Error analysis for a non-standard class of differential quasi-interpolants

Given a B-spline M on Rs, s≥1 we consider a classical discrete quasi-interpolant Qd written in the formQdf=∑i∈Zsf(i)L(⋅−i),where L(x)≔∑j∈JcjM(x−j) for some finite subset J⊂Zs and cj∈R. This fundamental function is determined to produce a quasi-interpolation operator exact on the space of polynomials of maximal total degree included in the space spanned by the integer translates of M, say Pm. By replacing f(i) in the expression defining Qdf by a modified Taylor polynomial of degree r at i, we derive non-standard differential quasi-interpolants QD,rf of f satisfying the reproduction propertyQD,rp=p,for allp∈Pm+r.We fully analyze the quasi-interpolation error QD,rf−f for f∈Cm+2(Rs), and we get a two term expression for the error. The leading part of that expression involves a function on the sequence c≔(cj)j∈J defining the discrete and the differential quasi-interpolation operators. It measures how well the non-reproduced monomials are approximated, and then we propose a minimization problem based on this function.