Convexity results and sharp error estimates in approximate multivariate integration

Authors:
Allal Guessab and Gerhard Schmeisser

Translated by:

Journal:
Math. Comp. **73** (2004), 1365-1384

MSC (2000):
Primary 65D30, 65D32, 41A63, 41A44, 41A80; Secondary 26B25, 26D15, 52A40

DOI:
https://doi.org/10.1090/S0025-5718-03-01622-3

Published electronically:
December 19, 2003

MathSciNet review:
2047091

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Abstract | References | Similar Articles | Additional Information

Abstract: An interesting property of the midpoint rule and the trapezoidal rule, which is expressed by the so-called Hermite-Hadamard inequalities, is that they provide one-sided approximations to the integral of a convex function. We establish multivariate analogues of the Hermite-Hadamard inequalities and obtain access to multivariate integration formulae via convexity, in analogy to the univariate case. In particular, for simplices of arbitrary dimension, we present two families of integration formulae which both contain a multivariate analogue of the midpoint rule and the trapezoidal rule as boundary cases. The first family also includes a multivariate analogue of a Maclaurin formula and of the two-point Gaussian quadrature formula; the second family includes a multivariate analogue of a formula by P.C. Hammer and of Simpson's rule. In both families, we trace out those formulae which satisfy a Hermite-Hadamard inequality. As an immediate consequence of the latter, we obtain sharp error estimates for twice continuously differentiable functions.

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Additional Information

**Allal Guessab**

Affiliation:
Department of Applied Mathematics, University of Pau, 64000 Pau, France

Email:
allal.guessab@univ-pau.fr

**Gerhard Schmeisser**

Affiliation:
Mathematical Institute, University of Erlangen-Nuremberg, 91054 Erlangen, Germany

Email:
schmeisser@mi.uni-erlangen.de

DOI:
https://doi.org/10.1090/S0025-5718-03-01622-3

Keywords:
Multivariate approximate integration,
convex functions,
Hermite--Hadamard inequality,
error estimates

Received by editor(s):
October 24, 2002

Published electronically:
December 19, 2003

Article copyright:
© Copyright 2003
American Mathematical Society