Approximation in rough native spaces by shifts of smooth kernels on spheres

Within the conventional framework of a native space structure, a smooth kernel generates a small native space, and “radial basis functions” stemming from the smooth kernel are intended to approximate only functions from this small native space. Therefore their approximation power is quite limited. Recently, Narcowich et al. (J. Approx. Theory 114 (2002) 70), and Narcowich and Ward (SIAM J. Math. Anal., to appear), respectively, have studied two approaches that have led to the empowerment of smooth radial basis functions in a larger native space. In the approach of [NW], the radial basis function interpolates the target function at some scattered (prescribed) points. In both approaches, approximation power of the smooth radial basis functions is achieved by utilizing spherical polynomials of a (possibly) large degree to form an intermediate approximation between the radial basis approximation and the target function. In this paper, we take a new approach. We embed the smooth radial basis functions in a larger native space generated by a less smooth kernel, and use them to approximate functions from the larger native space. Among other results, we characterize the best approximant with respect to the metric of the larger native space to be the radial basis function that interpolates the target function on a set of finite scattered points after the action of a certain multiplier operator. We also establish the error bounds between the best approximant and the target function.