radial polynomials as alternatives to flat radial basis functions

Due to the high approximation power and simplicity of computation of smooth radial basis functions (rbfs), in recent decades they have received much attention for function approximation. these rbfs contain a shape parameter that regulates their approximation power and stability but its optimal selection is challenging. to avoid this difficulty, this paper follows a novel and computationally efficient strategy to propose a space of radial polynomials with even degree that well approximates flat rbfs. the proposed space, mathcal{h}_n, is the shifted radial polynomials of degree 2n. by obtaining the dimension of mathcal{h}_n and introducing a basis set, it is shown that mathcal{h}_n is considerably smaller than mathcal{p}_{2n} while the distances from rbfs to both mathcal{h}_n and mathcal{p}_{2n} are nearly equal. for computation, by introducing new basis functions, two computationally efficient approaches are proposed. finally, the presented theoretical studies are verified by the numerical results.