Kevin Coulembier (Ghent University, Belgium)

Fourier theory on quantum spaces and q-Hankel transform

Abstract:
   We study Fourier theory on quantum Euclidean space ℝ_q^m. Our approach is related to the general abstract theory of Fourier transforms on quantum spaces in the Hopf-algebraic approach as developed by Kempf and Majid. We study ℝ_q^m from the point of harmonic analysis, which is described here by the Howe dual pair (O_q(m), U_q(sl₂)). The O_q(m)-invariance of the Fourier transform implies that the transform is in essence radial. These radial transforms are given by a q-deformation of the Hankel transforms. We obtain Hankel transforms using the first and second q-Bessel functions. This completes the theory of q-Hankel transforms started by Koornwinder and Swarttouw, who studied the q-Hankel transforms for the third q-Bessel function. Then we present some important properties of the obtained Fourier transform and finally link it to the harmonic oscillator on quantum Euclidean space.