= Zero-frequency cancellation of a causal response kernel
{title2=$\int I(t)dt=\widehat K(0)\widehat V(0)$}
For an integrable <linear response> kernel $K$ and input $V$, the <convolution theorem> gives $\widehat I(\omega)=\widehat K(\omega)\widehat V(\omega)$. The integrated output is therefore zero for every integrable input if and only if $\widehat K(0)=\int K(t)dt=0$. For a sum of causal damped sines, $K(t)=\Theta(t)\sum_j a_je^{-\gamma_jt}\sin(p_jt)$, the condition is $\sum_j a_jp_j/(\gamma_j^2+p_j^2)=0$. If all $a_j$ are positive, nonzero terms of both signs are needed; a term with $p_j=0$ vanishes identically.
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