Stephan Paukner :: syslog :: Entries from March 2006

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 10:22 | Comments (0) | Trackbacks (0)

Defined tags for this entry: mathematics

Tuesday, March 28. 2006

I don't car about it

Just some facts: Deregistration confirmed, car insuration cancelled by today, keeping current bonus level for next car. Bought car at May 29th, 2001. Drove about 14,600 km since March 2005. Sale price nearly covers all repair costs since November 2005. Obsolescence of €3,224 within 5 years means a yearly amount of about €644 and a monthly amount of €55. Including the uncovered repair costs of last year, which are about €900, I had a monthly expense of €71, which is quite the amount of an old Austrian “Tausender”. Including all repair costs, gasoline and insurance, this raises to about €190-200 per month. Cheers!

Posted by Stephan Paukner in Personal at 15:16 | Comments (0) | Trackbacks (0)

Defined tags for this entry: automobile

Monday, March 27. 2006

Car sold!

I didn’t think that it would go so quickly: I received a call yesterday evening at 19:00h, and 2 hours later my car was sold. I agreed to drive it to the potential buyer, Doris ascorted me with her Smart, and we arrived shortly after 20:00h. The potential buyer and a friend of him had a look at my car, they didn’t even want to take a test drive, as I just had gone over the highway, and that was proof enough for them. The buyer wanted to beat the price down to €700, and I finally agreed, as it was obvious that the car wasn’t in its best shape.

So, we filled out the act of sale in his livingroom and did the deal. We met once again this morning, where I handed him out the latest ÖAMTC-attestation. With this, he will care for the deregistration tomorrow.

Posted by Stephan Paukner in Personal at 13:24 | Comments (0) | Trackback (1)

Defined tags for this entry: automobile

Saturday, March 25. 2006

Migration successful

Today I finished migrating my server to a new system. The LAMP-site consisting of 3 domains, which also runs an MTA and mailing list software, has been moved from Debian ‘woody’ to Debian ‘testing’ (alias ‘etch’). Insiders know that this is quite a big jump, leaving out the ‘sarge’ release. The main reason for this move was dying hardware: I expect either the graphics board or the SCSI-controller to be the cause, as they both are from 1998. Once in a while some system processes ended up in a segmentation fault, so I looked for a new (used) PC, which should become the successor.

It was a good opportunity to clean up the system, as the old one had its roots in a time when I made my first steps with Debian and wanted to maintain everything using ./configure && make && make install. Now, I want to avoid this and use the original Debian packages, because these days some commonly used additions are now officially available. Also, with Debian ‘testing’, there won’t be so gigantic delays between new releases of packages.

The interesting increments in the version numbers are: MySQL 4.0.20→5.0.18, Postfix 1.1.11→2.2.9, Courier-IMAP 1.4.3→3.0.8, Apache 1.3.31→2.0.55 and PHP 4.3.8→5.1.2.

Posted by Stephan Paukner in Information Technology at 15:59 | Comments (0) | Trackback (1)

Defined tags for this entry: debian

Wednesday, March 22. 2006

Car for sale

After a photo session on the weekend, my sale ad can already be accessed at car4you.at, I hope somebody wants to have it for €999.-, as the ÖAMTC didn’t give it a good attestation: Corrosion of brake disk, crack in the headlight glass, minimal oil leakage, and some minor corrosion deficiencies.

Posted by Stephan Paukner in Personal at 11:28 | Comments (0) | Trackbacks (0)

Defined tags for this entry: automobile

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 10:43 | 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 11:18 | Comments (0) | Trackbacks (0)

Defined tags for this entry: mathematics

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 $\textstyle x\xi=\sum_{i\in I}x_i\xi_i$ .