The Jacobi-Dunkl transform on \(\mathbb R\) and the convolution product on new spaces of distributions (Q2270204)

The authors define the Jacobi-Dunkl transform by means of the eigenfunction of the Jacobi-Dunkl operator \(\Lambda _{\alpha ,\beta}\), \(\alpha\geq \beta\geq -1/2\), \(\alpha \not=-1/2\), on \(\mathbb{R}\). First, they collect the main properties related to the harmonic analysis associated with \(\Lambda _{\alpha ,\beta}\), giving some results on the Jacobi-Dunkl kernel and the Jacobi-Dunkl transform and convolution. Then they introduce and define the Jacobi-Dunkl transform on new distribution spaces \(A_m'\) by using the kernel method and establish boundedness, uniqueness, smoothness and inversion theorems for this transformation. Finally, they analyze the Jacobi-Dunkl convolution on \(A_m'\), proving algebraic properties and an interchange formula involving this convolution product and the Jacobi-Dunkl transform.