Fast Algorithms for Periodic Spline Wavelets on Sparse Grids

We consider Boolean sums of univariate interpolation operators which define multivariate jth order blending interpolation operators on sparse grids. Sample spaces are defined as range of the blending operators. Sample and wavelet spaces have significantly lower dimension and good approximation order for certain function spaces. Fast decomposition and reconstruction algorithms for bivariate spline wavelets, based on algorithms for univariate functions, are described. Operation counts for the algorithms are given and it is shown that the complexity depends linearly on the dimension of sample spaces.