Exact polynomial reproduction for oscillatory radial basis functions on infinite lattices

Until now, only nonoscillatory radial basis functions (RBFs) have been considered in the literature. It has recently been shown that a certain family of oscillatory RBFs based on J-Bessel functions gives rise to nonsingular interpolation problems and seems to be the only class of functions not to diverge in the limit of flat basis functions for any node layout. This paper proves another interesting feature of these functions: exact polynomial reproduction of arbitrary order on an infinite lattice in n. First, a closed form expression is derived for calculating the expansion coefficients for any order polynomial in any dimension. Then, a proof is given showing that the resulting interpolant, using this class of oscillatory RBFs, will give exact polynomial reproduction. Examples in one and two dimensions are presented. It is specifically noted that such closed form expressions cannot be derived for other classes of RBFs due to the fact that J-Bessel RBFS reproduce polynomials via a different mechanism.