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 mm-harmonic splines in RdRd that starts from a simple generator ϕ0ϕ0 and recursively defines generators ϕ1,ϕ2,…,ϕm−1ϕ1,ϕ2,…,ϕm−1 with corresponding quasi-interpolation operators reproducing polynomials of degrees 33, 5,…,2m−15,…,2m−1 respectively. In this paper we study the properties of generators ϕjϕj, and we prove that these new generators are positive definite functions, and are scaling functions whenever ϕ0ϕ0 has those properties. Moreover ϕ0ϕ0 and ϕjϕj 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.