Fourier
Jean Marie Joseph Fourier, Laurent Schwartz
|
|
Instructors
-
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
- Dr. S.K. (Samir) Bhowmik
Email S.K.Bhowmik "at" uva dot nl
Intended for: Master Mathematics students
Prerequisits: Bachelor mathematics, preferably with linear analysis or Fourier analysis.
Course Evaluation
Please be so kind to fill out the course evaluation form at the
mastermath site. Then mail it to me or hand it over at the day of presentations.
Presentation or Oral exam
Presentations are in Science Park!!
On Friday june 18 we organize a day for final oral presentations. Topic to be decided in consultation with one of the instructors. In case there are many presentations, or there are problems with the date, we take Monday June 14 as alternative.
You can also do an oral exam about the material in the course.
It may be that you need a grade (for e.g. Erasmus) earlier.
You can do the oral in such a case a bit before thee final class.
Contact us!
Schedule presentations
| Monday June 14
|
|---|
| Science park Room A1.14
|
|---|
| Time | Student | Title
|
|---|
| 13.00 | Sniekers | TBA
|
|---|
| 14.00 | Oomens | Laplace Transform
|
|---|
| 15.00 | Gang | Distribution on a half plane and on a manifold with
boundary
|
|---|
| 16.00 | Streekstra | Kakeya conjecture and restriction problem
|
|---|
|
|---|
| Friday June 18
|
|---|
| Science park Room A1.08
|
|---|
| Time | Student | Title
|
|---|
| 9.00 | | TBA
|
|---|
| 10.00 | Halvorsen | TBA
|
|---|
| 11.00 | Sierko | TBA
|
|---|
| 12.00 | Jansons | Mellin Transforms
|
|---|
| 14.00 | Comelli | Airy functions
|
|---|
| 15.00 | Ciapponi | TBA
|
|---|
For solidarity with your fellow students and because the talks should be interesting, we urge you all to attend your fellowstudents talks!
Overview
Tuesdays 10-13.00, starting February 9.
Place Building "Euclides" Plantage Muidergracht 24.
Room: P 0:19
See also DATANOSE
WEEKLY OVERVIEW + TAKE HOME EXAM PROBLEMS AT THE BOTTOM OF THE PAGE
In Fourier analysis one studies functions on R, Rn, or more general spaces by writing them as linear superpositions of elementary functions. This is very clear in the situation of periodic functions.
These may often be expressed as a linear superpositions of exponentials with the same period. It is interesting to study the connection between functions and their Fourier transform for its own sake, but it has turned out that Fourier analysis is a powerful tool to handle problems in differential equations, signal analysis etcetera. When one studies
the classes of functions that permit Fourier analysis, it soon becomes clear that one should study a larger class of objects, the so called distributions or generalized functions. In this course all these aspects will be dealt with to some extent.
Classical L2-theory of Fourier series. Convergence problems, periodic distributions and their Fourier theory, Distributions in Rn, Fourier transform, L2-theory, Schwartz class and tempered distributions, Fourier transforms of distributions, Sobolev space interpolation, microlocal theory, applications to differential operators, wavelets, lacunary series
Take home exercises, in class presentation, and or oral exam.
Tentatively lectures from 10-12, exercises 12-13. Instruction First weeks, Usually Wiegerinck, Last weeks Wiegerinck and Stolk will vary regularly.
Important
One hour of exercise class is not enough to do all the exercises. prepare at home, so that we can discuss what is difficult or unclear!
- Week 6. Material discussed Chapter 1 and 2 of the book + discussion of seminorms to put topology on the space of distributions.
Exercises to do: Chap 1. no 1,2,3,4,5,8,9,10,11,12; Chap 2. 1,2,3,4, 17,20.
To hand in on March 2 2: Chap 1,no 6,7; Chap 2 no5, 18,19,21. (see below; modification of handed in work is allowed)
- Week 7. Material discussed: Chapter 3 of the book.
Exercises to do: Chap 3. no 1,2,4,6,7, 11.
