Univariate Multiquadric Approximation: Reproduction of Linear Polynomials

Abstract

It is known that multiquadric radial basis function approximations can reproduce low order polynomials when the centres form an infinite regular lattice. We make a start on the interesting question of extending this result in a way that allows the centres to be in less restrictive positions. Specifically, univariate multiquadric approximations are studied when the only conditions on the centres are that they are not bounded above or below. We find that all linear polynomials can be reproduced on IR, which is a simple conclusion if the multiquadrics degenerate to piecewise linear functions. Our method of analysis depends on a Peano kernel formulation of linear combinations of second divided differences, a crucial point being that it is necessary to employ differences in order that certain infinite sums are absolutely convergent. It seems that standard methods cannot be used to identify the linear space that is spanned by the multiquadric functions, partly because it is shown that this space provides uniform convergence to any continuous function on any finite interval of the real line.