On the fractional Opdam–Cherednik transform

In the present paper, we study a new harmonic analysis in the setting of the Opdam–Cherednik. We establish the fractional Opdam–Cherednik transform in order to generalize the classical fractional Fourier transform. At first, 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 fractional Opdam–Cherednik transform on the corresponding Schwartz space and the tempered distribution. In the end, we develop the scope of this work by investigating the fractional Opdam–Cherednik transform for transport equation.