I am looking for a reliable and fast way of evaluating an integral like $$ F(r, \phi)= \int_0^1 \int_0^{2\pi} f(\rho, \theta) e^{2\pi i \rho r \cos(\theta - \phi)}\rho d\theta d\rho, $$ where $f$ is a complex-valued function defined on the unit disk $\mathbb{D}=\{|z|\leq 1 \} $, for many values of $(r, \phi)$ and for several functions $f$. So, quadrature-based methods shouldn't work well here. Apparently neither the approximation by a discrete Fourier transform (evaluated by the FFT) works well. There is a so-called Extended Nijboer-Zernike (ENZ) Analysis (http://www.nijboerzernike.nl/) that tries to do it semi-analytically, approximating $f$ by Zernike polynomials, but still it is not totally satisfactory.

So, my question is: what is the state of the art of the numerical computation of such type of integrals?