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

The paper presents a simple procedure for the construction of quasi-interpolation operators in spaces of mm-harmonic splines in RdRd, which reproduce polynomials of high degree. The procedure starts from a generator ϕ0ϕ0, which is easy to derive but with corresponding quasi-interpolation operator reproducing only linear polynomials, and recursively defines generators ϕ1,ϕ2,…,ϕm−1ϕ1,ϕ2,…,ϕm−1 with corresponding quasi-interpolation operators reproducing polynomials of degree up to 3,5,…,2m−13,5,…,2m−1 respectively. The construction of ϕjϕj from ϕj−1ϕj−1 is explicit, simple and independent of mm. The special case d=1d=1 and the special cases d=2d=2,m=2,3,4m=2,3,4 are discussed in details.