I read that the Poisson Kernel for the upper half-plane, $K_y(x)=\frac{1}{\pi}\frac{y}{y^2+x^2}$ is an approximation to the identity. The text states this without proof and I am hoping to see a proof of this fact. Thank you in advance for the help.
Asked
Active
Viewed 235 times
0
-
Numerator should be $y$ not $y^2$. Take a look at Dirac's $\delta$ distribution smooth approximation. Use what I explained there with $n=1$, $\zeta(x)=\frac{1}{\pi}\frac{1}{1+x^2}$ and $c=1$. – peek-a-boo Nov 06 '21 at 02:34
-
@peek-a-boo you are right! My mistake. Also, I am having trouble following your other answer. I think it’s just a bit over my head. I corrected the error above. – Abdul Nov 06 '21 at 03:31