Hermite functions and Fourier series

TL;DR

This paper develops Hermite function-based orthonormal bases for functions on the circle and sequences, exploring their relations via Fourier transforms, and introduces ladder operators and riggings to analyze their properties.

Contribution

It introduces new bases in $L^2$ spaces related by Fourier transforms, along with ladder operators and riggings, extending quantum oscillator concepts to these function spaces.

Findings

Constructed orthonormal bases using Hermite functions.

Established unitary relations between bases via Fourier transforms.

Developed ladder operators with properties similar to quantum harmonic oscillators.

Abstract

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle ($L^2(\mathcal C)$) and in $l_2(\mathbb Z)$, which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm--Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both $L^2(\mathcal C)$ and $l_2(\mathbb Z)$, so that all the mentioned operators are continuous.