Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 047, 30 pages      arXiv:1105.0585      https://doi.org/10.3842/SIGMA.2011.047
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

The Fourier Transform on Quantum Euclidean Space

Kevin Coulembier
Gent University, Galglaan 2, 9000 Gent, Belgium

Received November 19, 2010, in final form April 21, 2011; Published online May 11, 2011

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.

Key words: quantum Euclidean space; Fourier transform; q-Hankel transform; harmonic analysis; q-polynomials; harmonic oscillator.

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