Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

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
Published electronically: December 19, 2003
MathSciNet review: 2047091
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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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  • 1. G. Allasia and C. Giordano, Bilateral approximations of double integrals of convex functions, Calcolo 20 (1983), no. 1, 73–83 (Italian, with English summary). MR 747009, 10.1007/BF02575894
  • 2. Helmut Brass, Quadraturverfahren, Vandenhoeck & Ruprecht, Göttingen, 1977 (German). Studia Mathematica, Skript 3. MR 0443305
  • 3. P. S. Bullen, Error estimates for some elementary quadrature rules, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 602-633 (1978), 97–103 (1979). MR 580428
  • 4. H. Busemann and G. C. Shephard, Convexity on nonconvex sets, Proc. Colloquium on Convexity (Copenhagen, 1965) Kobenhavns Univ. Mat. Inst., Copenhagen, 1967, pp. 20–33. MR 0221382
  • 5. S. S. Dragomir, On Hadamard’s inequality on a disk, JIPAM. J. Inequal. Pure Appl. Math. 1 (2000), no. 1, Article 2, 11. MR 1756653
  • 6. S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, Internet Publication, http://rgmia.vu.edu.au, 2000.
  • 7. I. J. Good and R. A. Gaskins, The centroid method of numerical integration, Numer. Math. 16 (1970/1971), 343–359. MR 0283986
  • 8. Allal Guessab and Gerhard Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory 115 (2002), no. 2, 260–288. MR 1901217, 10.1006/jath.2001.3658
  • 9. Preston C. Hammer, The midpoint method of numerical integration, Math. Mag. 31 (1957/1958), 193–195. MR 0099753
  • 10. F. B. Hildebrand, Introduction to numerical analysis, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. MR 0075670
  • 11. Paul J. Kelly and Max L. Weiss, Geometry and convexity, John Wiley & Sons, New York-Chichester-Brisbane, 1979. A study in mathematical methods; Pure and Applied Mathematics; A Wiley-Interscience publication. MR 534615
  • 12. Knut Petras, Quadrature theory of convex functions: a survey and additions, Numerical integration, IV (Oberwolfach, 1992) Internat. Ser. Numer. Math., vol. 112, Birkhäuser, Basel, 1993, pp. 315–329. MR 1248413
  • 13. A.W. Roberts and D.E. Varnberg, Convex Sets, Academic Press, New York, 1973.
  • 14. Frank Stenger, Integration formulae based on the trapezoidal formula, J. Inst. Math. Appl. 12 (1973), 103–114. MR 0381261
  • 15. Frank Stenger, Numerical methods based on sinc and analytic functions, Springer Series in Computational Mathematics, vol. 20, Springer-Verlag, New York, 1993. MR 1226236
  • 16. A. H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. Prentice-Hall Series in Automatic Computation. MR 0327006

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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: http://dx.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