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 $(M\log D)^2$. This result is almost optimal within the $(\log D)^2 $ factor. The simulations show that the deterministic sampling can offer reconstruction performance similar to the random sampling.