Closed form representations and properties of the generalised Wendland functions

In this paper we investigate the generalisation of Wendland’s compactly supported radial basis functions to the case where the smoothness parameter is not assumed to be a positive integer or half-integer and the parameter ℓℓ, which is chosen to ensure positive definiteness, need not take on the minimal value. We derive sufficient and necessary conditions for the generalised Wendland functions to be positive definite and deduce the native spaces that they generate. We also provide closed form representations for the generalised Wendland functions in the case when the smoothness parameter is an integer and where the parameter ℓℓ is any suitable value that ensures positive definiteness, as well as closed form representations for the Fourier transform when the smoothness parameter is a positive integer or half-integer.