Orthogonal polynomial expansions on sparse grids

We study the orthogonal polynomial expansion on sparse grids for a function of d variables in a weighted L space. A fast algorithm is developed to compute the orthogonal polynomial expansion by combining the fast cosine transform, a fast transform from the Chebyshev orthogonal polynomial basis to the orthogonal polynomial basis for the weighted L space, and a fast algorithm of computing hierarchically structured basis functions. The total number of arithmetic operations used in the algorithm is O(n log n) where n is the highest polynomial degree in one dimension. The exponential convergence of the approximation for the analytic function is investigated. Specifically, we show the sub-exponential convergence for analytic functions and moreover we prove the approximation order is optimal for the Chebyshev orthogonal polynomial expansion. We furthermore establish the fully exponential convergence for functions with a somewhat stronger analytic assumption. Numerical experiments confirm the theoretical results and demonstrate the efficiency of the proposed algorithm.