Multiscale radial kernels with high-order generalized Strang-Fix conditions

Abstract

The paper provides a general and simple approach for explicitly constructing multiscale radial kernels with high-order generalized Strang-Fix conditions from a given univariate generator. The resulting kernels are constructed by taking a linear functional to the scaled f -form of the generator with respect to the scale variable. Equivalent divided difference forms of the kernels are also derived; based on which, a pyramid-like algorithm for fast and stable computation of multiscale radial kernels is proposed. In addition, characterizations of the kernels in both the spatial and frequency domains are given, which show that the generalized Strang-Fix condition, the moment condition, and the condition of polynomial reproduction in the convolution sense are equivalent to each other. Hence, as a byproduct, the paper provides a unified view of these three classical concepts. These kernels can be used to construct quasi-interpolation with high approximation accuracy and construct convolution operators with high approximation orders, to name a few. As an example, we construct a quasi-interpolation scheme for irregularly spaced data and derived its error estimates and choices of scale parameters of multiscale radial kernels. Numerical results of approximating a bivariate Franke function using our quasi-interpolation are presented at the end of the paper. Both theoretical and numerical results show that quasi-interpolation with multiscale radial kernels satisfying high-order generalized Strang-Fix conditions usually provides high approximation orders.