Sharp inequalities for geometric Fourier transform and associated ambiguity function

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 importan...

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Published in:Journal of mathematical analysis and applications 2020-04, Vol.484 (2), p.123730, Article 123730
Main Author: Lian, Pan
Format: Article
Language:English
Subjects:
Clifford algebra
Geometric Fourier transform
Hausdorff-Young inequality
Quaternion
Uncertainty principle
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description 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.
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subjects Clifford algebra
Geometric Fourier transform
Hausdorff-Young inequality
Quaternion
Uncertainty principle
title Sharp inequalities for geometric Fourier transform and associated ambiguity function
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