Teacher: Kari Astala

Scope: 10 cr

Type: Advanced studies

Teaching: Lectures on weeks 36-42 and 44-50, in room C123, on Tuesdays 10-12 and on Wednesdays 12-14.

First lecture on Tuesday, September 1.

Exercise classes are part of the course; these are on Mondays 12-14, in room C122. Exercise assignments are delivered on Wednesdays of the previous week.

First exercise class on Monday, September 7.

Topics: The course is an introduction to modern Fourier Analysis. Fourier analysis is a basic tool both in pure and applied mathematics, from PDE's and harmonic analysis to stochastics, signal processing, physics,...

The first part of the course, roughly weeks 36-42, covers discrete Fourier analysis (Fourier series and also Fast Fourier transform), while the second period, weeks 44-50, studies the continuous Fourier analysis, i.e. Fourier transforms of functions and (Schwartz) distributions in Rⁿ. Throughout the course, and time allowing, many of the applications of Fourier analysis are discussed.

Prerequisites: Basic knowledge on measure theory and Lebesgue integration (e.g. the course "Mitta ja integraali") is required. Knowledge of basics of L^p-spaces will be advantageous (e.g. as discussed in the course Reaalianalyysi I; see below). In addition some facts from functional analysis will be used - if necessary, these can be briefly discussed during the course.

Contents:

• Representing a periodic function as a Fourier series
• Basic properties of Fourier series
• Dirichlet kernel; Fejer kernel and good convolution kernels
• Pointwise convergence, Convergence in L^p, C[0,1], etc. -norm
• Continuous functions with diverging Fourier series
• Discrete Fourier transform, Fast Fourier transform
• Continuous Fourier transform in L¹(Rⁿ)
• Fourier transform in L²(Rⁿ); Plancherel's theorem
• Interpolation and Fourier transform in L^p(Rⁿ)
• Tempered distributions
• Applications of Fourier Analysis

Course material:

The course will basically follow the lecture notes: 'Kari Astala, Fourier Analyysi' (course in 2012; notes are in Finnish)

Minor modifications will be made, and new Lecture notes (in Finnish) will be provided on this page as the course proceeds.

Here are the complete Lecture notes with Appendix.

NOTE:

• Section VII.1. was left for a self study.
• Proof of Riesz-Thorin Theorem 11.2 (pages 100-106) was left for a self study.
  (Riesz-Thorin theorem is proved also e.g. in "Rudin: Real and Complex Analysis")

Except for the distribution theory, the material covered in this course can be found in the books:

• Stein-Shakarchi: Fourier Analysis, An Introduction.
• Rudin: Real and Complex Analysis.

Other books related to the course:

• Grafakos: Classical and Modern Fourier Analysis.
• Rudin: Functional Analysis.
• Duoandikoetxea: Fourier Analysis.

Additional information on real analysis, in particular on basics of L^p-spaces, can be found from lecture notes by Ilkka Holopainen attached here: Reaalianalyysi.I (2011).pdf

The information on L^p-spaces needed can also be found e.g. in Rudin's book 'Real and Complex analysis' mentioned above; a brief sketch of L^p-properties needed is also given in the Appendix of the course notes above.