The real Eisenstein series 


$G\_s^\* = \frac{\Gamma(s)}{\pi^s} \sum'\_{m,n}\frac{Im(\tau)}{|m+n \tau|^{2s}}$ 


admits the following integral representation (their Mellin transform):


$G\_s^\* = \frac{1}{2} \int\_0^\infty (\Theta\_\tau(t)-1) t^{s-1} dt$


where $\Theta\_\tau(t)$ is the associated theta series $\Theta\_\tau(t) = \sum\_{m,n}e^{-\pi Q\_\tau(m,n)t}$ . In general, we can construct real modular invariant quantities by considering:


$\hat M(\tau) = \frac{1}{2} \int\_0^\infty (\Theta\_\tau(t)-1) M(t) dt$


Does anyone know if it is possible to invert this relation? I.e. given a real modular form $\hat M(\tau)$, is it possible to find $M(t)$? Are there any references I could look at? Thank you very much!



Real modular form, inverse transform
Yes, this is essentially the $L^2$ spectral decomposition of the orthogonal complement to cuspforms (waveforms). It is spanned (for example) by pseudo-Eisenstein series $E\_\phi$, formed by winding up functions of the form $\Phi(x+iy)=\phi(y)$ with $\phi\in C^o\_c(0,\infty)$. The more mundane spectral decomposition of $\phi(y)$ on $(0,+\infty)\sim \mathbb R$ by ordinary Fourier transform, and a few further manipulations, expresses pseudo-Eisenstein series as integrals nearly of the sort given in the question. (It may be better to remove the factor $\zeta(2s)$ from the expression as posed.)


The previous scenario produces functions $M({1\over 2}+it)$ in the Paley-Wiener space as functions of $t\in \mathbb R$; then one proves that the map is an $L^2$ isometry on this part of the spectrum, and extends to full Plancherel by extending by continuity.


Given an automorphic form $f$ in the orthogonal complement to the cuspforms, the $s$-th spectral component is $\langle f, E\_s\rangle$, where $E\_s$ is the Eisenstein series with the $\zeta(2s)$ and Gamma factor removed, so that its Fourier expansion begins $y^s+c\_sy^{1-s}+\ldots$ where $c\_s=\xi(2s-1)/\xi(2s)$, with $\xi(s)$ the completed zeta.


One version of that discussion is at
<http://www.math.umn.edu/~garrett/m/mfms/notes_c/cont_afc_spec.pdf>
I first saw the argument in Godement's articles in the Boulder Conference, AMS Proc Symp Pure Math IX.


