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).

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

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!

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 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
2017-05-18, torbjri