Baddour (2011) derive a general relation (i.e., without assumed symmetries) between the Fourier transform in polar coordinates and the Hankel transform. Specifically, for a function in physical space $f(r, \theta)$ the Fourier transform is
$$ \begin{align} F(\vec\omega) &= F(\rho, \psi) \\ &= \sum_{n=-\infty}^{\infty}F_n(\rho)e^{i n \psi}\label{a}\tag{1} \end{align} $$ where $$ F_n(\rho) = 2\pi i^{-n}\mathbb{H}_n \{ f_n(r) \} $$ and $$ f_n(r) = {{1}\over{2\pi}}\int_{0}^{2\pi}f(r,\theta)e^{-in\theta}d\theta\label{b}\tag{2} $$
are the Fourier coefficients for $F(\vec\omega)$ and $f(r,\theta)$, respectively (see Section 3.2 of Baddour for details).
Wang et al. (2008) showed that using a Fourier transform on a finite radial range required additional k-space terms in the Fourier coefficients (see Section 2.3.2 for details). Admittedly, I did not follow exactly how those terms were derived, and they may be specific to the Helmholtz equation they were solving.
Given all of this, I now wish to attempt to use Equation (1) on a range finite in both $r$ and $\theta$. In particular, my physical region is a sequence of concentric annuli each of thickness $2t$ centered on a circle of radius $R$ and discretized into $N$ equal azimuthal bins of size $\theta_m = 2\pi/N$ for $0 \le m \le N$. There are no symmetries in my physical region.
For this problem, my assumption is that Equation (2) would look something like
$$ f_{n,m}(r) = {{1}\over{2\pi}}\int_{\theta_m}^{\theta_{m+1}}f(r,\theta)e^{-in\theta}d\theta\label{c}\tag{3} $$
However, I hesitate to write this as it breaks the "infinite" transform formulation of Equation (1). At the end of the day, I am interested in plotting the Fourier amplitudes as a function of both $r$ and $\theta$ such that $A_{nk}(r,\theta)$ is the amplitude of the $n$th mode of the radial component and the $k$th mode of the azimuthal component (corresponding to a particular region $\theta_m$).
My questions are thus
How do I get started deriving the necessary coefficients for the finite range in $r$? (Even just a reference to a text would be great!)
Is it legal to formulate some equation that reflects the intention of Equation (3)?
Update
As an interim workaround, I decided to convolve my input with a simple windowing function that would select each finite reach. For now, I am ignoring the errors introduced by ringing (Gibbs_phenomenon). Using the windowing function
$$ W_{k}(r,\theta;R) = \begin{cases} 1, & \text{if} \left\vert R - r\right\vert \leq t \text{ and } \theta_{k} \leq \theta \leq \theta_{k+1} \\ 0, & \text{otherwise} \end{cases}, $$
my problem reduces to
$$ g(r,\theta) = f(r,\theta)\ast\ast W_{k}(r,\theta;R) $$
The Fourier coefficients for the windowing function are just
$$ W_{k}(r;R) = {1 \over {2\pi}} \int_{0}^{2\pi} W_{k}(r,\theta;R)e^{-ik\theta}d\theta $$
See Section 8 of Baddour (2011) for a proof that the convolution theorem holds for the 2D polar Fourier Transform.
Any thoughts or suggestions for the original problem are still very much welcome!
References
Baddour, Natalie. Two-Dimensional Fourier Transforms in Polar Coordinates. Advances in Imaging and Electron Physics 165. 2011.
Wang, Qing; Ronneberger, Olaf; Burkhardt, Hans. Fourier Analysis in Polar and Spherical Coordinates. ALBERT-LUDWIGS-UNIVERSITAT FREIBURG INSTITUT FUR INFORMATIK Internal Report. 2008.