Increasing the polynomial reproduction of a quasi-interpolation operator

Shayne Waldron


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