Can anyone give me a hand with the proof of this properties?
Prove that:
a) The linear span of the set $\left\{T_bh/b\in\mathbb{R}\right\}$ is dense in $L_2(\mathbb{R})$, where $h(x)=e^{-\pi x^2}$.
b)$\blacksquare$ $V_{\phi}f(t,w)=e^{-2\pi itw}V_{\hat{\phi}}\hat{f}(w,-t)$.
$\blacksquare$ $V_{\phi}(T_{u}E_{\eta}f)(t,w)=e^{-2\pi iwu}V_{\phi}f(t-u,w-\eta)$.
where $V_{\phi}f(t,w)=\int_{\mathbb{R}}f(s)\overline{\phi(s-t)}e^{-2\pi iws}ds$, for $f,\phi\in{L_2(\mathbb{R}})$, with $\phi\neq0$.
($E_a$ is the modulation $E_af(t)=e^{2\pi iat}f(t)$, and $T_a$ is the traslation $T_af(t)=f(t-a)$, and $V_{\phi}f$ is the Gabor tranform of $f$)
Thank you very much in advance.
Once question a) is done, use (sesqui)linearity of the involved operators to do this when $f=T_ae^{-\pi x^2}$ and $\phi=T_be^{-\pi x^2}$.
For question a), what we have to show is that if $f\in L^2$ and for each $c$, we have $\langle f,T_ce^{-x^2}\rangle=0$ then $f=0$. To do that, use Plancherel identity and $\widehat{T_c(x\mapsto e^{-\pi x^2})}=e^{icx}e^{-\pi x^2}$. The function $x\mapsto \widehat f(x)e^{-\pi x^2}$ is integrable, and its Fourier transform is $0$, hence $\widehat f=0$ and $f=0$.