Hermite Functions, Lie Groups and Fourier Analysis.

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 on the use of rigged Hilbert spaces. We find a relation between the universal enveloping algebra of the symmetry groups with the fractional Fourier transform.