Fourier 'rules' for using 2D Fourier transforms in polar coordinates

What is it about?

Part of clever use of the Fourier transform is using the 'rules' that go with it that tells us how one operation in one domain becomes another operation in the Fourier domain. For example, shift/modulation, convolution/multiplication are well known rules of the Fourier transform in 1D. This paper derives the equivalent set of rules for Fourier transforms in 2D POLAR coordinates.

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Why is it important?

The set of Fourier 'rules' make it easier to use 2F Fourier transforms in polar coordinates since often one complicated operation (e.g. convolution) can be replaced with an easier one in the other transform domain (e.g. convolution in space becomes multiplication in the Fourier domain). Another interesting aspect about this paper is an explanation of the subtleties of a convolution operation. A TRUE (properly defined) convolution will imply an equivalent multiplication in the Fourier domain.