= Sine-kernel Volterra resolvent
{title2=$x+\sin*x=F\quad\Longrightarrow\quad x=F-\tfrac1{\sqrt2}\sin(\sqrt2\,\cdot)*F$}
For causal <convolution> on $t\ge0$, the <Laplace transform convolution theorem> converts $x+\sin*x=F$ 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.
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