In the next two posts I will finish discussing the steady-state heat equation on the disc by giving the proof that the solution we were led to in A motivational problem to the study of Fourier series is the unique solution of the problem which satisfies some natural conditions.

In this first post, I’ll introduce a new way of summing infinite series. We saw in my last post that Cesàro summability can be used to assign a value to a divergent series. Here is another way: a series of complex numbers $\sum_{k=0}^{\infty} c_k$ is said to be Abel summable to $s$ if for every $0 \leq r < 1,$ the series

$A(r) := \displaystyle\sum_{k=0}^{\infty}c_kr^k$

converges, and

$\displaystyle\lim_{r\rightarrow 1} A(r) = s.$

We define the $A(r)$‘s as the Abel means or the series.  We’ll see in exercice 1 that, as with Cesàro summability, if a series converges to $s,$ then it is Abel summable to $s.$ In fact, every Cesàro summable series is Abel summable to the same sum but the converse isn’t true.

Define the Abel means of $f(\theta) \sim \sum_{-\infty}^{\infty} a_ne^{in\theta}$ by

$A_r(f)(\theta) = \displaystyle\sum_{n=-\infty}^{\infty}r^{|n|}a_ne^{in\theta}.$

As in the case of Cesàro means and Fejér kernels, the key fact is that we can write the Abel means as convolutions with the Poisson kernel:

$A_r(f)(\theta) = (f*P_r)(\theta),$

where

$P_r(\theta) = \displaystyle\sum_{n =-\infty}^{\infty}r^{|n|}e^{in\theta}.$

Indeed, we have

$A_r(f)(\theta) = \displaystyle\sum_{n = -\infty}^{\infty} r^{|n|}a_ne^{in\theta}$

$= \displaystyle\sum_{n=-\infty}^{\infty}r^{|n|}\left(\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\varphi)e^{-in\varphi}d\varphi\right)e^{in\theta}$

$= \displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\varphi)\left(\sum_{n=-\infty}^{\infty}r^{|n|}e^{-in(\varphi-\theta)}\right)d\varphi.$

Note that the last equality is justified since $P_r(\theta)$ converges uniformly for $0 \leq r < 1.$

Lemma: If $0 \leq r < 1,$ then the Poisson kernel is a good kernel as $r$ tends to $1$ from below.

Note that you need to adapt the definition of  good kernels from this post by changing $n$ by $r$ and taking the limit as $r \rightarrow 1$ in the third property. For the proof, we will see in exercice 2 that

$P_r(\theta) = \displaystyle\frac{1-r^2}{1-2r\cos\theta+r^2}.$

Note, then, that

$1-2r\cos\theta + r^2 = (1-r)^2 + 2r(1-\cos\theta).$

So for $r > 0$ and $0 < \delta \leq |\theta| \leq \pi,$ we have

$1-2r\cos\theta + r^2 \geq c_{\delta} > 0$

for some $c_{\delta} \in \textbf{R}.$ Thus $P_r(\theta) \leq (1-r^2)/c_{\delta}$ for $\delta \leq |\theta| \leq \pi,$ giving us the third property of good kernels. By the uniform convergence of $P_r(\theta),$ we can integrate it term by term and get

$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}P_r(\theta)d\theta = \frac{1}{2\pi}\sum_{n=-\infty}^{\infty}r^{|n|}\int_{-\pi}^{\pi}e^{in\theta}d\theta$

$= r^0 = 1$

since $\int_{-\pi}^{\pi}e^{in\theta}d\theta = 0$ for all integer $n \neq 0.$ Moreover, since $P_r(\theta) \geq 0,$ this gives us the first and second property for good kernels.

QED

Recalling the theorem about good kernels of this post, this gives us an analogue of Fejér’s theorem for Abel means:

Theorem: The Fourier series of an integrable function on the circle is Abel summable to $f$ at every point of continuity. Moreover, if $f$ is continuous on the circle, then the Fourier series of $f$ is uniformly Abel summable to $f.$

Maybe I’m still missing something but it seems like this could’ve been proved by using the fact that Cesàro summability implies Abel summability coupled with Fejér’s theorem. We now have the tools necessary to completely solve the steady-state heat equation. See next post for the solution.