# TMA4170 Fourier analysis 2019

## Messages

- The first lecture will take place in GL-SB2 S21 on Monday, January 7. Instead of the text book, we will use the lecture notes (see below: SYLLABUS-Notes). But this is not an online course, motivation and necessary details, missing from the notes, are lectured. You are recommended to read section 4 "Appendix 1" part of notes before the lecture.

- The first exercise session will take place in S21, 15th January, 16:15-17:00

- Exercise for week 2 can be found at the end of the notes, week 3 exercise will be handed out on Thursday, 10th January.

- Today (17th Jan) we finished section 1.6. The next lecture will be given by Helge Holden.

- On Tuesday, we went through Exercise 2. Since we will not provide the solutions for the exercise. We encourage you to attend the Exercise session or hand in the exercise. Exercise for week 5 is "Exercise set 1", see section 6 of the notes.

- There will be no Exercise Session on 22 Jan
- There will be no Exercise Session on 29 Jan
- Week 5 exercise has two parts and can be found in the notes now, solutions for Week 3 and 4 exercises will be presented on 5th Feb, 16:15-18:00, S21. There is an error in the last exercise for Week 4, sgn(x) should be -1 when x is negative
- There will be no Exercise Session on 12 Feb
- We will need some complex analysis results form the Exercise for week 6, please have a look if you haven't taken the complex analysis course.
- Exercise set for week 7 is online now.
- Exercise set for week 8 is online now.
- Part 2 of the notes is online now. Next week (11-17 March) we will do central limit theorem, equidistribution theorem and random walks.
- An errata for part 1 of the notes (also contains a list of errors in the weekly exercises).
- The Exercise session for 26th March will by in Xu's office (942 S2), the same time.
- Complete version of Part 2 of the notes is online now.
- The final week will be repetition based on the exercises (No Exercise Session on 9 April)
- A short summary of the course and solutions to several exercises

## Course Information

- Time Plan can be found here.

#### LECTURES

- Mondays 12:15-14:00, S21 Sentralbygg 2
- Thursdays 10:15-12:00, S21 Sentralbygg 2

#### EXERCISE SESSIONS

- Tuesdays 16:15-17:00, S21 Sentralbygg 2

#### EXAM

- Exam date: May 31, 09:00-13:00
- 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-Notes

The curriculum is covered mainly by the lectures based on the following notes and everything handed out during the lectures.

1. part one

2. part two (renewed from time to time)

3. Errata

#### Recommend books

The Fourier analysis part of the notes is based on the following two books:

**E. M. Stein & R. Shakarchi: Fourier Analysis, An Introduction, Princeton University Press, 2003**

**H. Dym & H. P. McKean: Fourier Series and Integrals, Academic Press, 1972**

The wavelet part of the notes is based on the following book:

**A. Boggess & F. Narcowich: A first course in Wavelets with Fourier Analysis, Wiley, 2nd Edition, 2009**.

We also recommend the following standard textbook on Fourier analysis and wavelets:

** Fourier Analysis and Applications, by G. Gasquet and P. Witomski, Springer.**

#### LECTURER

- Helge Holden: Office 1156 in SB II, helge [dot] holden [at] ntnu [dot] no
- Xu Wang: Office 942 in SB II, xu [dot] wang [at] ntnu [dot] no

#### REFERENCE GROUP

- Paul Duffin: pauldu [at] stud [dot] ntnu [dot] no
- Dennis Fimland: dennishf [at] stud [dot] ntnu [dot] no

#### CONSULTATION BEFORE THE EXAM

TBA

#### FINAL EXAM

TBA

## Exercises

Usually we will hand out the exercise, but you can also find the exercises at the end of the notes. The first five exercise sets are in page 53–59 in Part 1 of the notes. There are some mistakes, see the Errata for the corrections.

Wavelet Exercise (Week 12-14): from Boggess–Narcowich

Ch.4: 1, 3, 4, 5

Ch.5: 5, 6, 10, 12

Ch.5: 8, 17, 18

Solutions can be found in the 2018 TMA4170 course page.

#### 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 (according to the notes)

Week 2 | 1.1-1.5 | |

Week 3 | 1.5, Appendix 2 | |

Week 4 | 1.6-1.7 | |

Week 5 | 1.7-1.8 | |

Week 6 | 2.1-2.3 | |

Week 7 | 2.4 | |

Week 8 | 2.5-2.6 | |

Week 9 | 2.7-2.8 | |

Week 10 | B&N Ch.4 | |

Week 11 | B&N Ch.5 | |

Week 12 | B&N Ch.6 | |

Week 13 | B&N Ch.7 | |

Week 14 | Repetition | |

Week 15 | Repetition |

## Actual Lecture Plan

Week 2 | Fourier series, Pointwise and Mean square convergence | Xu |

Week 3 | Lebesgue integral, elementary solution of D+s | Xu |

Week 4 | Sec 1.7.1, 1.7.2 Sec 1.8.1 | Helge |

Week 5 | Sec 2.1-2.3 | Helge(Monday), Ho Cheung Pang(Thursday) |

Week 6 | 2.4.1-2.4.2 | Helge(Monday), Xu(Thursday) |

Week 7 | 2.4.3-2.4.5 | Xu |

Week 8 | Exercise set 6 on complex analysis | Xu |

Week 9 | 2.5 | Xu |

Week 10 | 2.6-2.7 | Xu |

Week 11 | 2.8, 1.7.3, 1.8.2 | Xu |

Week 12 | Mon: Ch 4 in BN, Thu: Se 5.1 | Xu(Monday), Ho Cheung Pang(Thursday) |

Week 13 | Mon: Sec 5.2 (p. 204-214) in Boggess & Narcowich. Thu: Sec. 5.3 (up to p. 222) | Helge |

Week 14 | Mon: The remaining part of Ch. 5, and Ch. 6 up to and incl. p. 237 , Thu: wavelet transform | Helge(Monday), Xu(Thursday) |

Week 15 | Repetition based on exercises | Xu |