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 ϕ in R s , s ≥ 1 , normalized by ∑ i ∈ Z s ϕ ( ⋅ − i ) = 1 , the classical structure Q ( f ) ≔ ∑ i ∈ Z s λ f ( ⋅ + 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 λ f 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 .