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

A general method for constructing quasi-interpolation operators based on B-splines is developed. Given a B-spline ϕϕ in RsRs, s≥1s≥1, normalized by ∑i∈Zsϕ(⋅−i)=1∑i∈Zsϕ(⋅−i)=1, the classical structure Q(f)≔∑i∈Zsλf(⋅+i)ϕ(⋅−i)Q(f)≔∑i∈Zsλf(⋅+i)ϕ(⋅−i), for a quasi-interpolation operator QQ is considered. A minimization problem is derived from an estimate of the quasi-interpolation error associated with QQ when λfλf is a linear combination of values of ff at points in some neighbourhood of the support of ϕϕ; or a linear combination of values of ff 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 ϕϕ. The solution of that minimization problem is characterized in terms of specific splines which do not depend on λλ but only on ϕϕ.