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Discrete Transforms and Orthogonal Polynomials of (Anti)symmetric Multivariate Sine Functions

Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Prague 1, Czech Republic
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Entropy 2018, 20(12), 938; https://doi.org/10.3390/e20120938
Received: 29 October 2018 / Revised: 30 November 2018 / Accepted: 3 December 2018 / Published: 6 December 2018
Sixteen types of the discrete multivariate transforms, induced by the multivariate antisymmetric and symmetric sine functions, are explicitly developed. Provided by the discrete transforms, inherent interpolation methods are formulated. The four generated classes of the corresponding orthogonal polynomials generalize the formation of the Chebyshev polynomials of the second and fourth kinds. Continuous orthogonality relations of the polynomials together with the inherent weight functions are deduced. Sixteen cubature rules, including the four Gaussian, are produced by the related discrete transforms. For the three-dimensional case, interpolation tests, unitary transform matrices and recursive algorithms for calculation of the polynomials are presented. View Full-Text
Keywords: discrete multivariate sine transforms; orthogonal polynomials; cubature formulas discrete multivariate sine transforms; orthogonal polynomials; cubature formulas
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Brus, A.; Hrivnák, J.; Motlochová, L. Discrete Transforms and Orthogonal Polynomials of (Anti)symmetric Multivariate Sine Functions. Entropy 2018, 20, 938.

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