Effective band-limited extrapolation relying on Slepian series and `1 regularization

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 Functionsʹʹ (PSWF, as originally proposed by Slepian), whereas the higher ones can be handled by the recent so-called ``Compressive Samplingʹʹ (CS, proposed by Candès) 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 signalsʹʹ, 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.