Enhancement of the algebraic precision of a linear operator and consequences under positivity

Abstract

Let Ω be a compact convex domain in ${\mathbb{R}}^{d}$ and let L be a bounded linear operator that maps a subspace of C(Ω) into C(Ω). Suppose that L reproduces polynomials up to degree m. We show that for appropriately defined coefficients a _mrj the operator

$$H_{mr}[f]({\bf x}):= L \left[\sum\limits_{j=0}^{r} \frac{a_{mrj}}{j!} D^{j}_{{\bf x}-\cdot}\,f \right] ({\bf x}) \qquad ({\bf x} \in \Omega)$$

reproduces polynomials up to degree m+r. This is an immediate consequence of the main result (Theorem 3.1) which provides an integral representation of the error f(x) − H _mr [f](x). Special emphasis is given to positive linear operators L. In this case, sharp error bounds are established (Theorem 4.4) and interpolation properties are pointed out (Theorem 4.5). We also discuss various classes of admissible operators L and show an interrelation (Theorem 5.1).