Effective band-limited extrapolation relying on Slepian series and l1 regularization

Authors:
Laurent Gosse
Affiliations:
IAC-CNR "Mauro Picone" (sezione di Bari), Via Amendola 122/I - 70126 Bari, Italy
Venue:
Computers & Mathematics with Applications
Year:
2010

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Abstract

We consider a rather simple algorithm to address the fascinating field of numerical extrapolation of (analytic) band-limited functions. It relies on two main elements: namely, the lower frequencies are treated by projecting the known part of the signal to be extended onto the space generated by ''Prolate Spheroidal Wave Functions'' (PSWF, as originally proposed by Slepian), whereas the higher ones can be handled by the recent so-called ''Compressive Sampling'' (CS, proposed by Candes) algorithms which are independent of the largeness of the bandwidth. Slepian functions are recalled and their numerical computation is explained in full detail, whereas @?^1 regularization techniques are summarized together with a recent iterative algorithm which has been proved to work efficiently on so-called ''compressible signals'', which appear to match rather well the class of smooth bandlimited functions. Numerical results are displayed for both numerical techniques and the accuracy of the process consisting of putting them all together is studied for some test-signals showing a quite fast Fourier decay.