# TMA4170 Fourier analysis, spring 2017

Course description can be found here.

## Messages

**IMPORTANT:**New exams have been added to the Previous Exams section. However, when solving these, be aware that the curriculum has changed and that some exercises may not be covered by the current curriculum.- We will solve exam problems on Thu, April 20, and Tue, April 25, starting with the regular exam, 2016, which can be found from last year's course.
- There will be
**no lectures or exercise session**on Tuesday, March 7 and Thursday, March 9! To make up for this, there will be an**extra lecture**on Tuesday, February 28 in R30 at 15:15-17:00, replacing the exercise session. - The first lecture will take place in R30 on Tuesday, January 10.

- The first exercise session take place in R30 on Tuesday, January 17.

## Course Information

#### LECTURES

- Tuesdays 10:15-12:00, Auditorium R30
- Thursdays 10:15-12:00, Auditorium R30

#### EXERCISE SESSIONS

- Tuesdays 16:15-17:00, Auditorium R30

#### EXAM

- Exam date: May 18, 2017.
- 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).
- The Exam and a Solution proposal.

#### SYLLABUS

From the book by Boggess & Narcowich

- Chapter 1

- Chapter 2

- Chapter 3

- Chapter 4

- Chapter 5
- Chapter 6
- Section 7.1, 7.2 and 7.4
- Appendix A
- The notes on distributions found below

#### 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.

#### LECTURERS

- Helge Holden: Office 1156 in SB II, helge [dot] holden [at] ntnu [dot] no

- Torbjørn Ringholm: Office 1338 in SB II, torbjorn [dot] ringholm [at] ntnu [dot] no

Please write TMA4170 in the subject line on all e-mails related to the course.

## Exercises

The exercises are NOT mandatory, but strongly recommended.

Week 3 | B&N Chapter 0: 4, 5, 11, 12, 17, 23 | Solution Proposal |

Week 4 | B&N Chapter 1: 1, 18, 20, 22, 25. Hard: 40 | Solution Proposal |

Week 5 | B&N Chapter 2: 4, 5, 6, 10, 11 | Solution Proposal |

Week 6 | B&N Chapter 2: 8, 9, 13, 14 | Solution Proposal |

Week 7 | B&N Chapter 3: 3, 10, 11, 12 | Solution Proposal |

Week 8 | B&N Chapter 3: 7, 14, 16 | Solution Proposal |

Week 9 | B&N Chapter 4: 1, 3, 5, 9 (No ex. session, lecture instead) | Solution Proposal |

Week 10 | B&N Chapter 5: 4, 8, 9 (No ex. session) | Solution Proposal |

Week 11 | B&N Chapter 5: 5, 6, 7 | Solution Proposal |

Week 12 | B&N Chapter 5: 10, 13 | Solution Proposal |

Week 13 | B&N Chapter 6: 2, 4, 6 | Solution Proposal |

#### 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 | The Hilbert Space. Orthonormal bases. Fourier Coefficients. The Trigonometric System. |

Week 3 | Ch 1 | Convergence. Dirichlet's Kernel. |

Week 4 | Ch 2 | (Parseval's formula Ch 1), Gibb's phenomenon. Fourier transform |

Week 5 | Ch 2 | Fourier transform in L2, Plancherel's formula, convolutions. Sampling Theorem. |

Week 6 | Ch 2 | Oversampling. Heisenberg's Principle. Discrete Fourier Transform. |

Week 7 | Ch 3 | Z Transform. Haar Basis. Wavelets. |

Week 8 | Ch.4 | Haar basis |

Week 9 | Ch 4 | Wavelets |

Week 10 | Ch 5 | Wavelets. |

Week 11 | Ch 6 | Daubechies' Wavelets. |

Week 12 | Ch 6 | Daubechies' wavelets. Continuous Wavelet Transform. |

Week 13 | … | Various topics |

Week 14 | … | Repetition, Fourier transform of distributions |

Week 16 | … | Repetition, Fourier transform of distributions |

## Actual Lecture Plan

Date | Pages lectured |

10.01 | pp. 1-21 |

12.01 | pp. 38-64 |

17.01 | pp. 64-76 |

19.01 | pp. 76-94 + Gibbs' phenomenon |

24.01 | pp. 94-104 + Appendix A.1 |

26.01 | pp. 104–112 + a simple proof of convergence of Fourier-series |

31.01 | pp. 112-120 |

02.02 | pp. 120-135 |

07.02 | pp. 135-142 |

09.02 | pp. 143-149 |

14.02 | pp. 149-161 |

16.02 | pp. 161-175 |

21.02 | pp. 175-189 |

23.02 | pp. 190-196 |

28.02 | pp. 196-204 + Appendix A.2.2 |

02.03 | pp. 204-214 |

14.03 | pp. 214-217 + Appendix A.2.1 |

16.03 | pp. 217-225 |

21.03 | pp. 226-237 |

23.03 | pp. 238-242, 244-253 |

28.03 | pp. 253-258 |

30.03 | pp. 266-272 |

04.04 | Repetition |

06.04 | Theory of distributions. (pages 37-45). A text on tempered distributions can be found on (from UBC, Canada, by Feldman) |

20.04 | Exam 2016, problems 1,2,3,5 |

25.04 | Exam 2015, Exam 2016, problem 4 |

## Reference group

- Amra Buzaljko: amrab(at)stud.ntnu.no
- Ferenc Székely: ferencs(at)stud.ntnu.no