B-spline quasi-interpolation on sparse grids

We propose a periodic B-spline quasi-interpolation for multivariate functions on sparse grids and develop a fast scheme for the evaluation of a linear combination of B-splines on sparse grids. We prove that both of these operations require only O(nlogd−1n)O(nlogd−1n) number of multiplications, where nn is the number of univariate B-spline basis functions used in each coordinate direction and dd is the number of variables of the functions. We also establish the optimal approximation order of the periodic B-spline quasi-interpolation. Numerical examples are presented to confirm the theoretical estimates.