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Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2)

Hendrik De Bie (UGent) , Roy Oste (UGent) and Joris Van der Jeugt (UGent)
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Abstract
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.
Keywords
DUNKL OPERATORS, UNCERTAINTY PRINCIPLE, REPRESENTATION, SUPERSPACE, FORMULAS, integer values polynomials, Fourier Transform, Helmholtz relations

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Chicago
De Bie, Hendrik, Roy Oste, and Joris Van der Jeugt. 2016. “Generalized Fourier Transforms Arising from the Enveloping Algebras of Sl(2) and Osp(1|2).” International Mathematics Research Notices (15): 4649–4705.
APA
De Bie, H., Oste, R., & Van der Jeugt, J. (2016). Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2). INTERNATIONAL MATHEMATICS RESEARCH NOTICES, (15), 4649–4705.
Vancouver
1.
De Bie H, Oste R, Van der Jeugt J. Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2). INTERNATIONAL MATHEMATICS RESEARCH NOTICES. 2016;(15):4649–705.
MLA
De Bie, Hendrik, Roy Oste, and Joris Van der Jeugt. “Generalized Fourier Transforms Arising from the Enveloping Algebras of Sl(2) and Osp(1|2).” INTERNATIONAL MATHEMATICS RESEARCH NOTICES 15 (2016): 4649–4705. Print.
@article{8114130,
  abstract     = {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.},
  author       = {De Bie, Hendrik and Oste, Roy and Van der Jeugt, Joris},
  issn         = {1073-7928},
  journal      = {INTERNATIONAL MATHEMATICS RESEARCH NOTICES},
  language     = {eng},
  number       = {15},
  pages        = {4649--4705},
  title        = {Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2)},
  url          = {http://dx.doi.org/10.1093/imrn/rnv293},
  year         = {2016},
}

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