Different types of Plancherel’s theorems for square integrable functions associated with quaternion offset linear canonical transforms

The offset linear canonical transform (OLCT) is an important tool in signal processing and optics . Recently, the quaternion offset linear canonical transform (QOLCT) has been introduced which is the quaternion extension of the OLCT and the generalized form of quaternion Fourier transform(QFT). In this article, the Plancherel’s theorem of the scalar inner product for the two-sided QOLCT is introduced. Also, the quaternion inner product theorems for the right sided and left sided QOLCT have been discussed. Further, as an application of the Plancherel’s theorem, the real Paley-Wiener theorem and Donoho-Stark uncertainty principle have been explored as well as the solution of particular type of quaternion differential equations are discussed using QOLCT. Additionally, the advantages of QOLCT over QLCT and QFT is illustrated graphically using example and the use of Plancherel’s theorem in filter analysis is demonstrated.