SELECTED TOPICS IN MATHEMATICAL ANALYSIS 4
(106937)
Spring Semester 2008/9.
New material will be inserted here.
So the older comments will be pushed downwards by new things.
I am considering two different possible topics for this course. In choosing between them, I will also take into consideration preferences expressed by students who plan to take the course. In my first lecture, on Monday 16/3/09 I will give a general survey of aspects of both of the topics. One possibility is that I will deal with 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. There are a number of older and newer books in our library about interpolation spaces. I can provide more details about them if necessary. I can also provide anyone interested with a copy of my own informal notes about interpolation spaces, based on a series of six lectures that I delivered some years ago in the Czech Republic. Much of the course this semester, if indeed it is about interpolation spaces, will be an expanded version of those notes. In the part of the course which deals with applications of interpolation spaces, it may turn out to be possible to choose applications having a connection with other particular topics of interest to students taking the course. (One possibility is to describe how interpolation methods enable one to obtain estimates for the number of bound states of Schrodinger operators.) Time permitting, I can also tell you about very recent research on interpolation of compact operators, including some progress on a 45 year old open problem. The second possibility is that I will give a course about 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 etc. If I choose this option, I will probably base most of the course on material in the book "A Mathematical Introduction to Wavelets" by P. Wojtaszczyk. PREREQUISITES. For either of the above two 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 be difficult but perhaps not completely impossible to manage without some knowledge of Lebesgue integration. 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 Mondays, 12:30-13:30.) I wish you a very interesting and successful semester in all your courses. KTBH ( = Kol Tuv, B-Hatzlakha ) Michael Cwikel
EARLIER COMMENTS FROM A RELATED COURSE GIVEN IN 2006/7
SELECTED TOPICS IN FUNCTIONAL ANALYSIS AND ITS APPLICATIONS
(106433)
Spring Semester 2006/7.
26.06.07 This is the version of
Taylor's formula and remainder which is needed for Proposition 3.1 on page 47.
Please note that the proof of the corollory of Proposition 3.1 which I stated
in today's lecture is an exercise which you are requested to do.
11.06.07 Here is one more exercise.
30.5.07. Here are some exercises which you should solve.
You may be interested to look at the slides which Jan-Olov Stromberg
prepared for a course that he gave in 2004. The approach is rather
different from ours in this course, and also apparently quite different
from his beautiful construction of a special wavelet which we have
been discussing recently. You can see these slides at
http://www.ipam.ucla.edu/publications/mgatut/mgatut_5135.pdf
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