The intertwining operator for the generalized Dunkl transform on the line

Abstract

In harmonic analysis on a line with power weight, the unitary Dunkl transform first appeared.
It depends on only one parameter 𝑘 ⩾ 0. Then the two-parameter (𝑘, 𝑎)-generalized Fourier transform appeared, a special case of which is the Dunkl transform (𝑎 = 2). The presence of the parameter 𝑎 > 0 at 𝑎 ̸= 2 leads to the appearance of deformation properties. For
example, for functions in Schwarz space, the generalized Fourier transform may not be infinitely differentiable or decay rapidly at infinity. In the case of the sequence 𝑎=2/(2𝑟 + 1), 𝑟 ∈ Z+, the deformation properties of the generalized Fourier transform are very weak and after some change of variables they disappear. The resulting unitary transform for 𝑟 = 0 gives the usual Dunkl transform and has many of its properties. It is called the generalized Dunkl transform.
We define the intertwining operator that establishes a connection between the second-order differential-difference operator, for which the kernel of the generalized Dunkl transform is an eigenfunction, and the one-dimensional Laplace operator and allows us to write the kernel in a form convenient for its estimates. Unlike the intertwining operator for the Dunkl transform,
it has a nonzero kernel. In the paper, also on the basis of the properties of the generalized Dunkl transform, the properties of the (𝑘, 𝑎)-generalized Fourier transform for 𝑎 = 2/(2𝑟 + 1) are established.