Entries from Stephan Paukner

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., , where and for . But, it is even a common property for any -function that , where we again use the symbolism . This becomes clear by using the substitution :

So, it’s clear () that this means , and not only for the Gaussian function . And, for , the substitution leads to , because if , then and therefore . 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.

What I hate is when books say that something is clear, what definitely is not clear! So, why is it clear that for and ? 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 , the proof of which also doesn’t seem to be too trivial. What still confuses me is the symbolism .

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:

1. Why is for and , i.e., why is
   
   
   
   for with compact support? Why do the border terms in the partial integration disappear in this case?
2. What’s so special about functions with compact support anyway?

Ich sehe immer öfter - vor allem im Winter - junge Menschen in Jacken mit pelzberandeten Kapuzen. Diese Jacken sind meist grau oder zumindest dunkel, manchmal aber auch weiß. Auffällig ist aber die Kapuze, die am Rand mit einem hellen, flauschigen Pelz abgeschlossen ist.

Das alleine ist ja noch nicht das Kuriose, sondern, dass diese Pelzkapuzen stets im Rudel auftreten: Wo eine ist, ist der Rest der Bande nicht weit. Pro Bande wird selbstverständlich nur exakt das gleiche Modell getragen. Allesamt, ob männlich oder weiblich, sind die Mitglieder etwa 180cm groß, schlank, und um die 23 Jahre alt. Hinzu kommt ein kleiner Tussi- bzw. Prolo-Faktor.

War ich auch mal so komisch? Nun gut, bei mir waren’s die unsäglichen “Grunger”-Hemden, aber bitte, da war ich 15/16. Und von Jeans brauchen wir sowieso nicht sprechen.

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.

I don’t have an iPod. I don’t need one. If I had one, I wouldn’t know what to do with it, except selling it. I don’t listen to music regularly anymore, for me, that time has been more than 10 years ago. I used a simple Sony Walkman, and all CDs I bought got copied over to music cassettes.

And I really don’t understand what’s the hype with podcasting. While I still understand high interest in music, I can’t imagine replacing music by spoken text in everyday life. People in underground trains and streetcars surely give their brains a rest and let their thoughts flow while listening to their favorite music. But can they really relax while following a (maybe even scientific) “radio broadcast” consisting of spoken text? Or have I misunderstood what a podcast is?

But, I have a weblog. And I’m on Linux. I’m still cool.

March 2005. My wreck, ahem... my car already has problems getting “the sticker”, i.e. the sticker which signs it as being technically okay. A repair cost of €350 occurs.

August 2005. During our drive into our holidays, the back of my exhaust pipe rust through and broke off. €130 in botch.

September 2005. Can’t drive backwards, something is blocking. The brakes break. €319.50 make them new.

November 2005. Actually, can’t remember what happened. €220 at mate’s rates at Roli’s garage.

December 2005. Just a few days later I’m back to Roli. Exhaust pipe front, rust through, €143.

January 2006. Engine is dying off after some minutes of drive. Roli thinks to have found the failure at the fuel injection. €136.

February 2006. Engine still dying off, this time at the beginning of our drive into a holiday weekend. ÖAMTC tows us to Roli. Rent a car. After days Roli finally found out, car didn’t start anymore, starting module replaced, €292.

After one year, the hell of €1590.50 have been thrown out of the window. First steps towards selling the car are coming up.

Heute Morgen sah ich in einem Schaufenster beim Vorbeigehen ein Kinderbuch mit dem Titel “Aufräumen macht Spaß”, darauf abgebildet ein junger Mann, aus einer offenen Kommodenlade herausblickend, inmitten eines Idylls von Kinderzimmer. Da scheint ja ein echter Meisterpädagoge am Werk gewesen zu sein! Auf welch einfache Weise man doch die Einstellung von Kindern zu einer bestimmten Sache ins Gegenteil verkehren kann! Ich sehe schon die Nachfolgetitel, diesmal als Zielpublikum die Erwachsenen: “Operiert werden macht Spaß”, “Zähneziehen macht Spaß”, “Krebs haben macht Spaß”, und weitere.