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In this paper we study the problem of computing wavelet coefficients ofcompactly supported functions from their Fourier samples. For this, we use therecently introduced framework of generalized sampling. Our first resultdemonstrates that using generalized sampling one obtains a stable and accuratereconstruction, provided the number of Fourier samples grows linearly in thenumber of wavelet coefficients recovered. For the class of Daubechies waveletswe derive the exact constant of proportionality.  Our second result concerns the optimality of generalized sampling for thisproblem. Under some mild assumptions we show that generalized sampling cannotbe outperformed in terms of approximation quality by more than a constantfactor. Moreover, for the class of so-called perfect methods, any attempt tolower the sampling ratio below a certain critical threshold necessarily resultsin exponential ill-conditioning. Thus generalized sampling provides anearly-optimal solution to this problem.

On optimal wavelet reconstructions from Fourier samples: linearity and universality of the stable sampling rate