Deterministic Sampling of Sparse Trigonometric Polynomials

One can recover sparse multivariate trigonometric polynomials from few randomly taken samples with high probability (as shown by Kunis and Rauhut). We give a deterministic sampling of multivariate trigonometric polynomials inspired by Weil’s exponential sum. Our sampling can produce a deterministic matrix satisfying the statistical restricted isometry property, and also nearly optimal Grassmannian frames. We show that one can exactly reconstruct every M -sparse multivariate trigonometric polynomial with fixed degree and of length D from the determinant sampling X, using the orthogonal matching pursuit, and |X| is a prime number greater than (M logD). This result is optimal within the (logD) factor. The simulations show that the deterministic sampling can offer reconstruction performance similar to the random sampling.