(Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms. (English) Zbl 1152.81513
Summary: Four families of special functions, depending on \(n\) variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix elements are sine or cosine functions of one variable each. These functions are eigenfunctions of the Laplace operator, satisfying specific conditions at the boundary of a certain domain \(F\) of the \(n\)-dimensional Euclidean space. Discrete and continuous orthogonality on \(F\) of the functions within each family, allows one to introduce symmetrized and antisymmetrized multivariate Fourier-like transforms, involving the symmetric and antisymmetric multivariate sine and cosine functions.
MSC:
| 33E20 | Other functions defined by series and integrals |
| 33B10 | Exponential and trigonometric functions |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
References:
| [1] | J. Patera, in Symmetry in Nonlinear Mathematical Physics, Proceedings of Institute of Mathematics of NAS of Ukraine, Kiev, Ukraine, edited by A. G. Nikitin, Vol. 50 (2004), pp. 1152–1160. · Zbl 1092.22010 |
| [2] | Patera J., Symmetry, Integr. Geom.: Methods Appl. 1 pp 025– (2005) |
| [3] | Klimyk A. U., Symmetry, Integr. Geom.: Methods Appl. 2 pp 06– (2006) |
| [4] | Klimyk A. U., Symmetry, Integr. Geom.: Methods Appl. 3 pp 023– (2007) |
| [5] | DOI: 10.1063/1.1738187 · Zbl 1071.94002 · doi:10.1063/1.1738187 |
| [6] | DOI: 10.1063/1.1897143 · Zbl 1110.22008 · doi:10.1063/1.1897143 |
| [7] | DOI: 10.1063/1.2109707 · Zbl 1111.43002 · doi:10.1063/1.2109707 |
| [8] | DOI: 10.1088/1751-8113/40/18/006 · Zbl 1123.33020 · doi:10.1088/1751-8113/40/18/006 |
| [9] | A. Atoyan and J. Patera, in Group Theory and Numerical Methods, CRM Proceedings and Lecture Notes, Providence, RI, Amer. Math. Soc. Vol. 39 (2003), pp. 1–16. |
| [10] | DOI: 10.1016/j.astropartphys.2004.11.007 · doi:10.1016/j.astropartphys.2004.11.007 |
| [11] | DOI: 10.1063/1.2191361 · Zbl 1111.22012 · doi:10.1063/1.2191361 |
| [12] | Germain M., SPIE Electronic Imaging 6065 pp 03– (2006) |
| [13] | Germain M., Proc. SPIE 6065 pp 387– (2006) |
| [14] | DOI: 10.1090/S0002-9904-1962-10749-6 · Zbl 0113.33602 · doi:10.1090/S0002-9904-1962-10749-6 |
| [15] | DOI: 10.1007/BF01193547 · Zbl 0814.41022 · doi:10.1007/BF01193547 |
| [16] | DOI: 10.1088/1751-8113/40/34/006 · Zbl 1139.42001 · doi:10.1088/1751-8113/40/34/006 |
| [17] | Rao K. R., Disrete Cosine Transform–Algorithms, Advantages, Applications (1990) |
| [18] | Moody R. V., Symmetry, Integr. Geom.: Methods Appl. 2 pp 76– (2006) |
| [19] | DOI: 10.1137/S0036144598336745 · Zbl 0939.42021 · doi:10.1137/S0036144598336745 |
| [20] | DOI: 10.1109/78.295213 · doi:10.1109/78.295213 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
