Hermite functions, Lie groups and Fourier analysis

TL;DR

This paper explores the connections between Hermite and Laguerre functions, their symmetry groups, and Fourier analysis within a unified framework, with applications in quantum mechanics and signal processing.

Contribution

It introduces a unified framework using rigged Hilbert spaces to relate Hermite and Laguerre functions, symmetry groups, and Fourier transforms, including new discretized Fourier transform results.

Findings

Relation between symmetry groups and Fourier analysis

New functions on the circle with Hermite-based properties

Discretized Fourier transform on the circle

Abstract

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.