# Increasing the polynomial reproduction of a quasi-interpolation operator

## Abstract:

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 \$\Pi_m=\Pi_m(\Rd)\$ the \$d\$--variate polynomials of degree \$\le m\$. In particular, the reproduction of \$\Pi_m\$ leads to an approximation order of \$m+1\$. Prominent examples include Lagrange and Bernstein type approximations by polynomials, the orthogonal projection onto \$\Pi_m\$ 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 \$\Pi_m(\Rd)\$ and any \$r\ge0\$, we give an explicit construction of a quasi-interpolant \$R_m^{r+m}L=L+A\$ which reproduces \$\Pi_{m+r}\$, together with an integral error formula which involves only the \$(m+r+1)\$--st derivative of the function approximated. The operator \$R_m^{m+r}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.

Keywords: Quasi-interpolation, Lagrange interpolation, Bernstein polynomial, finite element method, multivariate polynomial approximation, error formula, multipoint Taylor formula, divided differences, Chu--Vandermonde convolution

Math Review Classification: Primary primary 41A80, 65D05; Secondary 41A05, 41A10.

Length: 12 pages

Last Updated: 29 January 2008