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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.
The compulsory text for this course is
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.).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.
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
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