A general method for constructing quasi-interpolants from B-splines

摘要

A general method for constructing quasi-interpolation operators based on B-splines is developed. Given a B-spline ϕ in Rs, s≥1, normalized by ∑i∈Zsϕ(⋅−i)=1, the classical structure Q(f)≔∑i∈Zsλf(⋅+i)ϕ(⋅−i), for a quasi-interpolation operator Q is considered. A minimization problem is derived from an estimate of the quasi-interpolation error associated with Q when λf is a linear combination of values of f at points in some neighbourhood of the support of ϕ; or a linear combination of values of f and some of its derivatives at some points in this set; or a linear combination of weighted mean values of the function to be approximated. That linear functional is defined to produce a quasi-interpolant exact on the space of polynomials of maximal total degree included in the space spanned by the integer translates of ϕ. The solution of that minimization problem is characterized in terms of specific splines which do not depend on λ but only on ϕ.