Sparse Legendre expansions via l1-minimization

We consider the problem of recovering polynomials that are sparse with respect to the basis of Legendre polynomials from a small number of random samples. In particular, we show that a Legendre s-sparse polynomial of maximal degree N can be recovered from m asymptotic to s log(4)(N) random samples that are chosen independently according to the Chebyshev probability measure dv(x) = pi(-1)(1 - x(2))(-1/2)dx. As an efficient recovery method, l(1)-minimization can be used. We establish these results by verifying the restricted isometry property of a preconditioned random Legendre matrix. We then extend these results to a large class of orthogonal polynomial systems, including the Jacobi polynomials, of which the Legendre polynomials are a special case. Finally, we transpose these results into the setting of approximate recovery for functions in certain infinite-dimensional function spaces. (C) 2012 Elsevier Inc. All rights reserved.