Jean Marie Joseph Fourier,       Laurent Schwartz

   Fourier picture
Schwartz picture

Fourier Analysis

This is the course page for the course "Fourier Analysis" that is part of the Mastermath program, Fall 2014.

The first meeting is on September 9, see the schedule on mastermath.nl.

General

The instructors are
  • Prof. dr. J.J.O.O. (Jan) Wiegerinck
    Email j.j.o.o.wiegerinck "at" uva dot nl

  • Dr. C.C. (Chris) Stolk
    Email C.C.Stolk "at" uva dot nl
  • The meetings are on Tuesdays from 14.00-16.45 at the Vrije Universiteit. Check the schedule on mastermath.nl.

    This page will be updated regularly during the course.

    Literature

    The compulsory text for this course is

  • Gerd Grubb, "Distributions and operators", Springer 2009.
    N.B. check your library for free electronic access (link here, only works from university networks). Via this link it is also possible to buy the paper version for a reduced price.
  • In addition there will be handouts: Handout 1 (weeks 1-3),   handout 2 (weeks 8-9, book of Strichartz, Chapter 5),   handout 3 (updated to contain the material of weeks 12 and 15),   Handout 4 (2nd handout Jan W.).
    Other literature:
  • J. J. Duistermaat and J. A. C. Kolk, "Distributions: Theory and applications", Birkhäuser, 2010.
  • J. Korevaar, "Fourier analysis and related topics", 2011, available online here.
  • R. S. Strichartz, "A guide to distribution theory and Fourier transforms", World Scientific, 2003.
  • Written exam

    A written exam will be scheduled. The exam will count for 75 % of the grade and the homework will count for 25 % of the grade. The written exam will be based on the exercises.

    Homework, exercises and course notes

    Approximately 5 homework sets will be scheduled as part of the grading. Other exercises will be recommended for the student to do by him/herself.

    Exercises Chapter 1: Preparation for exam: 1.7.2, 1.7.3, 1.7.4, 1.7.9
    Exercises Chapter 2: Preparation for exam: 2.7.1, 2.7.5, 2.7.10
    Homework 1: 1.7.5, 1.7.6, 2.7.6, due October 6 before class

    Exercises Grubb Chapter 3: Preparation for exam: 2, 3, 4, 5, 7, 8, 10, 17
    Homework 2: Grubb Chapter 3, exercises 6 and 11. Due 28 October before class.

    Exercises Grubb Chapter 5: 1, 3, 4, 7, 9, 11, 12. Note that in 3, 11 and 12 also n=1 is assumed.
    Homework 3: Grubb chapter 5, exercise 9, due by November 11.

    Exercises week 8 and 9: Some exercises come from chapter 5 of the book of Strichartz (this is handout 2).
    Exercises: Grubb 5.8, 3.12; Strichartz 5.1, 5.3, 5.10, 5.11, 5.12, 5.13, 5.17.
    Homework 4: Strichartz 5.3 and 5.11, due by November 25.

    Exercises week 10 and 11: Grubb 6.6, 6.7, 6.11, 6.18, 6.21, 6.22
    Homework 5: Grubb 6.22, due by December 9.

    Exercises week 12: Exercises 1.1, 1.2, 1.3 of the Exercises Radon transform part 1

    Exercises week 15: Exercises 2.1 of the Exercises Radon transform part 2

    Topics of the lectures

    The planned schedule is as follows

    Lecture 1-3. September 9, 16, 23. Jan Wiegerinck on Sep. 9 and 23, Chris Stolk on Sep. 16
    In these lectures we study Fourier series and distribution theory on the circle. The material is treated in lecture notes. We start with some basic results on convergence of the Fourier series, using the Dirichlet and Fejer kernels. Then we discuss lacunary series, that is, series where very few coefficients are nonzero. We then discuss distributions (generalized functions) on the circle. Distributions on the circle are connected with Fourier series, and are simpler than distributions on Rn, which are to be discussed next.

    Lecture 4-7. September 30, October 7,14,21. Jan Wiegerinck
    In the next two to three lectures the basics of distribution theory will be treated. This material is in Grubb, chapters 1-3 and appendix B. We study the space of test functions, the definition of distributions, examples, and the basic operations of differentiation, convolution and transformation under coordinate changes. It is shown that distributions are indeed a generalization of functions.
    Armed with these tools we then continue with the study of the continuous Fourier transform. The Schwarz class of test function is introduced followed by the temperate distributions. The Fourier transform is introduced, first as a transform of L^1 functions, and then on the general class of temperate distributions. The Parseval-Plancherel theorem is proved. This material is in Grub chapter 5.

    Lecture 8. October 28. Chris Stolk
    In the first part of this lecture we will continue with our discussion of the Fourier transform of distributions. In the second part we will start with the application to partial differential equations. We first discuss rotation and dilation of functions and how to obtain the Fourier transform of a rotated or dilated function in terms of that of the original function. We then discuss functions with rotational symmetry and homogeneous functions. The theory is applied to the function |x|^-r. We then continue with the applications to PDE. This is done using chapter 5 of the book of Strichartz. We discuss the heat equation on R^n and the Laplace equation.
    Literature: Grubb section 5.4, Strichartz Chapter 5 (this is handout 2). An alternative discussion of the function |x|^-r is in Strichartz Chapter 4, section 4.2, example 5.

    Lecture 9. November 4. Chris Stolk
    In the first part of the lecture we continued with the solution formulas for the standard PDE (the Laplace equation, the heat equation and the wave equation), using the material of handout 2 (chapter 5 of Strichartz). Next week we will start with chapter 6 of Grubb where Sobolev spaces are introduced and are used in the study of the solutions of certain partial differential equations. As a preparation section 4.1 of Grubb was presented. This explains how to define partial differential operators as operators on function spaces.

    Lecture 10-11. November 11, 18. Chris Stolk
    In these lectures we treat applications to differential operators, following Grubb chapter 6. We start by defining a class of operators denoted by Op(p(xi)) that acts by multiplication in the Fourier domain, using the class of slowly increasing functions O_M defined on page 97. The following topics are discussed: Definition of the Sobolev space H^s; the Sobolev embedding; duality between H^s and H^{-s}; the structure theorem; the definition of elliptic partial differential operators and the regularity results for elliptic operators given in theorem 6.22 and corollary 6.23. This material is in Grubb chapter 6, until page 139.

    Lectures 12 and 15. November 25, Dec. 16. Chris Stolk
    In these lectures we discussed the Radon transform. The Radon transform maps a function to its integral over hyperplanes. It has an important application in computerized tomography (medical imaging). We treat among others the Fourier slice theorem, and the inversion formula. The support theorem and a result on the range are mentioned but not proved. In the second lecture we discuss the ill-posedness of the inverse problem for the Radon transform. Related to this is the construction of the SVD of the Radon transform in 2 dimensions. A handout is put on the web, exercises will appear soon.

    Lecture 13-14.
    TBA


    Last modified: Mon Sep 15 08:11:11 CEST 2014