Celeghini Enrico, Gadella Manuel, Del Olmo Mariano A
Dipartimento di Fisica, Università di Firenze and INFN-Sezione di Firenze, 50019 Sesto Fiorentino, Firenze, Italy.
Departamento de Física Teórica, Atómica y Optica, Universidad de Valladolid, 47011 Valladolid, Spain.
Entropy (Basel). 2018 Oct 23;20(11):816. doi: 10.3390/e20110816.
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. The results obtained are relevant in quantum mechanics as well as in signal processing as Fourier analysis has a close relation with signal filters. In addition, 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.
在本文中，我们展示了在实数线R和半直线R +上的调和分析的最新结果，这些结果分别表明了埃尔米特函数和拉盖尔函数、它们的对称群以及傅里叶分析之间的紧密关系。这可以通过基于使用装配希尔伯特空间的统一框架来实现。我们发现对称群的泛包络代数与分数傅里叶变换之间的关系。由于傅里叶分析与信号滤波器密切相关，因此所获得的结果在量子力学以及信号处理中都具有重要意义。此外，我们介绍了一些关于圆上离散傅里叶变换的新结果。我们引入了在圆上使用埃尔米特函数构造的具有在傅里叶变换下有趣性质的新函数。
