On estimates for the quaternion linear canonical transform in the space \(L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\)

This paper is an exposition of some estimates which have a number of applications to interpolation theory. In particular some recent problems in image processing and singular integral operators require the computation of suitable estimates. In Abilov et al. (Comput Math Math Phys 48:2146, 2008) , Abilov et al. proved two useful estimates for the Fourier transform in the space of square integral multivariable functions on certain classes of functions characterized by the generalized continuity modulus, and these estimates are proved by Abilov for only two variables, using a translation operator. The purpose of this paper is to study these estimates for measurable sets from complex domain to hyper complex domain by using quaternion algebras, associated with the quaternion linear canonical transform, constructed by the generalized Steklov function.