Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions

Efficient recovery of smooth functions which are s-sparse with respect to the basis of so-called prolate spheroidal wave functions from a small number of random sampling points is considered. The main ingredient in the design of both the algorithms we propose here consists in establishing a uniform L? bound on the measurement ensembles which constitute the columns of the sensingmatrix. Such a bound provides us with the restricted isometry property for this rectangular random matrix, which leads to either the exact recovery property or the "best s-term approximation" of the original signal by means of the 1 minimization program. The first algorithm considers only a restricted number of columns for which the L? holds as a consequence of the fact that eigenvalues of the Bergman's restriction operator are close to 1 whereas the second one allows for a wider system of PSWF by taking advantage of a preconditioning technique. Numerical examples are spread throughout the text to illustrate the results.