Harmonic Analysis Lecture Notes

TL;DR

This graduate course notes introduce harmonic analysis fundamentals, emphasizing Fourier series, transforms, and key tools like maximal functions, culminating in applications to band-limited functions and uncertainty principles.

Contribution

Provides a comprehensive, structured introduction to harmonic analysis, integrating classical Fourier analysis with modern tools and applications in a pedagogical format.

Findings

Fourier series convergence on the circle

Application of Hilbert transform in L^p spaces

Analysis of band-limited functions and uncertainty principles

Abstract

These notes present a first graduate course in harmonic analysis. The first part emphasizes Fourier series, since so many aspects of harmonic analysis arise already in that classical context. The Hilbert transform is treated on the circle, for example, where it is used to prove L^p convergence of Fourier series. Maximal functions and Calderon--Zygmund decompositions are treated in R^d, so that they can be applied again in the second part of the course, where the Fourier transform is studied. The final part of the course treats band limited functions, Poisson summation and uncertainty principles. Distribution functions and interpolation are covered in the Appendices. The references at the beginning of each chapter provide guidance to students who wish to delve more deeply, or roam more widely, in the subject.