= Uniform Poisson summability of continuous circle functions
{title2=$\|f*P_r-f\|_\infty\to0$}
The <Poisson kernel on the circle> is nonnegative, has normalized integral one, and its mass outside any fixed arc about zero tends to zero as $r\uparrow1$. Split the convolution error into a short arc, controlled by the <modulus of continuity>, and its complement, controlled by the vanishing kernel mass. This proves <uniform convergence> for every continuous periodic function. Its Fourier multiplier is $r^{|n|}$, so the convolution is the Abel sum of the <Fourier series>.
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