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

given a b-spline m on r^s, s≥1 we consider a classical discrete quasi-interpolant q_d written in the form

qdf = ∑_{i ∈zs}f (i)l(· − i),

where l(x) := ∑_j∈jc_jm(x−j) for some finite subset j ⊂ z^s and c_j ∈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 p_m. 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 q_d,rf of f satisfying the reproduction property q_d,rp = p, for all p ∈ p_m+r .

we fully analyze the quasi-interpolation error q_d,rf−f for f ∈c^m+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.