Inverse Radon transform approximation and natural spaces of Fourier transformation In a CT reconstruction, the inverse Radon transformation $R^{-1}$ is realized using  "Fourier slice theorem/Projection slice theorem" and is covered in virtually every CT book or course. We have an unknown two dimensional scalar function $f(x,y)$ with the compact support. Its known integrals over the lines in plane are realized by means of functions $h^\alpha(s)$ $\alpha \in[0,\pi)$, $s \in \mathbb{R}$, they are supposed to be given by the Radon transform of $f$,
$$R(f(x,y)) = h^\alpha(s) = \int_{-\infty}^{\infty} f(s\cos \alpha + u \sin \alpha, s \sin \alpha - u \cos \alpha) du. $$
We can assume any meaningful behavior of $f$ and/or $h$ as being square integrable and sufficiently continuous. I am interested if it is sufficient for invertibility of its Radon transform. In particular I am interested in a conditions on two dimensional function $h^\alpha(s)$, for which exists a function $f(x,y)$ \begin{equation}f(x,y) = R^{-1} h^\alpha(s), \mbox{ that fullfils } h^\alpha(s) = R(f(x,y)),\end{equation} where $R$ is a Radon transform and $R^{-1}$ is its inverse by means of "filtered backprojection".
To formalize meaning of $R^{-1}$, we define a Fourier transform of $h^\alpha$ by $H^\alpha(w) = FT(h^\alpha(s))$ and introduce
$$\hat{h}^\alpha(s) = FT^{-1} (|w| H^\alpha(w)) = FT^{-1} (|w|) * h^\alpha(s) .$$
There is a formula
$$f(x,y) = R^{-1}(h^\alpha(s)) = \int_0^\pi \hat{h}^\alpha(x \cos\alpha + y \sin \alpha) d\alpha.$$
Now the "Fourier slice theorem/Projection slice theorem" can be used to show that $R$ and $R^{-1}$ are inverse operations.
So my problem is that CT textbooks do not really pay attention to the mathematical issues and assume that it is somehow working, we just have a set of $h^\alpha(s)$ and assume its a Radon transform of some $f$ that could be recovered by this inverse transformation.
What they pay attention to is that $FT^{-1} (|w|)$ has pole in $s=0$ and the convolution with such object is hard to perform, so they usually truncate it by considering $FT^{-1} (abs^a(w))$, where $abs^a(w)=|w|$ for $|w|<a$ and $0$ otherwise.
I am interested if the "CT book" approximation works for all functions $f(x,y)$ and/or $h$ and the following is true:
For every $f$ and/or $h$ with sufficient integrability, compact support or other conditions there exists some $a$ such that $FT^{-1} (|w| H^\alpha(w))  = FT^{-1}( abs^a(w) H^\alpha(w))$.
And can I at least claim that objects such as $|w| H^\alpha(w)$, $FT^{-1} (|w| H^\alpha(w))$ or $FT^{-1} (|w|) * h^\alpha(s)$ will in some sense be well-behaved functions?
I would also be interested in a theory of Fourier transformation of real valued functions on real domain. As it turns out that we often need a distributive sense of objects such as $FT^{-1} (|w|)$ I would like to know what are natural spaces of these distributions and if I have some nice theory for them such as the Hilbert space structure of the $L^2$ integrable functions on finite interval.
The definitions of Fourier transform that I use are as follows:
$$FT(f(s))(w) = F(w) = \int_{-\infty}^{\infty} f(s) \exp{(-2\pi i w s)}ds, $$
$$FT^{-1}(F(w))(s) = f(s) = \int_{-\infty}^{\infty} F(w) \exp{(2\pi i w s)}dw. $$
 A: There are many questions within your post, so I'm only giving an informal & partial answer:

*

*Does the Radon transform $Rf(a,b,t) = \int_{ax + by = t} f(x,y)dxdy$ always exist in a well-defined sense?  Yes, if f is continuous everywhere except perhaps at the origin where it may be $O((x^2+y^2)^{-\mu/2}), \mu>1$.     Also if f is $L^1$.  [See Rubin, 2015, p. 132].


*Does the inverse Radon transform always exist and is unique?  Here, things are more subtle because we are implicitly dealing with an inverse problem: given $h(a,b,t)$, can we find $f$ such that $h = Rf$ ? The answer very much depends on what you call a solution.  If a solution $f$ must be continuous, then the range theorem pretty much says that very few $h$ admit an inverse (see my post here for some details).  However, in $L^1$ solution space and a fortiori distributional spaces, the range of the Radon transform is much broader...  Regarding uniqueness, I think Helgason and his crowd found some esoteric $f_1$ and $f_2$ with identical $Rf_1=Rf_2$, but in CT practice engineers ignored this issue and never got into trouble.
It actually took me a while to wrap my head around the meaning of the range theorem.  Initially it seemed to mean that very few $h$ had inverses (potentially killing the non-CT problem I was working on), but luckily a colleague pointed out to me that standard theory is often limited to "boring" smooth functions when the "real" stuff we are often interested in is "kinky"...
I hope this helps putting you in the right direction.
References:
Rubin, Boris, Introduction to Radon transforms. With elements of fractional calculus and harmonic analysis, Encyclopedia of Mathematics and its Applications 160. Cambridge: Cambridge University Press (ISBN 978-0-521-85459-7/hbk). xvii, 576 p. (2015). ZBL1333.44002.
