On Quasi-interpolation with Radial Basis Functions

Abstract

It has been known since 1987 that quasi-interpolation with radial functions on the integer grid can be exact for certain order polynomials. If, however, we require that the basis functions of the quasi-interpolants be finite linear combinations of translates of the radial functions, then this can be done only in spaces whose dimension has a prescribed parity. In this paper we show how infinite linear combinations of translates of a given radial function can be found that provide polynomial exactness in spaces whose dimensions do not have this prescribed parity. These infinite linear combinations are of a simple form. They are, in particular, easier to find than the cardinal functions of radial basis function interpolation, which provide polynomial exactness in all dimensions. The techniques that are used in this work also give rise to some remarks about interpolation with radial functions both on the integers and on the nonnegative integers.