In order to compute $$ \hat{f}_k = \sum_{m = 0}^{M - 1} e^{-2\pi i k m \theta} f_m, \ \ k = 0, ..., M - 1$$
for any $\theta$, my book states that this can be done using fractional Fourier transform as follows.
So, as you can see, they define the $c$-vector, then the $g$-vector, and then the final equation is a 2-times application of the FFT and 1-time application of the inverse FFT.
Is this equation correct? I have implemented it in code using some simple sequence $f_m$, and it does not seem to work.
The resulting vector $\hat{f}_k$ does indeed work ... but not for all $k$! It is wrong for the first few $k$ ($k = 0, 1, ...$) and only equals the correct values for the bigger $k$ ($k = M - 1, M - 2, ...$). So I am not sure what is going wrong here since the equation clearly is doing half the job but ...
I used R with its FFT-function.
