Hello,


I am stuck with the following (hopefully not too trivial) problem.
I want to know, if the map
$${\cal D}(\mathbb{R}^2)\to L^2(H\_m,d\Omega\_m)\qquad f \mapsto \hat{f}|\_{H\_m}$$
has dense range.


Here $H\_m$ is the "upper mass shell" $\{ p\in \mathbb{R}^2:p\_0>0, p^2=m \}$ in the 2-dimensional Minkowski space, $d\Omega\_m=dp\_1/\sqrt{p\_1^2+m^2}$ the Lorentz invariant measure,
$\hat{f}$ is the Fourier Transform.


In Reed-Simon II, Chapter X, Exercise 44 one has to show that the Schwartz functions have dense range.
I think I have proved this, but I don't know how to extend it, to $\mathcal{D}$, if this is possible at all.


For the Schwartz functions I would show, that there are functions s.t.
$\hat{f}, p\_1\hat{f},\dots,p\_1^n\hat{f},\cdots$ 
is a complete system in $L^2(H\_m,d\Omega\_m)$.
But for these functions I need 
$| \hat{f}(\sqrt{p\_1^2 + m^2}, p\_1) | \leq c \exp(-a |p\_1|)$, 
with $c,a>0$ some constants.


Is there any function in $\mathcal{D}$ that fullfills this?
Is there another way to prove this?


Any suggestions are very welcome.



Fourier Transforms restricted to mass shell
I don't have any book in front of me at the moment, so I am a bit improvising here, but you'll find this in eg. Reed Simon 2,
Streater Wightman, or Josts book.


Let us call the map $$E:{\cal D}(\mathbb{R}^2)\to H=L^2(H\_m,d\Omega\_m)\qquad f \mapsto \hat{f}|\_{H\_m}.$$ I claim that already $E(\mathcal D(O))$ with $O$, let's say, compact with non-trivial interior is dense in the Hilbert space, where $\mathcal D(O)$ are the smooth functions with support in $O$. Let us take a vector $v\in H$ orthogonal to $\mathcal D(O)$, i.e. $(v,E(\mathcal D(O)))=0$. We have to show that $v=0$ for which it is actually enough to show that 
$(v,E(\mathcal S(\mathbb R))=0$, because the Schwartz functions are dense. 
Now let us define the map
$$
T:\mathcal S(\mathbb R^2) \to H: f \mapsto T(f):=(v,E(f))
$$
which turns out to be continous, i.e. a tempered distribution. We have to show that $T=0$.


The Fourier transform of this distribution has support on the forward light cone (even on the mass shell),
so it turn out that $T$ is the boundary value of an analytic function $\tilde T$ on $\mathcal T=\mathbb R^2+ i V\_+$ (this depends a bit on your convention) where $V\_+$ is the forward light-cone. It means that $$T(f) = \lim\_{\mathcal T \ni z\to 0} \int\_{\mathbb R^2} \tilde T(x+z)f(x)\mathrm dx.$$ But because $T \restriction\_{\mathcal D(O)}=0$ it follows that $\tilde T \restriction\_O=0$. By the edge of the wedge theorem $\tilde T \equiv 0$ on whole $\mathcal D$ and therefore $T=0$, what we wanted to show.


