Sharp inequalities for geometric Fourier transform and associated ambiguity function

Abstract

The geometric Fourier transform defined using Clifford algebra plays an increasingly active role in modern data analysis, in particular for color image processing. However, due to the non-commutativity, it is hard to obtain many important analytic properties. In this paper, we prove several important sharp inequalities, including the Hausdorff-Young inequality and its converse, Pitt's inequality and Lieb's inequality for Clifford ambiguity functions. With these inequalities, several uncertainty principles with optimal constants are derived for the geometric Fourier transform. Most results in this paper are new even in the easier case, i.e. the quaternion setting. The method developed here also works for more general geometric Fourier transforms.