= Conjugate Poisson kernel on the circle
{title2=$Q_r(t)=2r\sin t/(1-2r\cos t+r^2)$}
The <geometric series> gives $Q_r(t)=2\sum_{n\ge1}r^n\sin nt$. Its <convolution> multiplier is $-i\operatorname{sgn}(n)r^{|n|}$ for nonzero frequencies and zero at frequency zero. Since the series is uniformly absolutely convergent for $r<1$, these identities hold for every <Lebesgue integrable function> by termwise integration. If all negative <Fourier coefficients> vanish, $Q_r*f=-i(P_r*f-\widehat f(0))$.
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