I am currently working on computing convolution by Fast Fourier Transform (FFT). The functions are in 2D however with circular symmetry $f(\mathbf{r}) = f(r)$. I have already implemented 2D FFT to solve this problem.

If I change the real space and Fourier space variables into cylindrical coordinates, I will end up with a Hankel transform in 1D.

$$ F(\mathbf{k})=\iint f(\mathbf{r})e^{i\mathbf{k}\cdot\mathbf{r}}\operatorname{d}\!\mathbf{r}$$

$$F(\mathbf{k})=\int_{r=0}^\infty \int_{\theta=0}^{2\pi}f(r)e^{ikr\cos(\theta)}\,r\operatorname{d}\!\theta\operatorname{d}\!r$$

$$F(\mathbf{k})=F(k)= 2\pi\int_0^\infty f(r) J_0(kr) r\operatorname{d}\!r$$

I am wondering if I could carry out this 1D Hankel Transform by FFT?


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