Properties and applications of quaternion quadratic phase Fourier transforms

The quaternion quadratic phase Fourier transform (QQPFT), an extension of the well-known quaternion Fourier transform (QFT), has emerged as a significant advancement in signal processing and optics. In this study, we aim to provide a direct proof of the Plancherel theorem within the context of the QQPFT. Specifically, we establish the theorem of the scalar inner product for the two-sided QQPFT and explore the quaternion inner product concept for the right-sided QQPFT. Additionally, we present a proof of the Plancherel theorem for quaternion values in the left-sided QQPFT. Also, we discuss the asymptotic behavior of the two-sided QQPFT and the right-sided QQPFT. Finally, as an application, we discuss the solution of some generalized quaternion differential equations.