Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Minimal cubature formulae of
trigonometric degree


Authors: Ronald Cools and Ian H. Sloan
Journal: Math. Comp. 65 (1996), 1583-1600
MSC (1991): Primary 41A55, 41A63; Secondary 65D32
DOI: https://doi.org/10.1090/S0025-5718-96-00767-3
MathSciNet review: 1361806
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we construct minimal cubature formulae of trigonometric degree: we obtain explicit formulae for low dimensions of arbitrary degree and for low degrees in all dimensions. A useful tool is a closed form expression for the reproducing kernels in two dimensions.


References [Enhancements On Off] (What's this?)

  • 1. M. Beckers and R. Cools, A relation between cubature formulae of trigonometric degree and lattice rules, Numerical Integration IV (H. Brass and G. Hämmerlin, eds.), Birkhäuser Verlag, Basel, 1993, pp. 13--24. MR 95b:65034 (see also Erratum, MR 95m, p. xxv at the end of the issue and p. 7641 at the end of Section 65).
  • 2. K.K. Frolov, On the connection between quadrature formulas and sublattices of the lattice of integral vectors, Dokl. Akad. Nauk SSSR 232 (1977), 40--43; English transl. in Soviet Math. Dokl. 18 (1977), 37--41. MR 55:272
  • 3. J.G. Herriot, Nörlund summability of multiple Fourier series, Duke Math. J. 11 (1944), 735--754. MR 6:126d
  • 4. H.M. Möller, Polynomideale und Kubaturformeln, Ph.D. thesis, Universität Dortmund, 1973.
  • 5. I.P. Mysovskikh, On the construction of cubature formulas with fewest nodes, Dokl. Akad. Nauk SSSR 178 (1968), 1252--1254; English transl. in Soviet Math. Dokl. 9 (1968), 277--280. MR 36:7328
  • 6. ------, The approximation of multiple integrals by using interpolatory cubature formulae, Quantitative Approximation (R.A. De Vore and K. Scherer, eds.), Academic Press, New York, 1980, pp. 217--243. MR 82b:41031
  • 7. ------, Quadrature formulae of the highest trigonometric degree of accuracy, Zh. Vychisl. Mat. i Mat. Fiz. 25 (1985), 1246--1252; English transl. in U.S.S.R. Comput. Math. and Math. Phys. 25 (1985), 180--184. MR 87b:65030
  • 8. ------, On cubature formulas that are exact for trigonometric polynomials, Dokl. Akad. Nauk SSSR 296 (1987), 28--31; English transl. in Soviet Math. Dokl. 36 (1988), 229--232. MR 89b:41038
  • 9. ------, Cubature formulas that are exact for trigonometric polynomials, Metody Vychisl. 15 (1988), 7--19. (Russian) MR 90a:65050
  • 10. ------, On the construction of cubature formulas that are exact for trigonometric polynomials, Numerical Analysis and Mathematical Modelling (A. Wakulicz, ed.), Banach Center Publications, vol. 24, PWN -- Polish Scientific Publishers, Warsaw, 1990, pp. 29--38. (Russian) CMP 91:09
  • 11. M.V. Noskov, Cubature formulae for the approximate integration of periodic functions, Metody Vychisl. 14 (1985), 15--23. (Russian) MR 90f:65038
  • 12. ------, Cubature formulae for the approximate integration of functions of three variables, Zh. Vychisl. Mat. i Mat. Fiz. 28 (1988), 1583--1586; English transl. in U.S.S.R. Comput. Math. and Math. Phys. 28 (1988), 200--202. MR 90j:65042
  • 13. ------, Formulas for the approximate integration of periodic functions, Metody Vychisl. 15 (1988), 19--22. (Russian) CMP 21:03
  • 14. J. Radon, Zur mechanischen Kubatur, Monatsh. Math. 52 (1948), 286--300. MR 11:405b
  • 15. A.V. Reztsov, On cubature formulas of Gaussian type with an asymptotic minimal number of nodes, Mathematicheskie Zametki 48 (1990), 151--152. (Russian) CMP 91:04
  • 16. ------, Estimating the number of interpolation points for Gaussian-type cubature formulae, Zh. Vychisl. Mat. i Mat. Fiz. 31 (1991), 451--453; English transl. in U.S.S.R. Comput. Math. and Math. Phys. 31 (1991), 84--86. MR 92a:65082
  • 17. I.H. Sloan, Lattice rules for multiple integration, J. Comput. Appl. Math. 12 & 13 (1985), 131--143. MR 86f:65045
  • 18. I.H. Sloan and S. Joe, Lattice methods for multiple integration, Oxford University Press, Oxford, 1994.
  • 19. I.H. Sloan and J.N. Lyness, The representation of lattice quadrature rules as multiple sums, Math. Comp. 52 (1989), 81--94. MR 90a:65053

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 41A55, 41A63, 65D32

Retrieve articles in all journals with MSC (1991): 41A55, 41A63, 65D32


Additional Information

Ronald Cools
Affiliation: Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200 A, B-3001 Heverlee, Belgium
Email: Ronald.Cools@cs.kuleuven.ac.be

Ian H. Sloan
Affiliation: School of Mathematics, University of New South Wales, Sydney NSW 2033, Australia
Email: i.sloan@unsw.edu.au

DOI: https://doi.org/10.1090/S0025-5718-96-00767-3
Keywords: Cubature, trigonometric degree, lattice rules
Received by editor(s): September 15, 1993
Received by editor(s) in revised form: September 22, 1994, and August 28, 1995
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society