The FFT of f function that you seek (i.e. F) is two-dimensional in frequency space, in a trivial manner, even though your spatial function f is one-dimensional. To compute the desired F* using ffts, note that the second fourier frequency f2 is a phase ramp applied to f before the f1-frequency fft, which can be interpreted using the Fourier shift theorem. F* can then be tconstructed by a single initial fft on the function f, followed by a sequence of circshifts to replicate the effect of the f2 phase ramp. Rather than loop through a sequence of circshifts, it is faster to note that the structure of F looks like a Toeplitz matrix. Hence this psuedo-code ought to do the trick:
temp = fft(arr);
Fstar = conj(fliplr(toeplitz([temp(1) fliplr(temp(2:end))], temp)));
Bjorn Gustavsson's toolbox advice is worth following, as there's a decent body of good literature around the bispectrum and triple correlation (mostly in IEEE journals) which present a variety of robust statistical methods for estimating these quantities.
My comment here is just to address your specific question, as this particular fft query had also troubled me when I wanted to improve my basic understanding of the bispectrum (which remains basic, despite having read the literature).
