Sparse Legendre expansions via ℓ1ℓ1-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 ss-sparse polynomial of maximal degree NN can be recovered from m≍slog4(N)m≍slog4(N) random samples that are chosen independently according to the Chebyshev probability measure dν(x)=π−1(1−x2)−1/2dxdν(x)=π−1(1−x2)−1/2dx. As an efficient recovery method, ℓ1ℓ1-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.