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

AbstractThe paper presents a simple procedure for the construction of quasi-interpolation operators in spaces of m-harmonic splines in Rd, which reproduce polynomials of high degree. The procedure starts from a generator ϕ0, which is easy to derive but with corresponding quasi-interpolation operator reproducing only linear polynomials, and recursively defines generators ϕ1,ϕ2,…,ϕm−1 with corresponding quasi-interpolation operators reproducing polynomials of degree up to 3,5,…,2m−1 respectively. The construction of ϕj from ϕj−1 is explicit, simple and independent of m. The special case d=1 and the special cases d=2,m=2,3,4 are discussed in details