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11.2.2019 at 09:00 - 2.5.2019 at 23:59

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Tuomas Hytönen's picture

Tuomas Hytönen

Published, 8.4.2019 at 15:10

No lecture on Tue 9 April!

Timetable

Here is the course’s teaching schedule. Check the description for possible other schedules.

DateTimeLocation
Mon 11.3.2019
10:15 - 12:00
Tue 12.3.2019
10:15 - 12:00
Thu 14.3.2019
14:15 - 16:00
Mon 18.3.2019
10:15 - 12:00
Tue 19.3.2019
10:15 - 12:00
Thu 21.3.2019
14:15 - 16:00
Mon 25.3.2019
10:15 - 12:00
Tue 26.3.2019
10:15 - 12:00
Thu 28.3.2019
14:15 - 16:00
Mon 1.4.2019
10:15 - 12:00
Tue 2.4.2019
10:15 - 12:00
Thu 4.4.2019
14:15 - 16:00
Mon 8.4.2019
10:15 - 12:00
Thu 11.4.2019
14:15 - 16:00
Mon 15.4.2019
10:15 - 12:00
Tue 16.4.2019
10:15 - 12:00
Thu 25.4.2019
14:15 - 16:00
Mon 29.4.2019
10:15 - 12:00
Tue 30.4.2019
10:15 - 12:00
Thu 2.5.2019
14:15 - 16:00
Mon 6.5.2019
09:15 - 12:00

Material

Chapters IX-XIII of the Finnish lecture notes "Fourier analyysi", by Kari Astala and Eero Saksman, cover roughly, but not exactly, the same material as presented by the lecturer on the blackboard. If you are not attending the lectures, it is advisable that you ask a fellow student for a copy of their notes.

Lecture material

Tasks

1. exercise set for 14.3.

Tuomas Hytönen

2. exercise set for 21.3.

Tuomas Hytönen

3. exercise set for 28.3.

Tuomas Hytönen

4. exercise set for 4.4.

Tuomas Hytönen

5. exercise set for 11.4.

Tuomas Hytönen

6. exercise set for 25.4.

Tuomas Hytönen

7. exercise set for 2.5.

Tuomas Hytönen

Course diary

Mon 11.3. - Overview, basics of the Fourier transform in L^1.
Tue 12.3. - The Fourier transform of the Gaussian function, Fourier inversion formula in L^1.
Mon 18.3. - Plancherel's theorem, the Fourier(-Plancherel) transform in L^2.
Tue 19.3. - Riesz-Thorin interpolation theorem. Hausdorff-Young inequality on Fourier transform in L^p.
Mon 25.3. - Complex analysis tools (maximum principle, three lines lemma) behind Riesz-Thorin interpolation theorem.
Tue 26.3. - Intro to Schwartz test functions. Their invariance under the Fourier transform.
Mon 1.4. - Schwartz test functions as a normed space. Intro to tempered distributions and their Fourier transforms.
Tue 2.4. - Examples of tempered distributions (Dirac delta, p.v. 1/x) and operating with them.
Mon 8.4. - Computation of the Fourier transform of p.v. 1/x. Convolution with p.v. 1/x and its properties on L^2.
(Tue 9.4. - Lecture cancelled.)
Mon 15.4. - The Poisson summation formula with variations. Solving the heat equation with Fourier methods.
Tue 16.4. - The heat equation with Fourier methods continued. An application of Poisson summation in number theory.
Mon 29.4. - The support of a distribution. The structure of distributions with one-point support.

Conduct of the course

Course exam on 6 May. (See the timetable for details.)

Description

Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Fourier Analysis I, Real Analysis I

Master studies

Continuous Fourier transform and tempered distributions

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

Continuous Fourier transform on L^p-spaces and on tempered distributions

Lecture notes

Lectures and exercise classes

Exam and excercises, Course will be graded with grades 1-5

Exam, other methods will be described later