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

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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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
\Bibitem{LiSunXu12} \by Huiyuan Li, Jiachang Sun, Yuan Xu \paper Discrete Fourier analysis and Chebyshev polynomials with $G_2$ group \jour SIGMA \yr 2012 \vol 8 \papernumber 067 \totalpages 29 \mathnet{http://mi.mathnet.ru/sigma744} \crossref{https://doi.org/10.3842/SIGMA.2012.067} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2988027} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309390300001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84867512667} 

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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
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
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
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
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
6. Hrivnak J., Motlochova L., Patera J., “Cubature Formulas of Multivariate Polynomials Arising from Symmetric Orbit Functions”, Symmetry-Basel, 8:7 (2016), 63
7. Hakova, L.; Hrivnak, J.; Motlochova, L., “On cubature rules associated to Weyl group orbit functions”, Acta Polytechnica, 56:3 (2016), 202-213
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.
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
10. Brus A., Hrivnak J., Motlochova L., “Discrete Transforms and Orthogonal Polynomials of (Anti)Symmetric Multivariate Sine Functions”, Entropy, 20:12 (2018), 938
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