Increasing the polynomial reproduction of a quasi-interpolation operator

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