Abel–Poisson summation of Fourier series

Summary

Abel–Poisson summation applies Abel summation to the Fourier series of a periodic function. The $k$-th term is damped by the factor $r^{\abs{k}}$, and $r$ tends to $1$ from below. The damped series equals the convolution of the function with the Poisson kernel. As a function of the point $r e^{ix}$ of the unit disc, the damped sum is a harmonic function. Its boundary values, in the limit $r \to 1^-$, are the given function. For a continuous function the Abel–Poisson means converge uniformly to it.