SELECTED TOPICS IN ANALYSIS 2
(106929)
Spring Semester 2009/10.
I am considering three different possible topics for this course.
1. WAVELETS. (This will apparently be the chosen topic.)
2. Interpolation spaces
3. The function space BMO and its applications.
More details about these topics appear below in this document.
The main criterion for choosing between them will be the
preferences expressed by students who take the course.
There have been three hours of the course so far, and these were mainly
an introduction to wavelets.
If any other students still wish to join the course I can easily give them
repeat lectures of the material that they missed so far and/or refer them
to where they can read that material. You can contact me by email to fix a
time for this.
PREREQUISITES.
For each of the above three options, you will need some basic knowledge
of Hilbert spaces and Banach spaces. You also need to know about Fourier
series and Fourier transforms, in particular if the course will be about
wavelets.
I.e. you should have taken the course 104276 "Mavo l'Analiza
Funktsionalit" or something similar.
For some parts of the course it might sometimes be difficult but not
completely impossible to manage without some knowledge of Lebesgue
integration.
WAVELETS.
For a brief description of wavelets you can see, for example, the
document
http://www.math.technion.ac.il/~mcwikel/wavelets/WhatAreWavelets.txt
which I wrote when I taught a course about them some years ago. There are
of course many books about wavelets, (and they are mentioned in nearly
three million websites). Wavelets are a kind of modern alternative to
Fourier series and Fourier transforms, with many applications in both
theoretical and applied mathematics and engineering (image compression,
signal processing) etc.
If we choose this option for all or part of the course, as we apparently will,
I will probably base most of the treatment of it on material in the book
"A Mathematical Introduction to Wavelets" by P. Wojtaszczyk.
But I may also be able to consider other sources if there is special interest in them.
INTERPOLATION SPACES
This is a topic in functional analysis which is central in my own
research. Apart from its own intrinsic beauty, it has applications to
quite a number of branches of analysis, including approximation theory,
geometry of Banach spaces, harmonic analysis and mathematical physics.
Depending on the background of the students in the course, I may also
provide some additional background in harmonic analysis so that we can
better appreciate some of the ways in which interpolation spaces are
used. It may perhaps be possible to discuss applications to fields of
particular interest to participating students.
There are a number of older and newer books in our library about
interpolation spaces and I can also provide copies of lecture notes which
I wrote for earlier courses here and abroad. See also
http://www.math.technion.ac.il/~mcwikel/fa/index.html
BMO
The space of functions of Bounded Mean Oscillation was introduced and
first studied in the 1960s by the famous mathematicians Fritz John and
Louis Nirenberg, with motivations from the field of partial differential
equations. It has since been shown to be connected to many other topics,
including the two mentioned above, and been associated with work of
other very famous mathematicians, such as Fields Medallist Charles
Fefferman.
I can discuss its theory and applications, possibly including some recent
research.
HOW TO GET A GRADE.
The examination at the end of the course will be based, in large part,
but not completely, on problems which will be given to you to solve
during the semester. If students wish to prepare and give a short series
of lectures on one of the topics of the course, it may perhaps be
possible to offer them this as an alternative to doing the examination.
------------------
You can reach me at:
<mcwikel@math.technion.ac.il>
(Room 730, Telephone (829)4179. My office hour for this semester is on
Wednesdays, 10:30-11:30, or by appointment.)
I wish you a very interesting and successful semester in all your courses.
KTBH ( = Kol Tuv, B'Hatzlakha )
Michael Cwikel
Lecture times (until further notice):
Mondays 13:30-15:30, Wednesdays 8:30 - 9:30. Both in Amado 915.
[For those of you who do not like such an early time on Wednesdays, we will try to find
an alternative solution.]