(**UPDATED** for rapid decay considerations + new question)

In dimension 2, the Radon transform range theorem states that a *rapidly decaying* (Schwartz) function $g(t,\theta)$ can be represented as a Radon transform of some function $f(x,y)$ (i.e. $g=R[f]$) if and only if, for all integers $n\geq0$
$$P_n(\theta) := \int\limits_{-\infty}^{\infty} t^n g(t,\theta) dt$$ is a homogeneous polynomial *of degree* $n$ in $\cos\theta$ and $\sin\theta$. This is often referred to as the *moment* or *Cavalieri conditions*. See e.g. Helgason's book p.5, Lemma 2.2 (2011 ed.) for the property that $P_n$ must be a homogeneous polynomial of degree $n$.

**Question 1**: If $f$ is a radial function then its Radon transform $g=R[f]$ is known to be independent from $\theta$. Therefore all moments $P_n(\theta)$ are also independent from $\theta$, in apparent violation of the property that $P_n$ is a homogeneous polynomial in $\cos\theta, \sin\theta$ of degree $n$ when $n\geq1$. What am I missing?

For example, consider $f(x,y) = e^{-x^2-y^2} / \sqrt\pi$. Then $g(t,\theta) = e^{-t^2}$ which is independent from $\theta$ as expected of radial functions. The first moment is 0 which is NOT a homogeneous polynomial of degree 1, the second moment is $\sqrt\pi/2$ which is NOT a homogeneous polynomial of degree 2, and so forth.

**Question 2:** What happens when the above integral does not converge? Usually this happens when there is no solution to the Radon transform inverse problem, but consider $g(t,\theta) = (1-e^{-1/t^2})/|t|$ which is independent from $\theta$. After calculations, the inversion formula for radial function gives $$f(x,y) = f_0\!\left(\sqrt{x^2+y^2}\right), \qquad f_0(r) = \frac2{\pi r^3}\mathfrak D\!\left(\frac 1r\right)$$ where $\mathfrak D(x) := e^{-x^2}\int_0^x e^{t^2} dt$ is Dawson's function. So there exists $f$ such that $g=R[f]$, and yet
$$
P_n(\theta) = \int\limits_{-\infty}^{\infty} t^n \frac{1-e^{-1/t^2}}{|t|} dt
$$
does not converge for $n\geq 2$.

**My partial answer to Q2**: This specific example is not a Schwartz function. Any references to range theorems for non-Schwartz functions appreciated. I found "A Range Theorem for the Radon Transform" (Madych and Solmon, 1988), other suggestions very appreciated.

Thanks! p.