Titchmarsh’s-type theorem for two-sided quaternion Fourier transform and sharp Hausdorff–Young inequality for quaternion linear canonical transform

Abstract

In this work, we first introduce the two-sided quaternion Fourier transform and demonstrate its essential properties. We generalize Titchmarsh’s-type theorem in the framework of the two-sided quaternion Fourier transform. Based on the interaction between the quaternion Fourier transform and quaternion linear canonical transform we explore sharp Hausdorff–Young inequality for the quaternion linear canonical transform. The obtained result can be considered as a generalized version of sharp Hausdorff–Young inequality for the two-dimensional quaternion Fourier transformation in the literature.