rt.representation theory – Density of the Mellin transform inside the direct integral of induced representations

I’m trying to better understand the continuous spectrum of $G = operatorname{GL}_2(mathbb A_{mathbb Q})$, which is the direct integral of induced representations $mathbf H(s) = operatorname{Ind}_{Z(mathbb A)N(mathbb A)T(mathbb R)^0T(mathbb Q)}^{G(mathbb A)} omega_s$. The reference I’m using is Forms of GL(2) from the analytic point of view by Gelbart and Jacquet in Corvallis. In the beginning, we fix a character $omega$ of $mathbb A^{ast 1}/mathbb Q^{ast}$, extended to a character of the center $Z(mathbb A)$ of $G(mathbb A)$, and define $mathbf H(s)$ to be the space of measurable functions $f: G(mathbb A) rightarrow mathbb C$ satisfying

$$phi(begin{pmatrix} a u alpha & x \ & a v beta end{pmatrix} g) = |u/v|^{s+1/2}omega(a)phi(g)$$
for all $a in mathbb A^{ast}, u,v in (0,infty), alpha, beta in mathbb Q^{ast}, x in mathbb A$, and $g in G$, satisfying the norm condition

$$|phi|_s^2 = intlimits_K intlimits_{mathbb A^{ast 1}/mathbb Q^{ast}} phi(begin{pmatrix} x \ & 1end{pmatrix}k)d^{ast}xdk < infty$$

The continuous spectrum $mathscr L = L^2(G(mathbb Q)Z(mathbb A)backslash G(mathbb A),omega)_{textrm{cont}}$ is supposed to be the direct integral of the Hilbert spaces $mathbf H(iy)$ along the line $y > 0$, with the Lebesgue measure. That is, $mathscr L$ is the space of measurable sections $a$ from $(0,infty)$ into the spaces $mathbf H(iy) : y > 0$ satisfying the norm condition

$$|a|^2 = int_0^{infty} |a(y)|_{iy}^2 dy < infty.$$

To prove that this is so, one needs to produce a dense subspace of $mathscr L$ by means of functions $f: G(mathbb A) rightarrow mathbb C$, compactly supported modulo $Z(mathbb A)N(mathbb A)P(mathbb Q)$, satisfying the condition $f(zn gamma g) = omega(z) f(g)$. One can define the Mellin transform

$$hat{f}(g,s) = int_0^{infty} f(begin{pmatrix} t \ & 1 end{pmatrix} g)|t|^{-s-1/2}d^{ast }t in mathbf H(s)$$

and define $a_f in mathscr L$ by

$$a_f(y) = hat{f}(g,iy).$$

The argument that $mathscr L$ is indeed isomorphic to a closed subrepresentation of $L^2(G(mathbb Q)Z(mathbb A)backslash G(mathbb A))$ seems to rely on something like the claim that such functions $a_f$ are dense in $mathscr L$. How do we actually know that this is the case? Do we actually have enough control over the functions $f$ so that their Mellin transforms $hat{f}(g,iy)$ can together approximate any square-integrable section in $mathscr L$?

I have not seen this claim argued anywhere that discusses the continuous spectrum of $operatorname{GL}(2)$, but something like this seems essential to making the whole thing work. This claim does seem very reasonable, and similar to the assertion that, say, the Fourier transforms of Schwarz functions on $mathbb R$ make a dense subspace of $L^2(mathbb R)$.