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.