The following is for $D=4$. The correlators at a fixed point are power laws of the form $x^{-2\Delta}$, where $\Delta$ is the scaling dimension. Suppose I wish to find the nature of the spectrum at the fixed point, for which I calculate the spectral function $\rho(p^2)$ which is defined so that $$\langle\phi(x)\phi(0)\rangle=\int \frac{d^4p}{2\pi}^4e^{-ipx}\rho(p^2)$$
Now, for $\Delta=1$, I expect this to be the same as that of a fundamental scalar field, with $\rho(p^2)=\delta(p^2)$.
$\Delta=2$ should correspond to a composite operator of 2 massless fields, and thus I expect $\rho(p^2)=\int d^4k\delta[(k-p)^2]\delta(k^2)$ and so on.
However, I am unable to derive these relations formally. Any help would be appreciated.
As an example of the kind of attempts I made-none of which I claim to be rigorous in any way-note that the problem reduces to finding the fourier transform of $\frac{1}{x^2}$ and $\frac{1}{x^4}$. I tried introducing a regulator to control $x\to 0$, but got nowhere. Another approach was to call $\int d^4x e^{ipx}\frac{1}{x^4}=f(p)$, and find the differential equation for $f(p)$ by differentiating both sides with respect to $p$ until the LHS reduces to something like $\int d^4p e^{ipx}=\delta(x)$. It is likely that this is infact the right way to proceed, but I'm a little lost and frustrated by what should have been a simple calculation.
