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SIGMA, 2012, Volume 8, 067, 29 pages (Mi sigma744)  

This article is cited in 10 scientific papers (total in 10 papers)

Discrete Fourier analysis and Chebyshev polynomials with $G_2$ group

Huiyuan Lia, Jiachang Suna, Yuan Xub

a Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
b Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222, USA

Abstract: The discrete Fourier analysis on the $30^{\circ}$$60^{\circ}$$90^{\circ}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm–Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of $m$-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type.

Keywords: discrete Fourier series; trigonometric; group $G_2$; PDE; orthogonal polynomials.

DOI: https://doi.org/10.3842/SIGMA.2012.067

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Full text: http://emis.mi.ras.ru/.../067
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Bibliographic databases:

ArXiv: 1204.4501
MSC: 41A05; 41A10
Received: May 4, 2012; in final form September 6, 2012; Published online October 3, 2012
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Citation: Huiyuan Li, Jiachang Sun, Yuan Xu, “Discrete Fourier analysis and Chebyshev polynomials with $G_2$ group”, SIGMA, 8 (2012), 067, 29 pp.

Citation in format AMSBIB
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\by Huiyuan Li, Jiachang Sun, Yuan Xu
\paper Discrete Fourier analysis and Chebyshev polynomials with $G_2$ group
\jour SIGMA
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\vol 8
\papernumber 067
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. D. Lyakhovsky, “Multivariate Chebyshev polynomials in terms of singular elements”, Theoret. and Math. Phys., 175:3 (2013), 797–805  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Moody R.V. Motlochova L. Patera J., “Gaussian Cubature Arising From Hybrid Characters of Simple Lie Groups”, J. Fourier Anal. Appl., 20:6 (2014), 1257–1290  crossref  mathscinet  zmath  isi  scopus
    3. Hrivnak J. Motlochova L., “Discrete Transforms and Orthogonal Polynomials of (Anti) Symmetric Multivariate Cosine Functions”, SIAM J. Numer. Anal., 52:6 (2014), 3021–3055  crossref  mathscinet  zmath  isi  elib  scopus
    4. Lemire F.W. Patera J. Szajewska M., “Dominant Weight Multiplicities in Hybrid Characters of B-N , C-N , F-4, G(2)”, Int. J. Theor. Phys., 54:11 (2015), 4011–4026  crossref  mathscinet  zmath  isi  elib  scopus
    5. Hrivnak, J.; Walton, M. A., “Discretized Weyl-orbit functions: modified multiplication and Galois symmetry”, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 48:17 (2015), 175205  crossref  mathscinet  zmath  scopus
    6. Hrivnak J., Motlochova L., Patera J., “Cubature Formulas of Multivariate Polynomials Arising from Symmetric Orbit Functions”, Symmetry-Basel, 8:7 (2016), 63  crossref  mathscinet  isi  scopus
    7. Hakova, L.; Hrivnak, J.; Motlochova, L., “On cubature rules associated to Weyl group orbit functions”, Acta Polytechnica, 56:3 (2016), 202-213  crossref  scopus
    8. Hrivnak, J.; Motlochova, L., “On connecting Weyl-orbit functions to Jacobi polynomials and multivariate (Anti)symmetric trigonometric functions”, Acta Polytechnica, 56:4 (2016), 283-290.  crossref  scopus
    9. Czyzycki T. Hrivnak J. Patera J., “Generating Functions For Orthogonal Polynomials of a(2), C-2 and G(2)”, Symmetry-Basel, 10:8 (2018), 354  crossref  isi  scopus
    10. Brus A., Hrivnak J., Motlochova L., “Discrete Transforms and Orthogonal Polynomials of (Anti)Symmetric Multivariate Sine Functions”, Entropy, 20:12 (2018), 938  crossref  isi
  • Symmetry, Integrability and Geometry: Methods and Applications
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