To hand in on March 2: Chap 3. no 14,15,16.
- Week 8. Material discussed: Chapter 4 of the book, except convolutions and example 3 and 5. All examples are valuable for the exercises. Let us know in time if you have trouble understanding them .
Exercises to do: Chap 4. no 1,2,3,4,5,6,11,12,13.
To hand in on March 9: Chap 4. no 9,10,14
- Week 8. Material discussed: Chapter 4.3 of the book (convolutions), Chapter 5.
Exercises to do: It looks like we need a bit more time for the exercises so far.
To hand in on March 16: No new homework, get everything (i.e. Chapter 1-4)
finished.
- Week 9. We finished chapter 5 (wave equation in particular). Next we will turned to the notes, mostly Chapter 3 and discussed the notion of support and local equality of distributions. Distributions with compact support were identified as functionals on the space of smooth functions. This covers part of the material in chapter 6 of the book.
Exercises to do: Chapter 5. no 1,2,4,5,8,9,10,11,14,15.
To hand in on March 23: Chap 5: no 3, 12, 13.
- Week 10. More on the structure of distributions, Singular support. (Chapter 6 from the book, Chapter 3 and 4 from the notes).
- Week 11 Chapter 7.3 Paley Wiener Theorem and applications Exercises to hand in from the notes 3.6.14. Notice that in (i) one should read locally in L^1. Try to hand in April 8.
- Week 12 Distributions with compact support in Section 3 from the notes. Exercises to do chapter 6, problems Chap 6: 7, 27, 28, 29. To hand in April 6,
do not bother with 4 and 15.
- week 13 Chapter 7 Section 1,3,4. Exercises Chap 7, 2 to 23. Hand in 6, 8, 12, 17, 18.
- week 14 Chapter 7 section 5, Heisenberg Uncertainty and 6, Hermite functions. Some details about hermite functions (completeness, structure of Tempered distributions were scetched. Exercises 24 to 27. Hand in 28 -32.
- Hand in all "to do" exercises from chapter 1-7 on April 20 at the latest, or talk to us about a later date!!
- Week 16. Section 8.1 on Sobolev inequalities except the part on Holder continuity on page 170. Section 8.2 on Sobolev spaces. Exercises: 1,2,5,7. Hand in: 6,9,10.
- Week 17. Section 8.3 on elliptic operators, except the proof of the existence of a fundamental solution that starts halfway on page 183. Exercises: 11,17. Hand in: 12,15
- Week 18. Section 8.4 from Strichartz on pseudodifferential operators and the construction of a parametrix.
- Week 19. We will treat the properties of pseudodifferential operators in detail using a different text. Chapter 1, sections 1 - 3, 8.1 and 8.2 are the topic of this week. .
Hand in: Additional exercises for week 18-19 nos. 1 and 2
-
Week 20: In this lecture we finished the material of week 19. Hand in: Additional exercise for week 20-21 no 1.
-
Week 21: Applications: Radon transform, imaging with waves. No exercises.
- Exercise Chapter 2, no 19. Do the case n=2 as follows. Apply the case n=1 to obtain a decomposition of f(x,y) where y is considered parameter. Then the constant also depends on the parameter y. Next apply the case n=1 to f as it occurs in the "constant" on the second variable.
This you can do for bigger n, but an induction is not apparent (at least, not to me.)
Hörmander solves this by showing that a distribution L of which the n-th derivative is 0 can be written as a distribution of the first (n-1) variables acting on f and integratingthe result against a constant in the last variable.
- Chap 4, 9,10. The book is a bit easy going on statements about distributions that are
(should be) valid for distributions. About Hilbert Transform, check the notes, and perhaps 10 should be done before 9...
- Chap 5 no 12, 13. The point is to show that the suggested solution satisfy the right boundere values at x=0 and x=1.
Back to top
to the home page of Faculteit WINS
to the home page of Vakgroep
Wiskunde
to
the home page of Jan Wiegerinck