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In a series of recent papers (Adcock, Hansen and Poon, 2013, Appl. Comput.Harm. Anal. 45(5):3132-3167), (Adcock, Gataric and Hansen, 2014, SIAM J.Imaging Sci. 7(3):1690-1723) and (Adcock, Hansen, Kutyniok and Ma, 2015, SIAMJ. Math. Anal. 47(2):1196-1233), it was shown that one can optimally recoverthe wavelet coefficients of an unknown compactly supported function frompointwise evaluations of its Fourier transform via the method of generalizedsampling. While these papers focused on the optimality of generalized samplingin terms of its stability and error bounds, the current paper explains how thisoptimal method can be implemented to yield a computationally efficientalgorithm. In particular, we show that generalized sampling has a computationalcomplexity of $\mathcal{O}(M(N)\log N)$ when recovering the first $N$boundary-corrected wavelet coefficients of an unknown compactly supportedfunction from $M(N)$ Fourier samples. Therefore, due to the linearcorrespondences between the number of samples $M$ and number of coefficients$N$ shown previously, generalized sampling offers a computationally optimal wayof recovering wavelet coefficients from Fourier data.

A practical guide to the recovery of wavelet coefficients from Fourier measurements