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The Howe dual pair   $(\mathfrak {sl}(2),O(m))$   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   $\mathfrak {sl}(2)$   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   $\mathfrak {sl}(2)$  . 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   $(\mathfrak {osp}(1|2),{\rm Spin}(m))$  , in the context of the Dirac operator on   $\mathbb {R}^m$  . This connects our results with the Clifford-FT studied in previous work.

Generalized Fourier Transforms Arising from the Enveloping Algebras of (2) and (1∣2)