Master's Thesis - 3 :: Stephan Paukner

Thursday, March 30. 2006

Separability of Banach spaces

I mulled repeatedly for some weeks over the fact, why a Banach space is separable if any element can be approximated by a sequence of finite dimensional elements. It has been formulated as $S_nx\to x$ , where $S_n\in F(X)$ is a continuous linear operator with finite rank, and this conclusion was never obvious to me. For another time, there was a fact which should be clear and not difficult to show.

Finally, I remembered: Every (finite dimensional) vector space has a (finite) basis. I.e., if $\dim(S_my)=k(m)$ , then there is a basis $\{b_1,\ldots,b_{k(m)}\}$ which can be extended as soon as $\dim(S_{m+\ell}\,y)>k(m)$ for any $\ell\in\mathbb{N}$ . Any basis is countable, and a theorem says that a space $X$ is separable iff there is a countable set $A$ with $X=\overline{\operatorname{lin}\,A}$ , where $\overline{\operatorname{lin}\,A}$ is the closure of the set of all (finite) linear combinations of elements in $A$ .

Posted by Stephan Paukner in Master's Thesis at 08:22 | Comments (0) | Trackbacks (0)

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Wednesday, March 22. 2006

Progress in functional analysis, II

I finished on Hilbert space theory and will continue with chapter VI today, which covers spectral theory of compact operators. I don’t think I’ll even do until the end of chapter VI by the end of March, although I originally also wanted to have read through chapter VII (Spectral decomposition of self-adjoint operators). But, chapter VIII (Locally convex spaces) still covers some important topics.

I noticed that all those foundations in functional analysis are really very important for applications in Fourier analysis and Gabor analysis. Besides the special books, I’ll work through Blatter’s “Wavelets”, because it provides a really good approach to that topic. Also, Heil’s script about bases in Hilbert spaces and a syllabus about Fourier analysis will give a good occasion for repetition.

As another milestone, my employer agreed to reduce my working times. I wanted to have reached this with April, but the agreement won’t be valid until May. So, I’ll make my Fridays to free-days for another month. Beginning with May, I’ll only have to work for 28 hours/week, which was my threshold of pain. I hope I still can reduce this amound a bit. If I start working at 8:00h, I want to leave at about 15:00h to be able to sit at my desk at 16:00h. I’ll have to comply with my working hours strictly.

Posted by Stephan Paukner in Master's Thesis at 09:43 | Comments (0) | Trackbacks (0)

Friday, March 17. 2006

Properties of the Fourier transform, II

I think I found out where the mentioned property comes from: The solution lies in the application of a substitution in the mentioned integral. It is well known that a translation of any (Lebesgue integrable) function leads to a modulation of its Fourier transform, i.e., $\mathscr{F}(\mathrm{T}_a f) = \mathrm{M}_{-a}\mathscr{F}f$ , where $\mathrm{T}_a f(x) := f(x-a)$ and $\mathrm{M}_b f(x) := e^{2\pi\mathrm{i}bx}f(x)$ for $x\in\mathbb{R}$ . But, it is even a common property for any $\mathrm{L}^1$ -function $f$ that $\mathscr{F}f_{1/a}(\xi) = |a|\mathscr{F}f(a\xi)$ , where we again use the symbolism $f_a(x):=f(ax)$ . This becomes clear by using the substitution $x:=ay,\quad dx=|a|dy$ :

$\begin{eqnarray*}\displaystyle\mathscr{F}f_{1/a}(\xi) &=& \int_{\mathbb{R}}f\left(\frac{x}{a}\right)e^{-2\pi\mathrm{i}x\xi}\,dx\\ &=& |a|\int_{\mathbb{R}}f(y)e^{-2\pi\mathrm{i}ay\xi}\,dy\\ &=& |a|\mathscr{F}f(a\xi)\end{eqnarray*}$

So, it’s clear () that this means $\mathscr{F}f_a(\xi) = \frac{1}{|a|}\mathscr{F}f\left(\frac{\xi}{a}\right)\quad\forall f\in\mathrm{L}^1$ , and not only for the Gaussian function $\gamma(x) = e^{-x^2/2}$ . And, for $x\in\mathbb{R}^n$ , the substitution $x:=ay$ leads to $dx = |a|^n dy$ , because if $x_i := ay_i$ , then $dx_i = |a|dy_i$ and therefore $\textstyle\prod_{i=1}^ndx_i = |a|^n\prod_{i=1}^n dy_i$ . I’ll have to recall what the “value of the functional determinant” is.

