# Hermite Functions, Lie Groups and Fourier Analysis

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## Abstract

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## 1. Introduction

## 2. Harmonic Analysis on $\mathbb{R}$

#### 2.1. Hermite Functions and the Group $IO(2)$

#### 2.2. $UEA[io(2)]$ and Fractional Fourier Transform

## 3. Harmonic Analysis on ${\mathbb{R}}^{+}$

#### 3.1. Harmonic Analysis on $su(1,1)$

#### 3.2. Fourier-Like Transformations on ${\mathbb{R}}^{+}$

## 4. A New Harmonic Analysis on the Circle

#### A Discretized Fourier Transform on the Circle

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Celeghini, E.; Gadella, M.; Del Olmo, M.A.
Hermite Functions, Lie Groups and Fourier Analysis. *Entropy* **2018**, *20*, 816.
https://doi.org/10.3390/e20110816

**AMA Style**

Celeghini E, Gadella M, Del Olmo MA.
Hermite Functions, Lie Groups and Fourier Analysis. *Entropy*. 2018; 20(11):816.
https://doi.org/10.3390/e20110816

**Chicago/Turabian Style**

Celeghini, Enrico, Manuel Gadella, and Mariano A. Del Olmo.
2018. "Hermite Functions, Lie Groups and Fourier Analysis" *Entropy* 20, no. 11: 816.
https://doi.org/10.3390/e20110816