Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 

 

Discrete Fourier analysis on a dodecahedron and a tetrahedron


Authors: Huiyuan Li and Yuan Xu
Journal: Math. Comp. 78 (2009), 999-1029
MSC (2000): Primary 41A05, 41A10
Published electronically: August 27, 2008
MathSciNet review: 2476568
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron are deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the tetrahedron is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $ (\log n)^3$.


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

  • 1. L. P. Bos, Bounding the Lebesgue function for Lagrange interpolation in a simplex, J. Approx. Theory 38 (1983), no. 1, 43–59. MR 700876, 10.1016/0021-9045(83)90140-5
  • 2. J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447
  • 3. D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall. Inc., Englewood Cliffs, New Jersey, 1984.
  • 4. Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. MR 1827871
  • 5. Bent Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Analysis 16 (1974), 101–121. MR 0470754
  • 6. Thomas C. Hales, A proof of the Kepler conjecture, Ann. of Math. (2) 162 (2005), no. 3, 1065–1185. MR 2179728, 10.4007/annals.2005.162.1065
  • 7. J. R. Higgins, Sampling theory in Fourier and Signal Analysis, Foundations, Oxford Science Publications, New York, 1996.
  • 8. Tom H. Koornwinder, Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. III, Nederl. Akad. Wetensch. Proc. Ser. A. 77=Indag. Math. 36 (1974), 357–369. MR 0357905
  • 9. Huiyuan Li, Jiachang Sun, and Yuan Xu, Discrete Fourier analysis, cubature, and interpolation on a hexagon and a triangle, SIAM J. Numer. Anal. 46 (2008), no. 4, 1653–1681. MR 2399390, 10.1137/060671851
  • 10. Robert J. Marks II, Introduction to Shannon sampling and interpolation theory, Springer Texts in Electrical Engineering, Springer-Verlag, New York, 1991. MR 1077829
  • 11. Jiachang Sun, Multivariate Fourier series over a class of non tensor-product partition domains, J. Comput. Math. 21 (2003), no. 1, 53–62. Special issue dedicated to the 80th birthday of Professor Zhou Yulin. MR 1974272
  • 12. Jiachang Sun, Multivariate Fourier transform methods over simplex and super-simplex domains, J. Comput. Math. 24 (2006), no. 3, 305–322. MR 2229712
  • 13. A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 41A05, 41A10

Retrieve articles in all journals with MSC (2000): 41A05, 41A10


Additional Information

Huiyuan Li
Affiliation: Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
Email: hynli@mail.rdcps.ac.cn

Yuan Xu
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222.
Email: yuan@math.uoregon.edu

DOI: https://doi.org/10.1090/S0025-5718-08-02156-X
Keywords: Discrete Fourier series, trigonometric, Lagrange interpolation, dodecahedron, tetrahedron
Received by editor(s): April 10, 2007
Received by editor(s) in revised form: February 29, 2008
Published electronically: August 27, 2008
Additional Notes: The first authors were supported by NSFC Grants 10601056, 10431050 and 60573023. The second author was supported by NSF Grant DMS-0604056
Article copyright: © Copyright 2008 American Mathematical Society