TMA4170 Fourieranalyse våren 2016

Kursbeskrivelse finnes i studiehåndboka.

Beskjeder

OFFICE HOURS MONDAY 6 JUNE 15-16, TUESDAY 7 JUNE 14-16 IN 1152 SB II

*Extra exercises during the lecture Monday 25.IV.2016*

* THE FIRST LECTURE WILL TAKE PLACE ON MONDAY THE 11th of January 2016
* THE FIRST EXERCISE SESSION ON TUESDAY the 19th of January 2016

Kursinformasjon

LECTURES

  • Monday 10.15-12, Auditorium KJL24
  • Friday 14.15-16, Auditorium KJL24

EXERCISES

  • Tuesday 17.15-18. Auditorium KJE24

EXAM.

  • Exam date: 10th of June, 2016. TextSolutions
  • 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 pleases you).
  • 16.V.2015 Text
  • 16.V.2015 Solutions

SYLLABUS

From the book by Boggess& Narcowich

  • Chapter 1
  • Chapter 2
  • Section 3.1
  • Chapter 4
  • Chapter 5
  • Chapter 6
  • Section 7.4

Moreover, the syllabus contains:

  • the Fourier Transform of Distributions
  • the Poisson Summation Formula
  • the Fourier Transform in several variables (basic knowledge). Radial functions.
  • all the EXERCISES!

___

TEXT BOOK

Most of the material will be found in
A. Boggess & F. Narcowich: A first course in Wavelets with Fourier Analysis.
The Fourier Series and Transforms have to be complemented with other sources. The Fourier transforms of distributions have to be added.

Teacher

  • Office hours: ???day 1?:15-1?:00

EXERCISES

Week 3 Exercises 2016 Example 5 postponed
Week 4 Exercises 2016
week 5 Exercises2016
week 6 Exercises2016. Do example 5 for many functions
week 7 Exercises2016
week 8 Exercises2016 MOVED TO G038 23.II.2016
week 9 Exercises2016. Some helpdistribution.pdf. .Sign(x)
week 10 Exercises 2016. Not related to the examples:Laplace Eqn
week 11Exercises2016. Some solutions Appendix: some formulas in several variables
week 14Chapter V: 4,5,7,8,182016 See Shannon's wavelet below!
week 15Exercises2016. Obvious missprint in the range of integration of the Four.coeff.ex.4
week 16Exercises2016
week 17Extra exercisesMonday 25.IV.2016, 10-12 o"clock.
week 17ExercisesTuesday 26th of April

Notes

More "advanced" theory:

Additional Literature

  • E. Stein, R. Shakarchi: "Fourier Analysis", Princeton.
  • "A Guide to Distribution Theory and Fourier Transforms" by R. Strichartz, World Scientific.–Easy to read. Useful information.
  • "Wavelets -A Primer" by Ch. Blatter

Tentative lecture plan

usually according to Boggess & Narcowich

Week 3 Ch. 0 The Hilbert Space. Orthonormal bases. Fourier Coefficients. The Trigonometric System.Bessel's ineq.
Week 4 Ch 1 Convergence. Dirichlet's Kernel, Fejer's KernelSe notes on conv. below!
week 5 Ch 2 (Parseval's formula Ch 1), Gibb's phenomenon. Fourier transform
week 6 Ch 2 Fourier transform in L2, Plancherel's formula, convolutions. Sampling Theorem.
week 7 Ch 2 Oversampling. Heisenberg's Principle. Poisson's summation formula.Exercise 2.14 Oversampling
——–7 Ch 3 Discrete Fourier Transform.
week 8 Ch 3 Fast Fourier Transform. Theory of DistributionsNot in the text book.
week 9 Notes Fourier Transform of Distributions. The Schwartz class. Haar basisnot in the text book.
week 10 Ch 4 Haar Basis. Wavelets.
week 11 Ch 5 Wavelets.
week 13 Ch 6 Daubechies' Wavelets. Shannon's WaveletCalculation of 2nd moment Last 5 lines irrelevant. One 1/2 missing inside Q(w)
week 14 Ch 6 Daubechies's wavelet. Continuous Wavelet Transform.
week 15 …Random walks, various things, H. Weyl's theorem (from the book by Stein-Shakarchi)
week 16 …Repetition
week 17 …Extra exercises on Monday the 25.IV.2016

Referansegruppe

  • Erik Jørgensen ….. erijorg@stud.ntnu.no
  • Are Austad ….. aresa@stud.ntnu.no
2016-06-13, Lars Peter Lindqvist