Poisson kernel approaches Dirac delta proof

Question

I know that the Poisson kernel is given by $$ \frac{1}{2\pi}\frac{(1-r^2)}{(1-2r\cos\theta+r^2)} $$ I'm trying to prove this converges to Dirac delta $\delta(\theta)$ when $r \rightarrow 1^-$ and have no IDEA, can you please help ?

Thanks a lot!

Answer 1

Do you know the following practical criterion allowing to say that a sequence $f_n(x)$ that converges to $\delta$ ?

It is enough (sufficient condition, thanks to David C. Ullrich who has pointed this) for this sequence to verify three items:

• For any fixed $x \ne 0$: $ \ \lim_{n \to \infty}f_n(x)=0$.

• (if $x = 0$) $ \ \lim_{n \to \infty}f_n(0)=\infty$.

• For all $n$: $ \ \int_{-\infty}^{\infty} f_n(x)dx=1 \ $(normalization).

Now take a look at https://math.stackexchange.com/q/1022983 ...