---
tag: 6X94NPK782
kind: definition
name: Abel–Poisson summation of Fourier series
msc: ["42A24", "40G10", "31A05"]
defines: RZZ78GXP5F
uses: [G12KXDHPPS, EZXPY4A19C, TCYSWCTYYM, 0N1JQXZXHC, 089KWVYD69]
sources:
  - url: https://en.wikipedia.org/wiki/Poisson_kernel
---

## Definition

Let $(c_k)_{k \in \Z}$ be complex numbers, and consider the trigonometric series $\sum_{k \in \Z} c_k e^{ikx}$, summed symmetrically as $c_0 + \sum_{k \ge 1} (c_k e^{ikx} + c_{-k} e^{-ikx})$.
For $0 \le r < 1$ and real $x$ put
$$A_r(x) = \sum_{k \in \Z} r^{\abs{k}} c_k e^{ikx}$$
whenever this series converges.
The series is *Abel–Poisson summable* at $x$ to $s$ if $A_r(x)$ is defined for all $0 \le r < 1$ and $A_r(x) \to s$ as $r \to 1^-$.
This is [[Abel summation|EZXPY4A19C]] of the symmetric series at $x$.

## Remarks

If $f$ is a [[Riemann integrable|0N1JQXZXHC]] $2\pi$-[[periodic function|089KWVYD69]] and $c_k = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-ikt} \, dt$ are its [[Fourier coefficients|G12KXDHPPS]], then the $c_k$ are bounded, $A_r(x)$ converges absolutely, and
$$A_r(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} P_r(x - t) f(t) \, dt, \qquad P_r(\theta) = \sum_{k \in \Z} r^{\abs{k}} e^{ik\theta} = \frac{1 - r^2}{1 - 2r\cos\theta + r^2}.$$
Here $P_r$ is the [[Poisson kernel|TCYSWCTYYM]] of the unit disc.

Example: for $f(x) = \cos x$ one has $c_1 = c_{-1} = \frac12$ and $c_k = 0$ otherwise, so $A_r(x) = r \cos x \to \cos x$.

Non-example: for $c_k = 1$ for all $k$ and $x = 0$, one gets $A_r(0) = \sum_{k \in \Z} r^{\abs{k}} = \frac{1 + r}{1 - r} \to +\infty$.
So the series $\sum_{k \in \Z} e^{ikx}$ is not Abel–Poisson summable at $0$.