Optimal wavelet reconstructions from Fourier samples via generalized sampling

We consider the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples. For this, we use the recently introduced framework of generalized sampling in the context of compactly supported orthonormal wavelet bases. Our first result demonstrates that using generalized sampling one obtains a stable and accurate reconstruction, provided the number of Fourier samples grows linearly in the number of wavelet coefficients recovered. We also present the exact constant of proportionality for the class of Daubechies wavelets. Our second result concerns the optimality of generalized sampling for this problem. Under some mild assumptions generalized sampling cannot be outperformed in terms of approximation quality by more than a constant factor. Moreover, for the class of so-called perfect methods, any attempt to lower the sampling ratio below a certain critical threshold necessarily results in exponential ill-conditioning. Thus generalized sampling provides a nearly-optimal solution to this problem. I. GENERALIZED SAMPLING A fundamental problem of signal processing is the reconstruction of signals from a discrete set of measurements. This can be formulated in a Hilbert Space H with inner product 〈·, ·〉, where one seeks to reconstruct a function f ∈ H from measurements of the form 〈f, sj〉 for some {sj}j∈N ⊆ S ⊆ H. A key development is the Shannon-Nyquist Sampling Theorem, which stated that bandlimited or compactly supported signals to be fully described via measurements 〈 f, e j· 〉 , j ∈ Z, for some appropriate > 0. In particular, f and its Fourier transform f̂(·) = ∫ f(x)e−ix·dx can be approximated respectively as follows: