On uncertainty principles in the finite dimensional setting

TL;DR

This paper establishes a quantitative uncertainty principle for vector representations in two bases, extending classical results to finite-dimensional settings and applying them to discrete Fourier analysis and trigonometric polynomials.

Contribution

It introduces a new quantitative uncertainty principle for finite-dimensional vectors and demonstrates its application to the discrete Short Time Fourier Transform and trigonometric polynomials.

Findings

Quantitative uncertainty bounds for two-base representations

Extension to discrete Short Time Fourier Transform

Application to trigonometric polynomial analysis

Abstract

The aim of this paper is to prove an uncertainty principle for the representation of a vector in two bases. Our result extends previously known qualitative uncertainty principles into quantitative estimates. We then show how to transfer this result to the discrete version of the Short Time Fourier Transform. An application to trigonometric polynomials is also given.