The Uncertainty Principle in Harmonic Analysis -- Lecture Notes on Selected Topics

TL;DR

This paper explores key aspects of the uncertainty principle in harmonic analysis, focusing on Fourier analysis, spectral gaps, and quasi-analyticity, highlighting classical theorems and recent developments.

Contribution

It provides an overview of classical and modern results related to the uncertainty principle in harmonic analysis, emphasizing Fourier analysis on the circle and real line.

Findings

Discussion of the Paley–Wiener theorem and Beurling–Malliavin multiplier theorem

Analysis of spectral gaps and their influence on Fourier transforms

Examination of logarithmic integrability in approximation and quasi-analyticity

Abstract

These lecture notes are devoted to selected topics related to the uncertainty principle in harmonic analysis. Rather than attempting a systematic treatment, we emphasize only a number of both classical and deep manifestations of this principle, mainly from the perspective of Fourier analysis on the unit circle and on the real line. We consider problems of uniqueness and reconstruction for Fourier series and Fourier transforms, the influence of spectral gaps, and the role of logarithmic integrability in questions of approximation and quasi-analyticity. Central results discussed include the Paley--Wiener theorem, the Beurling--Malliavin multiplier theorem, and the Ivashev--Musatov theorem. These notes are intended as an entry point toward the research literature, with several sections pointing in the direction of more recent developments.