Increasing the polynomial reproduction of a quasi-interpolation operator

Quasi-interpolation is an important tool, used both in theory and in practice, for the approximation of smooth functions from univariate or multivariate spaces which contain Πm=Πm(Rd)Πm=Πm(Rd), the dd-variate polynomials of degree ≤m≤m. In particular, the reproduction of ΠmΠm leads to an approximation order of m+1m+1. Prominent examples include Lagrange and Bernstein type approximations by polynomials, the orthogonal projection onto ΠmΠm for some inner product, finite element methods of precision mm, and multivariate spline approximations based on macroelements or the translates of a single spline.For such a quasi-interpolation operator LL which reproduces Πm(Rd)Πm(Rd) and any r≥0r≥0, we give an explicit construction of a quasi-interpolant Rmr+mL=L+A which reproduces Πm+rΠm+r, together with an integral error formula which involves only the (m+r+1)(m+r+1)th derivative of the function approximated. The operator Rmm+rL is defined on functions with rr additional orders of smoothness than those on which LL is defined. This very general construction holds in all dimensions dd. A number of representative examples are considered.