Welcome to the course!

This course is taught by Kurt Johansson, kurtj@kth.se.

Course plan

A detailed plan of the course can be found below.

Course objectives

The aim of the course is to provide a rigorous introduction to Fourier analysis and its applications.

Intended learning outcomes

After passing the course, the students should be able to

formulate central definitions and theorems of Fourier analysis;
apply and generalize theorems and methods within Fourier analysis;
describe, analyze and formulate basic proofs within Fourier analysis.

Course Litterature

Primary: Dym, McKean “Fourier Series and Integrals”. Academic Press, 1985.
Secondary: Stein, Shakarchi "Princeton Lectures in Analysis I Fourier Analysis''. Princeton University Press, 2003.

A more elementary book that is not at the level of the course but may have some interest is "Fourier Analysis and Its Applications" by Anders Vretblad. A more advanced text, more advanced than the course, is "An Introduction to Harmonic Analysis" by Yitzhak Katznelson.

Examiner

Kurt Johansson

Administrative matter

Answers to administrative questions, concerning for example registration for the course, registration to the exams, re-exams, etc. can be found on the following web page: studentexpeditionen

For questions about this and other administrative inquires please contact studentexpeditionen, easiest and fastest, by email: studentoffice@math.kth.se .

Examination

There will be a written exam on Wednesday 7 jan 08:00-12:00. Grades C and lower are based just on the written exam. For higher grades (A and B) you also need to take an oral exam. You are allowed to take the oral exam if you received the grade C on the written exam. The grades A and B will be decided based on both the written and the oral exam. All exams are closed book with no aids allowed.

Support for students with disabilities

Students with disabilities may have the right to certain compensatory support for example during examination.

KTH has coordinators for students with disabilities, see Funka, who deals with issues relating to functional disabilities. You should turn to Funka at funka@kth.se for information about support.

Plan of the course

Note that the plan below is preliminary and that there may be changes. It will be updated during the course. The literature references are to the books [SS] Stein, Shakarchi "Princeton Lectures in Analysis I Fourier Analysis'' and [DM] Dym, McKean “Fourier Series and Integrals”. The lectures will not follow the presentation of any particular book. Note that the lecture notes may contain mistakes.

Lecture 1 (2/9) Introduction. Basic theory of Fourier series. Convolution. SS2.1-2.3, DM1.5. Lecture 1

Lecture 2 (9/9) Summation kernels. Riemann-Lebesgue lemma. DM1.5, SS2.4. Lecture 2

Lecture 3 (12/9) Pointwise convergence of Cesàro means. Size of Fourier coefficients. Pointwise convergence of Fourier series. DM1.4-1.5, SS2.2, SS2.5. Lecture 3

Lecture 4 (16/9) Orthonormal systems in Hilbert space. Fourier series in L². DM1.2-1.4, SS3.1-3.2.

Lecture 5 (19/9) More on convergence and divergence of Fourier series. Continuous but nowhere differentiable function. SS4.3

Lecture 6 (23/9) Jacobi's identity for theta function. The heat equation on the unit circle. Weyl equidistribution theorem, DM1.7.5-6, DM 1.8.3, SS4.2

Lecture 7 (26/9) Fourier transform on L¹(R). DM2.1, DM2.6

Lecture 8 (30/9) Fourier transform on Schwartz space. DM2.2, SS5.1.

Lecture 9 (3/10) Fourier transform on L²(R). DM2.3

Lecture 10 (7/10) Applications of the Fourier transform. Poisson summation formula. Central limit theorem. Heisenberg uncertainty relation. DM2.7.5, DM2.7.7, DM2.7.8, SS5.3, SS5.4.

Lecture 11 (28/10) Fourier transform in several variables. Applications to PDE. DM2.10, SS5.1-5.3.

Lecture 12 (31/10) PDE continued. Fourier analysis and complex analysis. DM3.2.

Lecture 13 (7/11) Fourier analysis and complex analysis. DM3.3-3.4, DM3.8.

Lecture 14 (11/11) Fourier series and commutative groups. DM4.1-4.2, DM4.4.

Lecture 15 (18/11) Discrete Fourier analysis. DM4.5, SS7.1-7.2.

Lecture 16 (25/11) Applications.

Lecture 17 (28/11) Applications.

Lecture 18 (2/12) Applications.

Lecture 19 (5/12) Recapitulation. Outlook.

Exercise sheets

Below you find sets of problems suitable for the course. The solutions to some of the you can find in books or online but it is a good idea too first try on your own, and write your own solutions. This will help you prepare for the exam.

Exercise sheet 1