# MAT6, Spring 2013

## Integration and Fourier Theory

There will be an individual, pass/fail oral examination with internal censorship. The procedure is described as follows.

1. Each student draws one of the five exam questions (see the list below), then he/she has a 30 minute preparation time. During this half an hour one may use all possible materials. The presentation should be prepared either in Danish, English or French.

2. The examination starts right after the preparation time. It is the student who has the initiative; one should be able to present at least one of the central issues related to the drawn question. The actual presentation should not take more than 20 minutes.

3. When the presentation is over, you will get a suplimentary question not related to the subject just presented. These secondary questions are not known in advance, but are only dealing with definitions and fundamental concepts of the theory; no proofs are expected to be given at this stage. The second question should be answered in not more than 5-7 minutes.

4. The student leaves the examination room, and the examinator and censor make their decision. When this is over, the student is informed about the outcome, with an explanation of the decision.

During the presentation it is allowed to have an A4 paper containing a plan of your talk. You are not allowed to use your notes.

### Exam questions

The list is subject to changes.
1. Abstract Integration (page 5-31; the most important results are Fatou's lemma, the monotone and dominated convergence theorems).
2. Lp Spaces (page 61-71; most important results are Jensen's inequality, Hoelder and Minkowski inequalities, completeness of Lp spaces).
3. Elementary Hilbert Space Theory (page 76-92; most important results are the decomposition theorem 4.11, the completeness of the trigonometric system.
4. Integration on Product Spaces (page 160-170; most important results are the construction of the product measure, and The Fubini theorem.
5. The Fourier transform (page 178-190; most important results are the inversion theorem, and the Plancherel theorem).

Opdateret d.21 januar 2013 af HC.