# 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).

#### LECTURER

- Kristian Seip: Office 956 in SB II, kristian [dot] seip [at] ntnu [dot] no

#### REFERENCE GROUP

#### 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!

## 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 |