Properties of generators of quasi-interpolation operators of high approximation orders in spaces of polyharmonic splines

We have presented in Bozzini et al. (2011) a procedure in spaces of m-harmonic splines in Rd that starts from a simple generator φ0 and recursively defines generators φ1, φ2,.,φm-1 with corresponding quasi-interpolation operators reproducing polynomials of degrees 3, 5,.,2m-1 respectively. In this paper we study the properties of generators φj, and we prove that these new generators are positive definite functions, and are scaling functions whenever φ0 has those properties. Moreover φ0 and φj generate the same multiresolution analysis. We show that it is possible to define a convergent subdivision scheme, and to provide in this way a fast computation of the quasi-interpolant.