School of Mathematics, Trinity College

MAU34206 - Harmonic Analysis I 2020

JS & SS Mathematics , SS Two-subject Moderatorship

Lecturer Dmitri Zaitsev

Annual Examinations: The format will be the same as in the year 2017/2018 exam, which can be considered a sample paper. Credit will be given for the best 3 questions out of total 4 questions.

Problem Sheets in PDF: Sheet 1 Sheet 2 Sheet 3

Module outline:

Introduction: Trigonometric polynomials. Complex-valued Fourier polynomials and sums of exponentials. Periodic functions. Topological groups. Characters. Finite and finitely generated abelian groups. The unit circle in the complex plane and the additive group of real numbers as topological groups. Dual groups. Inner products. Orthogonality, orthonormality and linear independence of characters. Fourier coefficients. Fourier series and Fourier transform in concrete and abstract settings.

Convergence results on the unit circle: Fourier series and partial sums. Bessel's inequality. Translations and convolutions. Homogeneous Banach spaces. Integrals valued in Banach spaces and their applications to convolutions. Summability kernels and convergence result for their convolutions. Cesàro averages of a sequence. Dirichlet and Fejér kernels. Their applications to partial sums of Fourier series and their Cesàro averages.

Fejér's theorem and applications: Uniform convergence for continuously differentiable functions. Density of trigonometric functions in the space of continuous functions on the unit circle. Unique determination of functions in L1 by their Fourier series. Convergence of Fourier series in L2. Parseval's identity. Fourier characterization of functions in L2. Riemann-Lebesgue lemma.

Textbooks (the "Google Books" links do not show the entire books but some books are available online, just copy-paste and search):

A. Deitmar. A First Course in Harmonic Analysis. Springer-Verlag New York. 2005. DOI 10.1007/0-387-27561-4.
Y. Katznelson, An Introduction to Harmonic Analysis. Dover 1976.
E. Hewitt & K. A. Ross. Abstract Harmonic Analysis 1. Springer 1963.
L. H. Loomis. An Introduction to Abstract Harmonic Analysis, Van Nostrand Company, Inc. 1953.
W. Rudin. Fourier Analysis on Groups. 1962. Print ISBN:9780470744819, Online ISBN:9781118165621, DOI:10.1002/9781118165621.
S. Axler. Measure, Integration & Real Analysis. Springer's Graduate Texts in Mathematics 2020.

Various Lecture Notes:

Chapter 1: Examples for Fourier analysis by Richard M. Timoney
Chapter 2: Convergence results for Fourier series by Richard M. Timoney
Chapter 3: LCA groups and their duals by Richard M. Timoney
Lecture notes, Harmonic Analysis minicourse by Joaquim Bruna
Lecture notes: harmonic analysis by Russell Brown
Mini-Course on Applied Harmonic Analysis by Philipp Christian Petersen
Fourier Analysis, MIT OpenCourseWare by David Jerison

Old Harmonic Analysis course web pages.
Old MA342A web page by Richard M. Timoney with Lecture Notes and Problem Sheets.

For exam-related problems look in TCD past examination papers and Mathematics department examination papers.

I will appreciate any (also critical) suggestions that you may have for the course. Let me know your opinion, what can/should be improved, avoided etc. and I will do my best to follow them. Feel free to come and see me if and when you have a question about anything in this course. Or use the feedback form from where you can also send me anonymous messages.