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

Abstract: Given a B-spline M on R^s, s>=1 we consider a classical discrete quasi-interpolant Q"d written in the formQ"df=@?i@?Z^sf(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 Q"df 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 propertyQ"D","rp=p,for allp@?P"m"+"r.We fully analyze the quasi-interpolation error Q"D","rf-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.

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