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Mathematica Slovaca

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Volume 67, Issue 2

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Three-variable symmetric and antisymmetric exponential functions and orthogonal polynomials

Agata Bezubik / Jiří Hrivnák
  • Department of physics Faculty of nuclear sciences and physical engineering Czech Technical University in Prague Břehová 7, 115 19 Prague Czech Republic
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/ Jiří Patera
  • Centre de recherches mathématiques Université de Montréal C. P. 6128 Centre ville Montréal, H3C 3J7, Québec Canada
  • MIND Research Institute Irvine, CA 92617 USA
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/ Severin Pošta
  • Department of mathematics Faculty of nuclear sciences and physical engineering Czech Technical University in Prague Trojanova 13, 120 00 Prague Czech Republic
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Published Online: 2017-04-28 | DOI: https://doi.org/10.1515/ms-2016-0280

Abstract

The common exponential functions whose exponents are the scalar products 〈λ,x〉, where x is a real variable and λ is an integer, admit two generalizations to any higher dimension, the symmetric and the antisymmetric ones [KLIMYK, A.—PATERA, J.: (Anti)symmetric multivariate exponential functions and corresponding Fourier transforms, J. Phys. A: Math. Theor. 40 (2007), 10473–10489]. Restriction in the paper to the three variables only allows us to work out many specific properties of the symmetric and antisymmetric functions useful in applications. Such are (i) the orthogonalities, both the continuous one and the discrete one on the 3D lattice of any density; (ii) corresponding discrete and continuous Fourier transforms; (iii) generating functions for the related polynomials in three variables, and others. Rapidly increasing precision of the interpolation with increasing density of the 3D lattice is shown in an example.

MSC 2010: Primary 33B10; 42C05; 42C10; 41A05

Keywords: exponential functions; orthogonal polynomials

We gratefully acknowledge the support of this work by the Natural Sciences and Engineering Research Council of Canada and by the Doppler Institute of the Czech Technical University in Prague. JH is grateful for the hospitality extended to him at the Centre de recherches mathématiques, Université de Montréal. JP expresses his gratitude for the hospitality of the Doppler Institute. AB is grateful for the hospitality extended to her at Department of mathematics FNSPE CTU. SP acknowledges the support of SGS15/215/OHK4/3T/14, project of the Czech Technical University in Prague. JH gratefully acknowledges support by RVO68407700.

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About the article


Received: 2014-04-02

Accepted: 2015-05-29

Published Online: 2017-04-28

Published in Print: 2017-04-25


Citation Information: Mathematica Slovaca, Volume 67, Issue 2, Pages 427–446, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2016-0280.

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