TY - JOUR UR - http://lib.ugent.be/catalog/pug01:8114130 ID - pug01:8114130 LA - eng TI - Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2) PY - 2016 JO - (2016) INTERNATIONAL MATHEMATICS RESEARCH NOTICES SN - 1073-7928 PB - 2016 AU - De Bie, Hendrik TW16 002000066440 AU - Oste, Roy WE02 000110755206 802001620845 0000-0002-3418-7067 AU - Van der Jeugt, Joris WE02 801000490010 0000-0003-1387-1676 AB - The Howe dual pair (sl(2), O(m)) allows the characterization of the classical Fourier transform (FT) on the space of rapidly decreasing functions as the exponential of a well-chosen element of sl(2) such that the Helmholtz relations are satisfied. In this paper, we first investigate what happens when instead we consider exponentials of elements of the universal enveloping algebra of sl(2). This leads to a complete class of generalized FTs, that all satisfy properties similar to the classical FT. There is moreover a finite subset of transforms which very closely resemble the FT. We obtain operator exponential expressions for all these transforms by making extensive use of the theory of integer-valued polynomials. We also find a plane wave decomposition of their integral kernel and establish uncertainty principles. In important special cases we even obtain closed formulas for the integral kernels. In the second part of the paper, the same problem is considered for the dual pair (osp(1 vertical bar 2), Spin(m)), in the context of the Dirac operator on R-m. This connects our results with the Clifford-FT studied in previous work. ER -Download RIS file
00000nam^a2200301^i^4500 | |||
001 | 8114130 | ||
005 | 20161219154540.0 | ||
008 | 161014s2016------------------------eng-- | ||
022 | a 1073-7928 | ||
024 | a 000383778200006 2 wos | ||
024 | a 1854/LU-8114130 2 handle | ||
024 | a 10.1093/imrn/rnv293 2 doi | ||
040 | a UGent | ||
245 | a Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2) | ||
246 | a Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1 vertical bar 2) | ||
260 | c 2016 | ||
520 | a The Howe dual pair (sl(2), O(m)) allows the characterization of the classical Fourier transform (FT) on the space of rapidly decreasing functions as the exponential of a well-chosen element of sl(2) such that the Helmholtz relations are satisfied. In this paper, we first investigate what happens when instead we consider exponentials of elements of the universal enveloping algebra of sl(2). This leads to a complete class of generalized FTs, that all satisfy properties similar to the classical FT. There is moreover a finite subset of transforms which very closely resemble the FT. We obtain operator exponential expressions for all these transforms by making extensive use of the theory of integer-valued polynomials. We also find a plane wave decomposition of their integral kernel and establish uncertainty principles. In important special cases we even obtain closed formulas for the integral kernels. In the second part of the paper, the same problem is considered for the dual pair (osp(1 vertical bar 2), Spin(m)), in the context of the Dirac operator on R-m. This connects our results with the Clifford-FT studied in previous work. | ||
598 | a A1 | ||
100 | a De Bie, Hendrik u TW16 0 002000066440 0 801002044535 | ||
700 | a Oste, Roy u WE02 0 000110755206 0 802001620845 0 0000-0002-3418-7067 | ||
700 | a Van der Jeugt, Joris u WE02 0 801000490010 0 0000-0003-1387-1676 | ||
650 | a Mathematics and Statistics | ||
653 | a DUNKL OPERATORS | ||
653 | a UNCERTAINTY PRINCIPLE | ||
653 | a REPRESENTATION | ||
653 | a SUPERSPACE | ||
653 | a FORMULAS | ||
653 | a integer values polynomials | ||
653 | a Fourier Transform | ||
653 | a Helmholtz relations | ||
773 | t INTERNATIONAL MATHEMATICS RESEARCH NOTICES g Int. Math. Res. Notices. 2016. (15) p.4649-4705 q :15<4649 | ||
856 | 3 Full Text u https://biblio.ugent.be/publication/8114130/file/8114150 z [ugent] y Int_Math_Res_Notices-2016-De_Bie-4649-705.pdf | ||
920 | a article | ||
852 | x WE b WE02 | ||
922 | a UGENT-WE | ||
852 | x EA b TW16 | ||
922 | a UGENT-EA |
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