Efficient and stable recovery of Legendre-sparse polynomials

We consider the recovery of polynomials that are sparse with respect to the basis of Legendre polynomials from a small number of random sampling points. We show that a Legendre s-sparse polynomial of maximal degree N can be recovered from m ≍ s log⁴(N) random samples that are chosen independently according to the Chebyshev probability measure π^Ȓ1(1 - x²)^Ȓdx on [Ȓ1; 1]. As an efficient recovery method, ℓ1-minimization can be used. We establish these results by showing the restricted isometry property of a preconditioned random Legendre matrix. Our results extend to a large class of orthogonal polynomial systems on [Ȓ1; 1]. As a byproduct, we obtain condition number estimates for preconditioned random Legendre matrices that should be of interest on their own.