Quasi-interpolation operators on hexagonal grids with high approximation orders in spaces of polyharmonic splines

In this paper we discuss the problem of approximating data given on hexagonal lattices. The construction of continuous representation from sampled data is an essential element of many applications as, for instance, image resampling, numerical solution of PDE boundary problems, etc. A useful tool is quasi-interpolation that doesn’t need the solution of a linear system. It is then important to have quasi-interpolation operators with high approximation orders, and capable to provide an efficient computation of the quasi interpolant function. To this end, we show that the idea proposed by Bozzini et al. [1] for the construction of quasi-interpolation operators in spaces of m-harmonic splines with knots in Z2 which reproduce polynomials of high degree, can be generalized to any spaces of m-harmonic splines with knots on a lattice Γ of R2 and in particular on hexagonal grids. Then by a simple procedure which starts from a generator ϕ0Γ with corresponding quasi-interpolation operator reproducing only linear polynomials, it is possible to define recursively generators ϕ1Γ,⋯,ϕm−1Γ with corresponding quasi-interpolation operators reproducing polynomials up to degree 3,5,⋯,2m−1. We show that this new generators of quasi-interpolation operators on a general lattice are positive definite functions, and are scaling functions whenever ϕ0Γ has those properties. Moreover we are able to associate with ϕjΓ a dyadic convergent subdivision scheme that allows a fast computation of the quasi-interpolant.