Unconditional convergence of iterative approximation methods

Abstract

In this paper, we propose several iterative formats for quasi-interpolation by using positive definite radial basis functions. Firstly, we present the methodology of approximate interpolation, and then employ the diagonally compensated reduction to preprocess the iterative method. We show that the proposed iterative methods are convergent no matter which shape parameter is used, therefore it can transform a nonconvergent iterative format into a convergent one. The proposed iterative methods can approximate the corresponding exact interpolant with arbitrary precision. Numerical results illustrate the effectiveness of our methods.