The Fourier Transform on Quantum Euclidean Space

Details The Fourier Transform on Quantum Euclidean Space The Fourier Transform on Quantum Euclidean Space

2011-05-03

arxiv.org/abs/1105.0585v2

SIGMA 7:047,2011

Physics

Mathematical Physics

Scientific paper

10.3842/SIGMA.2011.047

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.

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