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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
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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$.

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  • 1. L. Bos, Bounding the Lebesgue function for Lagrange interpolation in a simplex, J. Approx. Theory 38 (1983), 43-59. MR 700876 (84e:41007)
  • 2. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed. Springer, New York, 1999. MR 1662447 (2000b:11077)
  • 3. D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall. Inc., Englewood Cliffs, New Jersey, 1984.
  • 4. C. F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge Univ. Press, 2001. MR 1827871 (2002m:33001)
  • 5. B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Anal. 16 (1974), 101-121. MR 0470754 (57:10500)
  • 6. T. C. Hales, A proof of the Kepler conjecture. Ann. of Math. (2) 162 (2005), no. 3, 1065-1185. MR 2179728 (2006g:52029)
  • 7. J. R. Higgins, Sampling theory in Fourier and Signal Analysis, Foundations, Oxford Science Publications, New York, 1996.
  • 8. T. Koornwinder, Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators, Nederl. Acad. Wetensch. Proc. Ser. A77 = Indag. Math. 36 (1974), 357-381. MR 0357905 (50:10371a)
  • 9. H. Li, J. Sun and Yuan Xu, Discrete Fourier analysis, cubature and interpolation on a hexagon and a triangle, SIAM J. Numer. Anal. 46 (2008), 1653-1681. MR 2399390
  • 10. R. J. Marks II, Introduction to Shannon Sampling and Interpolation Theory, Springer-Verlag, New York, 1991. MR 1077829 (92j:41001)
  • 11. J. Sun, Multivariate Fourier series over a class of non tensor-product partition domains, J. Comput. Math. 21 (2003), 53-62. MR 1974272 (2004m:42011)
  • 12. J. Sun, Multivariate Fourier transform methods over simplex and super-simplex domains, J. Comput. Math. 24 (2006), 305-322. MR 2229712 (2007a:42019)
  • 13. A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, 1959. MR 0107776 (21:6498)

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Additional Information

Huiyuan Li
Affiliation: Institute of Software, Chinese Academy of Sciences, Beijing 100190, China

Yuan Xu
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222.

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

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