Let $\Omega > 0$ and denote by $\mathcal{B}_\Omega$ the subspace of $L^2(\Bbb R)$ consisting of signals that are bandlimited to $(-\Omega, \Omega)$. Denote $\mathcal{L}_{\Omega} : L^2(\Bbb R) \rightarrow L^2(\Bbb R)$ the ideal lowpass filter that projects signals $x \in L^2(\Bbb R)$ into $\mathcal{B}_\Omega$: i.e., $\mathcal{L}_\Omega$ takes $x$ to the signal $y$ with
\begin{equation} \hat{y} =\begin{cases} \hat{x} & \text{for } |\omega| < \Omega\\ 0 & \text{otherwise} \end{cases} \end{equation}
A function $\gamma: \Bbb R \rightarrow \Bbb R$ is monotone increasing if $\gamma (t_2) > \gamma (t_1)$. The companding of a signal $x: \Bbb R \rightarrow \Bbb R$ by $\gamma$ is the signal $[\gamma \circ x](t) = \gamma(x(t))$.
Denote $u(t) = sinc(2\Omega t)$. Find an explicit expression of $\mathcal{L}_\Omega (u)(t)$.
If $x \in \mathcal{B}_\Omega$ and $\gamma$ is monotone increasing, then $\gamma \circ x$ is generally not bandlimited. In particular, $\mathcal{L}_\Omega (\gamma \circ x) \neq \gamma \circ x$. Suppose $x_1, x_2 \in \mathcal{B}_\Omega$ and $\mathcal{L}_\Omega (\gamma \circ x_1) = (\gamma \circ x_2)$. Show that $x_1 = x_2$ almost everywhere on $\Bbb R$.
My Attempt/Input
So I know this is regarding the recovery of bandlimited signals after companding and running through a low pass filter.
For the first part, this is essentially asking for an expression for when the sinc function is run through the low pass filter, correct? But the frequency of the sinc function doesn't change so wouldn't the whole signal go through?
As for the second part, I know I start with the integral of the product of $(x_1 - x_2)$ with $(\gamma \circ x_1 - \gamma \circ x_2)$ but not sure how to go about it.