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

摘要

Given a B-spline M on R~s,s≥1 we consider a classical discrete quasi-interpolant Q_d written in the form Q_df = ∑iεZ~sf(i)L(·-i), where L(x):=∑jεJCjM(x—j) for some finite subset J C Z~s and cj εR. 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/(i) in the expression defining Gdfby a modified Taylor polynomial of degree r at i, we derive non-standard differential quasi-interpolants Q_D,rf off satisfying the reproduction property Q_D,rp = p, for all p ε P_(m+r). We fully analyze the quasi-interpolation error Q_drf—ffor f εC_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)_jεJ 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.