Polar Duality and the Donoho--Stark Uncertainty Principle

TL;DR

This paper explores the connection between polar duality in convex geometry and the Donoho-Stark uncertainty principle, establishing new concentration estimates for functions and their Fourier transforms, with implications for the Wigner function.

Contribution

It introduces a novel link between convex geometric duality and harmonic analysis uncertainty principles, providing new bounds on concentration trade-offs.

Findings

Derived estimates for concentration trade-offs between functions and their Fourier transforms.

Established a new concentration result for the Wigner function using the Donoho-Stark principle.

Connected convex geometric inequalities with harmonic analysis uncertainty principles.

Abstract

Polar duality is a fundamental geometric concept that can be interpreted as a form of Fourier transform between convex sets. Meanwhile, the Donoho-Stark uncertainty principle in harmonic analysis provides a framework for comparing the relative concentrations of a function and its Fourier transform. Combining the Blaschke--Santal\'o inequality from convex geometry with the Donoho--Stark principle, we establish estimates for the trade-off of concentration between a square integrable function in a symmetric convex body and that of its Fourier transform in the polar dual of that body. In passing, we use the Donoho-Stark uncertainty principle to establish a new concentration result for the Wigner function.