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

Abstract

Abstract Given a B-spline M on R s , s ≥ 1 we consider a classical discrete quasi-interpolant Q d written in the form Q d f = ∑ i ∈ Z s f ( i ) L ( ⋅ − i ) , where L( x) ≔ ∑ j∈ J c j M( 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 Q d f by a modified Taylor polynomial of degree r at i, we derive non-standard differential quasi-interpolants Q D, r f of f satisfying the reproduction property Q D , r p = p , for all p ∈ P m + r . We fully analyze the quasi-interpolation error Q D, r f − f for f ∈ C m + 2 ( R s ) , and we get a two term expression for the error. The leading part of that expression involves a function on the sequence c ≔ ( c j ) 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.