Generalized translation operator and uncertainty principles associated with the deformed Stockwell transform

Abstract

We study the generalized translation operator associated with the deformed Hankel transform on R. Firstly, we prove the trigonometric form of the generalized translation operator. Next, we derive the positivity of this operator on a suitable space of even functions. Making use of the positivity of the generalized translation operator we introduce and study the deformed Stockwell transform. Knowing the fact that the study of uncertainty principles is both theoretically interesting and practically useful, we formulate several qualitative uncertainty principles for this new integral transform. Firstly, we mainly establish various versions of Heisenberg's uncertainty principles. Secondly, we derive some weighted uncertainty inequalities such as Pitt's and Beckner's uncertainty inequalities for the deformed Stockwell transform. We culminate our study by formulating several concentration-based uncertainty principles, including the Amrein-Berthier-Benedicks and local inequalities for the deformed Stockwell transform.