Background: The Fourier Transform is - at least partly due to its factorizable nature (FFT) - arguably one of the most impactful integral transforms in science and engineering.
When performing a Fourier Transform on signals of more than one dimension, what is commonly done is to utilize the separability of the transform on a Cartesian grid. This way it conveniently becomes possible to perform the whole multidimensional FFT as a sequence of one-dimensional FFTs - one along each dimension.
However this only works on ON coordinate systems, in other words Cartesian grids. What happens if we want to perform on some other coordinate system - for example a polar coordinate system:
Example of basis functions on a polar grid in 2D.
Can we still derive a way to use the speed-ups provided by a FFT factorization ?
Clarification : my signal is still sampled on a Cartesian grid like a computer bitmap image is, but the Fourier basis functions should be separable like $\exp(-2\pi i r) \exp(-2\pi i \theta)$ with origo in the middle of the image.