdot )\ast F$)

For causal convolution on $t\ge 0$, the Laplace transform convolution theorem converts $x+\sin \ast x=F$ into $\bar{x}=[(p^2+1)/(p^2+2)]\bar{F}$. Since this multiplier is $1-1/(p^2+2)$, inversion gives the displayed resolvent kernel. The formula remains valid for locally integrable forcing on finite intervals: substitution and the Fubini theorem reduce verification to an identity between the two continuous sine kernels. Thus a global Laplace transform is a convenient derivation, rather than an essential growth restriction on every local forcing.