Three-variable symmetric and antisymmetric exponential functions and orthogonal polynomials

Year, pages: 2017, 427 - 446

The common exponential functions whose exponents are the scalar products $\langleλ,x\rangle$, 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.: \textit{(Anti)symmetric multivariate exponential functions and corresponding Fourier transforms}, J. Phys. A: Math. Theor. \textbf{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.

Bezubik, A., Hrivnák, J., Patera, J., Pošta, S. 2017. Three-variable symmetric and antisymmetric exponential functions and orthogonal polynomials. In Mathematica Slovaca, vol. 67, no.2, pp. 427-446. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0280

Bezubik, A., Hrivnák, J., Patera, J., Pošta, S. (2017). Three-variable symmetric and antisymmetric exponential functions and orthogonal polynomials. Mathematica Slovaca, 67(2), 427-446. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0280