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Generalized Fourier Transforms Arising from the Enveloping Algebras of 𝔰𝔩(2) and 𝔬𝔰𝔭(1∣2)
- Source :
- International Mathematics Research Notices. 2016:4649-4705
- Publication Year :
- 2015
- Publisher :
- Oxford University Press (OUP), 2015.
-
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 Fourier transforms, 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|2),Spin(m)), in the context of the Dirac operator. This connects our results with the Clifford-Fourier transform studied in previous work.<br />Comment: Second version, changes in title, introduction and section 2
- Subjects :
- Pure mathematics
Uncertainty principle
General Mathematics
Operator (physics)
010102 general mathematics
42B10, 13F20, 17B60
Universal enveloping algebra
Dirac operator
01 natural sciences
symbols.namesake
Kernel (algebra)
Fourier transform
Mathematics - Classical Analysis and ODEs
Helmholtz free energy
0103 physical sciences
Classical Analysis and ODEs (math.CA)
FOS: Mathematics
symbols
010307 mathematical physics
Representation Theory (math.RT)
0101 mathematics
Mathematics - Representation Theory
Mathematics
Dual pair
Subjects
Details
- ISSN :
- 16870247 and 10737928
- Volume :
- 2016
- Database :
- OpenAIRE
- Journal :
- International Mathematics Research Notices
- Accession number :
- edsair.doi.dedup.....715f461449f613fabcd95ad2c5808fee