Bargmann’s versus for fractional Fourier transforms and application to the quaternionic fractional Hankel transform

Abstract

We present a general formalism `a la Bargmann for constructing fractional Fourier transform associated to specific class of integral transforms on separable Hilbert spaces. As concrete application, we consider the quaternionic fractional Fourier transform on the real half-line and associated to the hyperholomorphic second Bargmann transform for the slice Bergman space of second kind. This leads to an extended version of the well-known fractional Hankel transform. Basic properties are derived including inversion formula and Plancherel identity.