## Course on Fourier analysis and Distribution theory: September - December 2002

### General information

Name of course:   Fourier analysis and Distribution theory

Course code:  OWIN W406

Time: Monday 13.15-16.00, room P.015B, Euclides building

First session: Monday 2 September 2002, 13.15 hour, room P.015B. This is only for discussing the way this course will be organized. Detailed treatment of material will start on 9 September.

Last session: Monday 6 Janaury 2003, 13.15 hour.

Lecturer:   T.H. Koornwinder

Prerequisites:
Required:   elementary real analysis, elementary Hilbert space theory
Helpful:   Lebesgue integration theory, elementary Banach space theory, elementary complex analysis

Contents:   Fourier series, Fourier integrals, wavelets, distributions, tempered distributions

Form of teaching:   Lecture/reading course (first two hours of each session), exercise class (last hour of each session), some additional exercises using a computer algebra package

Form of examination:   Take home exercises

Literature:
Fourier series and integrals:
Course notes on Fourier analysis by T.H. Koornwinder, with adaptations by J. Wiegerinck

Wavelets:   to be announced, see for a first orientation
T.H. Koornwinder (ed.), Wavelets: an elementary treatment of theory and applications, World Scientific, 1993
and/or
I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics 61, SIAM, 1992

Distributions:
G. Friedlander and M. Joshi, Introduction to the theory of distributions, Cambridge University Press, 1999, second printing (paperback)

Program
14 sessions, one per week:
• 5 sessions on Fourier analysis
• 3 sessions on wavelets
• 6 sessions on distributions
Students are expected to have read already the part of syllabus or book to be treated in a session, and to have made some of the exercises to be discussed during a session.

Take home exercises
For take home exercises, to be submitted, you may choose exercises which were on the list for discussion during class, but actually were not discussed. These are:
1.26, 1.27, 1.32, 2.7, 2.22, 2.25, 3.8, 3.16, 3.17, 4.16, 4.21, 5.2, 5.4, 5.12, 7.9, 7.12, 8.9, 9.4, 9.5.
Any other exercises from the syllabus, not yet discussed in class, may also be chosen.
There is an error in the formulation of Exercise 9.5 in the syllabus. Change the first line of Exercise 9.5 into:
"Apply (9.1) with f as in Exercise 7.9 in order to show that"

6 January
To be treated in Friedlander: Chapter 8

9 December
To be treated in Friedlander: Chapter 5

2 December
To be treated in Friedlander: Chapter 4

25 November
Treated in Friedlander: theory of Chapter 3, in particular section 3.1;
exercise 3.2; hints for exercises 3.1 and 3.3.

18 November
Treated in Friedlander: theory of sections 2.4 and 2.7; exercises 2.4, 2.9, 2.10.

11 November
Treated in Friedlander: section 1.3 and parts of Chapter 2.
We did also exercises 2.1, 2.2.

4 November
Treated: Parts of Chapter 1 of the book G. Friedlander and M. Joshi, Introduction to the theory of distributions, Cambridge University Press, 1999, second printing (paperback).
We did also exercises 1.2, 1.3, 1.4.

28 October (session was canceled)
To be treated: continuation of paper by Alm and Walker and accompanying software (see 14 October) with emphasis on wavelets.

14 October
Syll. Fourier Analysis, Chapter 10.2 (Fourier transform on finite cyclic groups),
section 10.13 (discetization of Fourier transform),
windowed Fourier transform (section 8 in the Chapter on The continuous wavelet transform in T.H. Koornwinder (ed.), Wavelets: an elementary treatment of theory and applications),
spectograms (sections 3.1 and 3.2 in J.F. Alm and J.S. Walker, Time-frequency analysis of musical instruments, SIAM Review 44 (2002), no.3, pp. 457-476.
Also computer demonstration:
approximation of periodic functions by finite Fourier sums (or Cesaro means of them) and of the Gibbs phenomenon,
part of the software (for Windows PC) accompanying the paper by Alm and Walker. We demonstrated Fourier transforms and spectograms of sound samples from flute, piano and guitar.

7 October
Syll. Fourier Analysis, chapters 8 (L2 theory) and 9 (Poisson summation formula).
Exercise: 8.12

30 September
Syll. Fourier Analysis, chapters 6 (Generalities about Fourier integrals) and 7 (Inversion formula).
Exercises: 6.11, 6.19, 6.20.

23 September
Syll. Fourier Analysis, chapters 3.5 (Uniform convergence of Fourier series), 3.6 (Gibbs phenomenon) and 4 (The Fejér kernel), 5.1 (The isoperimetric inequality).
Exercises: 3.27, 4.5, 4.22, 5.3.

16 September
Syll. Fourier Analysis, chapters 2.2 (Fubini), 2.3 (Convolution) and 3.1-3.4 (Dirichlet kernel).
Treated exercises: 2.18, 2.21, 3.7.

9 September
Syll. Fourier Analysis, chapters 1 (L2 theory) and 2.1 (growth rates of Fourier coefficients).
Treated exercises: 1.25, 2.24.
Recommended for take home exercises: a choice of 1.26, 1.27, 1.32, 2.7, 2.22.