Schwartz space invariance under the one-dimensional (k,a)-generalized Fourier transform

Establish whether the one-dimensional (k,a)-generalized Fourier transform F_{k,a} maps the Schwartz space S(R) into itself; equivalently, determine if S(R) is invariant under F_{k,a} for admissible parameters k and a in dimension N=1.

Background

The (k,a)-generalized Fourier transform F_{k,a}, introduced by Ben Saïd, Kobayashi, and Ørsted, deforms the classical Fourier analysis via the Hamiltonian involving the Dunkl Laplacian and a parameter a>0. In one dimension, it admits an integral representation with kernel B_{k,a} and shares several structural properties with the classical transform, including a Plancherel formula.

In the introduction, the authors note that despite recent progress, essential analytic properties of F_{k,a} remain unsettled even in one dimension. One such property is the invariance of the Schwartz space S(R) under F_{k,a), which is a cornerstone in classical Fourier analysis and many generalized transforms. Establishing S(R)-invariance would clarify the fine regularity and decay behavior preserved by F_{k,a}, enabling deeper harmonic analysis and PDE applications in this deformed setting.