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Discrete Fourier Analysis on Fundamental Domain and Simplex of A d Lattice in d-Variables

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Abstract

A discrete Fourier analysis on the fundamental domain Ω d of the d-dimensional lattice of type A d is studied, where Ω2 is the regular hexagon and Ω3 is the rhombic dodecahedron, and analogous results on d-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the simplex 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)d. The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature formulas, which provides only the second known example of such formulas.

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Authors and Affiliations

  1. Institute of Software, Chinese Academy of Sciences, Beijing, 100190, China

    Huiyuan Li

  2. Department of Mathematics, University of Oregon, Eugene, OR, 97403-1222, USA

    Yuan Xu

Authors
  1. Huiyuan Li
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  2. Yuan Xu
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Corresponding author

Correspondence to Yuan Xu.

Additional information

Communicated by Michael Frazier.

The first author was supported by NSFC Grants 10601056 and 10971212. The second author was supported by NSF Grant DMS-0604056.

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Li, H., Xu, Y. Discrete Fourier Analysis on Fundamental Domain and Simplex of A d Lattice in d-Variables. J Fourier Anal Appl 16, 383–433 (2010). https://doi.org/10.1007/s00041-009-9106-9

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  • DOI: https://doi.org/10.1007/s00041-009-9106-9

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