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

Ben Adcock, Anders C Hansen, Clarice Poon

Research output: Contribution to journalArticle

18 Citations (Scopus)


In this paper we study 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. 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. For the class of Daubechies wavelets we derive the exact constant of proportionality.
Our second result concerns the optimality of generalized sampling for this problem. Under some mild assumptions we show that 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.
Original languageEnglish
Pages (from-to)387-415
Number of pages29
JournalApplied and Computational Harmonic Analysis
Issue number3
Early online date2 Aug 2013
Publication statusPublished - 1 May 2014

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