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Increasing the polynomial reproduction of a quasi-interpolation operator

## Shayne Waldron

## 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

## Availability: