Master's Thesis - 4

Some time ago, I have finished Werner’s Functional Analysis, according to schedule. I’ve gone through the whole book, but I’m still insecure with spectral theory and weak*-topologies. I thought about repeating some chapters, but I want to try a different approach, and as Conway’s Course in Functional Analysis is cited so often in the literature I currently read, I finally bought it. I’m still going through my notes of Prof. Cigler’s lecture. Additionally, I want to go through the “real part” of Rudin’s Real and Complex Analysis, accompanied by selected topics from my more general books about analysis and linear algebra.

I also finished Blatter’s Wavelet Primer as planned, but skipped the last chapter. So now the road is open for Feichtinger/Strohmer’s Gabor Analysis and Algorithms. I’m already progressing in Gröchenig’s Foundations (about 1/3) and Christensen’s Frames (about 1/4). My plan is to have those books (almost, modulo some skipped chapters) finished by the end of August. From September on, I only want to focus on additional literature like papers and specialized books, which my thesis will deal with. Among these are topics about Banach Gelfand triples and Wiener amalgam spaces.

I contacted HGFei again and made clear that my original aims have changed: I don’t just want to get my studies finished anymore, I want to deliver a qualitative work. After my studies, I want to work in the area of signal processing; my current job is just an existential assurance. I don’t just want to have an engineering-level approach to my master thesis, it would be a pity in my eyes. He answered that we should talk in July the latest about a long-term proceeding. He didn’t mention a possible topic, so I’ll proceed doing general reading on the subject.

I thought about Karin’s master thesis, which wasn’t of such high quality to HGFei as she had expected. She did her master exam and now simply has finished and doesn’t want to get into contact with the surroundings of the university again. I simply find that a pity. She works in insurance mathematics.

This touches my project only at the margin, but my employer agreed to reduce my working hours to 26/week! It was my psychologically motivated target to be able to leave the office at 15:00h if I start working at 8:00h. As I sometimes might begin later and might have to stay a little longer if needed, the leaving time will often get postponed anyway.

The change of my employment contract leads to a “restart at zero”, i.e., all unused holidays and accumulated overtime will get paid off, which isn’t such a benefit for me. However, I have reached a milestone. If I really want to stay in my time plan, I’ll have to do 5 pages/day in Blatter as well as in Gröchenig, and this is only possible if I have enough time available. And there is much more literature.

Until the mid of May, i.e. in one month, I want to have finished reading the Wavelet Primer and the whole functional analysis, including the script of Prof. Cigler. Until the end of May, I want to fix the topic of my thesis. Until the end of June, I want to have finished reading the Foundations of Time-Frequency Analysis; the two weeks of holiday at the beginning of June are already taken into account. After my holidays, i.e. by the mid of June, I definitely want to start on my special topic.

One of my next steps is to check out if sampling theory could be the topic of choice. I stepped over the book Sampling, Wavelets and Tomography, which appeared quite interesting to me. The NuHAG library owns it, but it is currently borrowed by Tobias W. Karin’s CD includes his master thesis, among other papers, and I think I will ask him how I should proceed to get an overview and if he had an idea for a concrete topic. I will contact him after having read the brief summary of Shannon’s sampling theorem in Blatter’s Wavelet Primer, which I will reach within the next days. HGFei still hasn’t answered yet, and I don’t know what I should think about that.

Yesterday I was at Karin’s flat to talk about mathematics, master thesis, master exam, and much more. She wants to take the master exam at the beginning of May. We had a little overview about the most important theorems in functional analysis. She gave me a CD containing all her TeX files, some papers and of course her own master thesis, which she had finished recently.

I’ve finally done chapter VII of Werner’s book. At the end of that chapter, I didn’t follow the content in detail anymore, as it appeared more and more exotic to me and not too much related to TF-analysis. The two remaining chapters (Locally convex spaces incl. distribution theory, Banach algebras) will be done within one or two months, but without force; it will be enough to go through 2 pages per day.

The main focus now should be time-frequency analysis, repeating basics about Hilbert spaces and Fourier- as well as wavelet-analysis. I’ve already started to read Gröchenig’s Foundations some time ago. Also, I finally bought Blatter’s Wavelet Primer in its German original.

I contaced HGFei and told him about my current status. I asked him whether I should try to fix a topic or if I should continue reading. Also, I told him about my idea of taking measure and integration theory for my master exam. He hasn’t answered yet.

Just a note: We know that the inclusion yields for . A similar inclusion statement is valid for -spaces iff we look at . Here, for , the inclusion is the other way round: . This is not valid for , as is not finite. In Cigler’s script, the fact had just been written down quickly, and it wasn’t mentioned that this was only valid for bounded intervals. I already wondered why it was always said that the Fourier transform could be expanded to “” and not simply “”.

I will start with spectral decomposition of self-adjoint operators today. It will take me about one week to go through that last chapter of my reading plan of this book, so I won’t be so far behind my original time plan.

I think I’ll have to buy Blatter’s Wavelet primer, as I start wanting to write notices into the borrowed copy. It provides an easily readable approach and isn’t that expensive.