# TMA4170 Fourier analysis, spring 2018

Course description can be found here.

## Messages

• The first lecture will take place in R30 on Tuesday, January 9.
• The first exercise session will take place in Room 656, Sentralbygg 2, on Wednesday, January 17.
• NB! Starting from Thursday January 11, teaching will take place in Room 656 in Sentralbygg 2.

## Course Information

#### LECTURES

• Tuesdays 14:15-16:00, Auditorium 656 Sentralbygg 2
• Thursdays 08:15-10:00, Auditorium 656 Sentralbygg 2

#### EXERCISE SESSIONS

• Wednesdays 16:15-17:00, Auditorium 656 Sentralbygg 2

#### EXAM

• Exam date: May 23
• Permitted aids: One A-4 sized sheet of yellow paper stamped by the Department of Math. Sciences (on it you may in advance write whatever you like).

#### SYLLABUS

We plan to cover the following parts of the book by Boggess & Narcowich

• Chapter 0
• Chapter 1 (I recommend this paper by Paul Chernoff as a supplement to 1.3.2 in B&N)
• Chapter 2
• Chapter 3 (See here for the original paper of Cooley and Tukey on FFT, which as of February 6, 2018, has 13770 citations in Google Scholar)
• Chapter 4
• Chapter 5
• Chapter 6
• Section 7.1, 7.2 and 7.4
• Appendix A

During the last four weeks of the course (starting on March 13), we will discuss some classical examples of applications of Fourier series and integrals in other fields such as probability, approximation theory, geometry, number theory, differential equations, and physics. A good part of this material can be found in the book by Dym & McKean (see below). Be aware, however, that typically many different proofs can be given of the results discussed during the lectures. You may for example consult this paper for a nice complex analytic proof of the isoperimetric inequality.

Final note on the syllabus (May 3): You should consider all topics covered by the lectures and everything handed out during the lectures as part of the syllabus. Also the exercises should be considered part of the syllabus.

#### TEXT BOOK

A. Boggess & F. Narcowich: A first course in Wavelets with Fourier Analysis, Wiley, 2nd Edition, 2009.
The book is available as an eBook through the university library, although restricted to only one person at a time.

You may find the book H. Dym & H. P. McKean: Fourier Series and Integrals, Academic Press, 1972 an interesting supplement; it is a little more advanced and contains a variety of important applications of Fourier analysis. The same can be said about E. M. Stein & R. Shakarchi: Fourier Analysis, An Introduction, Princeton University Press, 2003. Ingrid Daubechies: Ten Lectures on Wavelets, SIAM, 1992 is a classic that goes more deeply into the various aspects of wavelet theory.

As mentioned on page 160 of Boggess & Narcowich, Yves Meyer, Wavelets, Algorithms and Applications, SIAM, 1993 is interesting reading. Yves Meyer is the 2018 Lars Onsager Lecturer at NTNU. See here for a recent interview with him, by Bjørn Dundas and Christian Skau. Here are the slides from a recent talk about Yves Meyer's achievements (in Norwegian).

#### CONSULTATION BEFORE THE EXAM

I will be available for questions on May 16 and May 18. You may also contact me on any workday before May 16. NB! I will be away on May 22, and May 17 and 21 are both holidays.

#### FINAL EXAM

Here is the final exam.

## Exercises

The exercises are NOT mandatory, but strongly recommended.

 Week 3 B&N Ch.0: 6, 7, 8, 11, 15, 23 Solution Week 4 The Fejér kernel + B&N Ch.1: 1, 22 Solution Week 5 The Schwartz class and the Poisson summation formula + B&N Ch.2: 4, 5 Solution Week 6 The four exercises here + B&N Ch.2: 14 Solution Week 7 B&N Ch.3: 2, 3, 10, 11, 12 (Extra: You may take a look at Ex. 14, Ch. 3, for an interesting application of FFT) Solution Week 8 B&N Ch.4: 1, 3, 4, 5 Solution Week 9 B&N Ch.5: 5, 10, 12 Solution Week 10 B&N Ch.5: 8, 17, 18 Solution Week 11 Construction of compactly supported wavelets Solution Week 12 Some applications of Fourier methods Solution Week 15 The isoperimetric inequality and equidistribution Solution Week 16 Miscellaneous applications and results Solution

#### Previous exams:

NOTE: The exams from 2014 and older are written with a slightly different curriculum in mind and as such may contain exercises not covered by the current curriculum!

 2017 Exam Solution 2016 Exam Misprint in ex. 2: Should be 1/abs(x)² not 1/abs(x) Solution 2015 Exam Solution 2014 Exam Solution 2013 continuation Exam Solution 2013 Exam Solution 2012 Exam Solution 2006 Exam Solution 2004 Exam Solution

## Tentative Lecture Plan

 Week 2 Ch. 0 Week 3 Ch 1 Week 4 Ch 2 Week 5 Ch 2 Week 6 Ch 2 Week 7 Ch 3 Week 8 Ch.4 Week 9 Ch 4 Week 10 Ch 5 Week 11 Ch 6 Week 12 Ch 6 Week 14 Ch 7 Week 15 Ch 7 Week 16 … Week 17 …

## Actual Lecture Plan

 Date Pages/topics lectured 09.01. 1–21 11.01. 38–64 16.01. 64–83 18.01. 87–88 (Gibb's phenomenon), 90–91 (Poisson's formula) 23.01. 92–110 25.01. 107–110 + Appendix A.1. 30.01. 110–127 01.02. Note on the Paley--Wiener theorem and Fourier analysis 06.02. 132–142 08.02. 143–153 13.02. 160–175 (Notice: Thm. 4.9 is not correct as stated) 15.02. 176–186 20.02. 190–197 22.02. 196–200 + 221–224 (Notice: Thm. 5.11 is not correct as stated) 27.02. 201–214 01.03. 225–227 + Appendix A.2.3 (where we use the Shannon scaling function instead of Haar) 06.03. 234–238 08.03. 238–241, 253–258 13.03. Fourier analysis and probability: Random Walks and the Central Limit Theorem 15.03. Wirtinger's inequality and the isoperimetric problem (and the relation between these) 20.03. Weyl's equidistribution theorem 22.03. A continuous but nowhere differentiable function 05.04. NO LECTURE 10.04 The Poisson summation formula revisited: The functional equation for $\zeta(s)$ 12.04 The heat equation (pp. 145–153 in Stein and Shakarchi) 17.04 Good kernels (ctd.): Heat, Poisson, Fejer (pp. 48–51, 156–158 in Stein and Shakarchi) 19.04 NO LECTURE 24.04 NO LECTURE 26.04 Repetition: Discussion of the syllabus based mainly on the first 6 exercises 03.05 Repetition: Discussion of the syllabus based mainly on the last 6 exercises