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

In a series of recent papers [B. Adcock, A. C. Hansen, and C. Poon, Appl. Comput. Harmon. Anal., 36 (2014), pp. 387--415; B. Adcock, M. Gataric, and A. C. Hansen, SIAM J. Imaging Sci., 7 (2014), pp. 1690--1723; Adcock et al., SIAM J. Math. Anal., 47 (2015), pp. 1196--1233], it was shown that one can optimally recover the wavelet coefficients of an unknown compactly supported function from pointwise evaluations of its Fourier transform via the method of generalized sampling. While these papers focused on the optimality of generalized sampling in terms of its stability and error bounds, the current paper explains how this optimal method can be implemented to yield a computationally efficient algorithm. In particular, we show that generalized sampling has a computational complexity of $\mathcal{O}\left(M(N)\log N\right)$ when recovering the first $N$ boundary-corrected wavelet coefficients of an unknown compactly supported function from $M(N)$ Fourier samples. Therefore, due to the linear correspondences between the number of samples $M$ and the number of coefficients $N$ shown previously, generalized sampling offers a computationally optimal way of recovering wavelet coefficients from Fourier data.