Elias Stein's book "Real Analysis, Measure Theory, Integration, Hilbert spaces" chapter4 Exercise 9:\
let $H_1 = L^2([-\pi,\pi])$ be the Hilbert space of functions $F(e^{i\theta})$ on the unit circle with inner product $(F,G)= \frac{1}{2\pi} \int_{-\pi}^{\pi} F(e^{i\theta}) \overline G(e^{i\theta}) \,d\theta$
let $H_2$ be the space $L^2(\mathbb{R})$.
using the mapping $x \to \frac{i-x}{i+x}$ of $\mathbb{R}$ to the unit circle show that:
(a)The correspondence $U: F \to f$ whit $f(x) = \frac{1}{\pi^\frac{1}{2}(i+x)}F(\frac{i-x}{i+x})$ gives a unitary mapping of $H_1$ to $H_2$.
(b)As a result $\{\frac{1}{\pi^\frac{1}{2}}(\frac{i-x}{i+x})^n\frac{1}{i+x}\}_{n=-\infty}^\infty$ >is an orthonormal basis of $L^2(\mathbb{R})$.
In order to solve this problem I've shown that defined $U$ is a bijection and linear. I tried to show that $||F||_{H_1} = ||U(F)||_{H_2}$ and only idea I got is to use change of variable which I know from Riemann integration theory but I'm not sure same argument holds in Lebesgue integral.Does it?
Also to solve second part, is it enough to show that image of an orthonormal basis say$\{e^{in\theta}\}_{n=-\infty}^\infty$ equals the given set?