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.
1. exercise set for 14.3.
2. exercise set for 21.3.
3. exercise set for 28.3.
4. exercise set for 4.4.
5. exercise set for 11.4.
6. exercise set for 25.4.
7. exercise set for 2.5.
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.
Course exam on 6 May. (See the timetable for details.)
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
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
Lectures and exercise classes
Exam and excercises, Course will be graded with grades 1-5
Exam, other methods will be described later