Stephan Paukner :: syslog :: Entries tagged as mathematics

Entries tagged as mathematics

Saturday, November 11. 2006

Raising questions, 1-62

When I started on my way to my Masters degree seriously in January 2006, several questions began to arise in the course of time, and they continue to do so. While there were some very fundamental questions at the beginning (because my last contact to FA and basic analysis had been quite some time ago), they matured to some more advanced questions today. Here, I give a list of them, in their order of appearance. The answers, however, and as far as they have already been provided anyway, did not come up in the same order. Some questions are so general that they are to be understood as a challenge to look into the books and get familiar with it.

This list will continue to grow in separate entries. Also, for the sake of documentation, currently missing answers will only be given separately.

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Posted by Stephan Paukner in Master's Thesis at 17:23 | Comments (0) | Trackbacks (6)

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Friday, November 10. 2006

A reading approach to time-frequency analysis

Here I give my personal reading approach to get familiar with the topics which the NuHAG is working on. This is just one way to get started, and I will mention some pros and cons I noticed. There is always some redundancy, but in my eyes that was of great worth.

A good way to get some first impressions is to read A Short Introduction to Gabor Analysis by Thomas Strohmer, which is the introductory chapter of [Feichtinger/Strohmer] and is also available online. You could also read the introductory chapters of [Blatter], which will make clear what the central ideas are in general. After having read that, you also want to understand it all.

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Posted by Stephan Paukner in Master's Thesis at 11:39 | Comments (0) | Trackbacks (2)

Defined tags for this entry: literature, mathematics

Sunday, April 9. 2006

Inclusions of $L^p$-spaces

Just a note: We know that the inclusion $\ell^1\subseteq\ell^{p_1}\subset\ell^{p_2}\subseteq\ell^\infty$ yields for $1\leq p_1<p_2\leq\infty$ . A similar inclusion statement is valid for $L^p$ -spaces iff we look at $L^p[a,b]$ . Here, for $p_1<p_2$ , the inclusion is the other way round: $L^{p_2}\subset L^{p_1}\subseteq L^1$ . This is not valid for $L^p(\mathbb{R})$ , as $\int_{\mathbb{R}}dt$ is not finite. In Cigler’s script, the fact $L^{p_2}\subset L^{p_1}$ 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 “ $L^2\cap L^1$ ” and not simply “ $L^2$ ”.

Posted by Stephan Paukner in Master's Thesis at 16:30 | Comments (0) | Trackbacks (0)

Defined tags for this entry: mathematics

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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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?

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

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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