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

Enhancement of the algebraic precision of a linear operator and consequences under positivity 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). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

, Volume 13 (4) – Feb 6, 2009
15 pages

/lp/springer_journal/enhancement-of-the-algebraic-precision-of-a-linear-operator-and-7J05Armwxz
Publisher
Springer Journals
Copyright © 2009 by Birkhäuser Verlag Basel/Switzerland
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-008-2253-4
Publisher site
See Article on Publisher Site

### 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).

### Journal

PositivitySpringer Journals

Published: Feb 6, 2009

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