Sharp Hausdorff-Young inequalities for the quaternion Fourier transforms

Abstract

The quaternion Fourier transforms are powerful tools in modern data analysis, in particular for color image processing. At present, there are mainly three different quaternion Fourier transforms widely used. In this paper, we prove the sharp Hausdorff-Young inequalities for these three transforms and the more general ones, i.e., the steerable quaternion Fourier transforms. Then Hirschman’s entropy uncertainty principle in the quaternion setting follows from the standard differential approach.