  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.
