The Fourier Transform on Quantum Euclidean Space

TL;DR

This paper develops a Fourier transform framework on quantum Euclidean space, introducing a modified transform, its inverse, and analyzing its properties, including relations to q-Hankel transforms and the harmonic oscillator.

Contribution

It presents a novel Fourier transform on quantum Euclidean space, including its inverse, Bochner's relations, and connections to q-Bessel functions and the harmonic oscillator.

Findings

Fourier transform defined via Bochner's relations and q-Hankel transforms.

Transform acts between harmonic oscillator representation spaces.

Transform is its own inverse and satisfies Parseval's theorem.

Abstract

We study Fourier theory on quantum Euclidean space. A modified version of the general definition of the Fourier transform on a quantum space is used and its inverse is constructed. The Fourier transforms can be defined by their Bochner's relations and a new type of q-Hankel transforms using the first and second q-Bessel functions. The behavior of the Fourier transforms with respect to partial derivatives and multiplication with variables is studied. The Fourier transform acts between the two representation spaces for the harmonic oscillator on quantum Euclidean space. By using this property it is possible to define a Fourier transform on the entire Hilbert space of the harmonic oscillator, which is its own inverse and satisfies the Parseval theorem.