The Fourier Transform on Quantum Euclidean Space

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.

References in corpus (14)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Algebraic {$q$}-Integration and Fourier Theory on Quantum and Braided Spaces
• Free Braided Differential Calculus, Braided Binomial Theorem and the Braided Exponential Map
• Spherical harmonics and integration in superspace
• Reality in the Differential Calculus on q-euclidean Spaces
• Integration in superspace using distribution theory
• The $S{O}_q(N,\mathbf{R})$-Symmetric Harmonic Oscillator on the Quantum Euclidean Space ${\mathbf{R}}_q^N$ and its Hilbert Space Structure
• q-Integration on Quantum Spaces
• Supersymmetry from a braided point of view
• Realization of $U_q(so(N))$ within the differntial algebra on ${\mathbf{R}}_q^N$
• The $q$-deformed Schrödinger Equation of the Harmonic Oscillator on the Quantum Euclidian Space
• q-deformed harmonic and Clifford analysis and the q-Hermite and Laguerre polynomials
• q-Exponentials on quantum spaces
• Generalized Q-Exponentials Related to Orthogonal Quantum Groups and Fourier Transformations of Noncommutative Spaces