  In this paper, we present recent results in harmonic analysis in the real
line R and in the half-line R^+, which show a closed relation between Hermite
and Laguerre functions, respectively, their symmetry groups and Fourier
analysis. This can be done in terms of a unified framework based in the use of
rigged Hilbert spaces. We find a relation between the universal enveloping
algebra of the symmetry groups with the fractional Fourier transform. The
results obtained are relevant in quantum mechanics as well as in signal
processing as Fourier analysis has a close relation with signal filters. Also,
we introduce some new results concerning a discretized Fourier transform on the
circle. We introduce new functions on the circle constructed with the use of
Hermite functions with interesting properties under Fourier transformations.
