= Poisson kernel on the circle
{c}
{title2=$P_r(t)=(1-r^2)/(1-2r\cos t+r^2)$}
For $0<r<1$, the <geometric series> gives $P_r(t)=1+2\sum_{n\ge1}r^n\cos nt$. Hence this positive even kernel has normalized integral one, and normalized <convolution> with it multiplies the $n$th <Fourier coefficient> by $r^{|n|}$. Its mass outside any fixed neighbourhood of zero tends to zero as $r\uparrow1$. If an integrable function has finite one-sided limits $A,B$ at zero, evenness gives half the mass to each side and the convolution tends to $(A+B)/2$.
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