In this paper, we study a new harmonic analysis in the Opdam-Cherednik setting. We establish the linear canonical Opdam-Cherednik transform in order to generalize the classical Opdam-Cherednik transform. Firstly, we establish the associated harmonic analysis and we introduce some of its basic properties, such as Parseval identity and inversion formula. Next, we study the linear canonical Opdam-Cherednik transform on the corresponding spaces of Schwartz and of tempered distributions. Additionally, an application in solving a generalized heat equation is given. Finally, we establish several uncertainty principles, including Miyachi-type and Hardy-type theorems, for this new transform.
Abdallah, Nabiha, Ben (Thu,) studied this question.