Regarding the proof that the Gaussian function is an eigenfunction of the Fourier transform, HGFei once handed us one out which seems to be more elegant than the proofs which use the well definedness of solutions for initial value problems of linear common differential equations.

Posted by Stephan Paukner in Master's Thesis at 10:18 | Comments (0) | Trackbacks (0)

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Thursday, March 16. 2006

Properties of the Fourier transform

What I hate is when books say that something is clear, what definitely is not clear! So, why is it clear that $(\mathscr{F}\gamma_a)(\xi) = \frac{1}{a^n}(\mathscr{F}\gamma)\left(\frac{\xi}{a}\right)$ for $\gamma(x)=e^{-x^2/2}$ and $\gamma_a(x)=\gamma(ax)$ ? It seems to have something to do with the differentiation rules of the Fourier transform, and I couldn’t verify the mentioned equation yet. As far as I can see, it really is not that trivial that a simple clear-ness is justified.

This claim came together with $(\mathscr{F}\gamma)(\xi) = e^{-\xi^2/2}$ , the proof of which also doesn’t seem to be too trivial. What still confuses me is the symbolism $\textstyle x\xi=\sum_{i\in I}x_i\xi_i$ .

Posted by Stephan Paukner in Master's Thesis at 09:25 | Comments (0) | Trackback (1)

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Tuesday, March 14. 2006

Functions with compact support

I finished chapter IV on friday and continued on Hilbert space theory. Unfortunately, when it came to connections with Fourier transformation (Sobolev and Schwartz spaces, distributions), my understanding began to slow down. Some questions occured:

Why is $\langle f^\prime,\varphi\rangle=-\langle f,\varphi^\prime\rangle$ for $\Omega\subset\mathbb{R}^n, \, f\in\mathrm{C}^1(\Omega)$ and $\varphi\in\mathscr{D}(\Omega)$ , i.e., why is

$\displaystyle\int_\Omega\, f^\prime(x)\overline{\varphi(x)}\,dx=-\int_\Omega\, f(x)\overline{\varphi^\prime(x)}\,dx$

for $\varphi$ with compact support? Why do the border terms in the partial integration disappear in this case?
What’s so special about functions with compact support anyway?

Continue reading "Functions with compact support"

Posted by Stephan Paukner in Master's Thesis at 07:54 | Comments (0) | Trackbacks (0)

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Friday, March 3. 2006

Progress in functional analysis

I tried to have chapter IV (Main theorems about operators on banach spaces) of Werner’s “Functional Analysis” done by the end of February, but I just managed to do until Chapter III (Theorem of Hahn-Banach and its consequences). It’s not so easy to follow the content without repeating the former theorems and results, but at least I regain the feeling for the topics. Karin will have functional analysis as her second exam topic, and even she told me about having forgotten most of the contents already. So, I’m not really badly off.

I even have to do some more, working through the script of Prof. Cigler and having a look at the basics of time-frequency analysis. In April, I want to be that far that I’ll only have to look into my books I bought from HGFei in January.

I also want to see if Rudin’s “Functional Analysis” could be of worth for me. I already possess his “Real and Complex Analysis” in a German translation, which I’ll need if I really take measure theory for my master exam.

Posted by Stephan Paukner in Master's Thesis at 14:56 | Comments (0) | Trackbacks (0)

Monday, February 6. 2006

Understanding the Fourier transform

I borrowed the book “Fouriertransformation für Ingenieur- und Naturwissenschaften” by Klingen from the NuHAG library to use it as a help for understanding the (multidimensional) Fourier transform. Unfortunately, it seems to be on a rather applied/engineer level and only seems to contain one- and two-dimensional functions. We’ll see if it is useful to me.

Posted by Stephan Paukner in Master's Thesis at 19:59 | Comments (0) | Trackbacks (0)

Wednesday, February 1. 2006

d-dimensional Fourier transform

I know what the factor $e^{2\pi\mathrm{i}x\omega}$ for $x,\omega\in\mathbb{R}$ in the definition of the Fourier transform

$\displaystyle(\mathscr{F}f)(\omega) = \int_{\mathbb{R}} f(x)e^{-2\pi\mathrm{i}x\omega}\,dx$

means, if that factor is seen as a function in $x$ : If $\omega$ is fixed, then the variation of $x$ delivers the circle function, returning to the starting point for every $x\omega\in\mathbb{N}$ .

But what if $x,\omega\in\mathbb{R}^d$ ? Here, $x\cdot\omega$ has to be understood as the inner product $\langle x,\omega\rangle$ and the integral turns into the $d$ -dimensional Lebesgue-integral, i.e., if $x=(x_1,\ldots, x_d)$ , this means