# Density of polynomials in Gaussian Hilbert spaces

$\newcommand{\D}{\,\mathrm{d}} \newcommand{\Ds}{\mathrm{d}} \newcommand{\intRd}{\int_{\R^d}} \newcommand{\intR}{\int_{-\infty}^\infty} \newcommand{\GH}{{L^2_\gamma}} \newcommand{\LHG}{{L^2_\gamma}} \newcommand{\F}{\mathbf{F}} \newcommand{\N}{\mathbf{N}} \newcommand{\R}{\mathbf{R}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\e}{\mathrm{e}} \newcommand{\CcRd}{{{C_{\mathrm{c}}^\infty}(\R^d)}} \newcommand{\GSH}{{W^{1, 2}_\gamma}}$

Hello there! Today I would like to tell you something about polynomials. That sounds great doesn’t it. I am planning to make this Part 1 of a series on Gaussian (harmonic analysis), but the current part is about something much more elementary: density of polynomials. Nate Eldredge ask on Math.SE whether the polynomials are dense in the Gaussian Sobolev space, and the answer -which he gives himself- is indeed “yes”. But, I think one can do this more elegantly using Hermite polynomials, of which most of the required things to understand this post are collected by me here.

Edit (27 February 2013): The style has been slightly updated as is explained here.

# The Ornstein-Uhlenbeck operator with the Gaussian measure

The setting we consider is the same as in [1] That is, we consider the Gaussian measure on $L^2$:

\label{eq:Gaussian-measure}
\Ds\gamma_d(x) := \Ds\gamma(x) = \e^{-x^2} \, \frac{\Ds{x}}{\pi^{\frac{d}2}}.
Continue reading “Density of polynomials in Gaussian Hilbert spaces” »

# Got moved!

Hello dearest!

Today is a good day. I’ve moved the WordPress blog to a VPS so that I can use MathJax on the blog instead of the filthy WordPress.com alternative. I have set up a new VPS for this, so that the PHP and MySQL installation does not poison my cute Python daemons. Settings might need some tweaking, and I’ve put everything behind a proxy. Perhaps more about that later.

For now, I have MathJax’ed the post A finitely but not countably additive measure. Looks lovely doesn’t it? It sure does.

Always yours, JT.

# A finitely but not countably additive measure

$\renewcommand{\bar}{\overline} \DeclareMathOperator{\Lim}{Lim} \newcommand{\N}{\mathbf{N}} \newcommand{\R}{\mathbf{R}}$

Hello dearest readers! How a joyful day it was, full of good math. Actually it sucked. In any case, I have tried to post some of the things I have written on my blog. This is my first attempt on pushing the document which gives a finitely, but not countable additive measure. It will require more explanation, but yours truly is truly unsatisfied with the typesetting and needs to start some decent scripting before this gets done. Hold on to your nickers, this will be updated soon.

Why then do I post it you say? Good question! I will attempt to answer it. The main reason is that it pushes me harder out of inertia, and hence improving myself. I like that, pushing my own boundaries such that they are no boundaries anymore and are then elsewhere. The other thing is that some might have some suggestions on how to improve the LaTeX -> WordPress thing.

Edit: The issues have been resolved! I now have my own WordPress setup, so I can use MathJax. I set up a different VPS for that as I do not want to poison my other servers with filthy things like PHP. Got it? Good! In any case, it looks better now. So much better, I removed the ‘try #1′ from the title. I even have footnotes! How cool is that?

# A finitely but not countably additive measure

Consider the space $\ell^\infty$ of all bounded real-valued sequences on the natural numbers $\N = \{1, 2, 3, \dots\}$. Let $c$ be the subspace of $\ell^\infty$ of all convergent sequences. Next, we define the operator $\Lim$ on $c$, which is given by:

\label{eq:Lim-operator}
\begin{split}
\Lim : c &\to \R\\
x &\mapsto \lim_{n \to \infty} x_n.
\end{split}

For the purpose of extending $\Lim$ to $\ell^\infty$, we define the mapping $p$:

\label{eq:Extension-of-Lim-Hahn-Banach}
p(x) := \inf\Bigl\{\Lim(y) : y \in c \, \text{and} \,
\limsup_{n \to \infty} x_n \leq \lim_{n \to \infty} y_n \Bigr\}.

Note that $p$ is defined for all in $\ell^\infty$ as certainly we can find for every bounded real-valued sequence a convergent sequence whose limit is above any term in the sequence. The subadditivity $p(x + y) \leq p(x) + p(y)$ is easily established. Whenever $x$ is in $c$, then we have that $p(x) = \Lim(x)$, hence we can apply the Hahn-Banach theorem. By this theorem there exists a linear extension of $\Lim$ which we will henceforth denote by $\Lim$ as well, or $\overline\Lim$ when confusion might arise. Additionally, there holds that:
\begin{equation*}
\overline\Lim(x) \leq p(x) \quad \text{for all $x$ in $\ell^\infty$.}
\end{equation*}
Setting $x \to -x$ gives $\overline\Lim(-x) \leq p(-x)$. Hence:
\begin{equation*}
-p(-x) \leq \overline\Lim(x) \leq p(x).
\end{equation*}
Whenever $x$ is in $c$, then ${\overline\Lim}(x) = \Lim(x)$, hence we consider $x$ in $\ell^\infty \setminus c$. Using \eqref{eq:Extension-of-Lim-Hahn-Banach} and $x \geq 0$ we get:
\begin{align*}
-p(-x) &= -\inf\Bigl\{-\Lim(-y) : y \in c \, \text{and} \, \limsup_{n \to \infty}
-x_n \leq \lim_{n \to \infty} y_n \Bigr\}\\
&= \sup\Bigl\{\Lim(-y) : y \in c \, \text{and} \, \limsup_{n \to \infty}
-x_n \leq \lim_{n \to \infty} y_n \Bigr\}.
\end{align*}
In the last line we can set $y = 0$ in the supremum to obtain a lower bound, and get:
\begin{equation*}
0 \leq -p(-x) \leq \overline\Lim(x).
\end{equation*}
In particular if $x \geq 0$, then $\Lim(x) \geq 0$, so $\Lim$ is a positive operator defined on all of $\ell^\infty$. Furthermore, $\Lim$ is a so-called Banach limit.

Next, we consider the $\sigma$-algebra on $\N$ which consists of all its subsets, that is $\sigma(\N) = 2^\N$. We now define a finitely additive measure $\mu : \sigma(\N) \to \{0, 1\}$ for every subset $S$ of $\N$ as follows1:
\begin{equation*}
\mu(S) = \overline\Lim \bigl([n \in S] \bigr).
\end{equation*}
We can now see that $\mu$ is a finitely additive measure. $\mu(\emptyset) = 0$ and $\mu \geq 0$ are easy to verify. For the finite additivity select a finite collection of disjoint sets $\{S_n\}_{n = 1}^k$. These $S_n$ are finite or infinite sequences. So, by linearity of $\Lim$ we get:
\begin{equation*}
\mu\biggl(\bigcup_{n = 1}^k S_n \biggr) = \Lim\biggl(\sum_{n = 1}^k
S_n \biggr) = \sum_{n = 1}^k \Lim(S_n) = 0.
\end{equation*}
To show $\mu$ is not $\sigma$-additive, consider the sets $S_n := \{n\}$. The singletons containing the natural number $n$. As these are finite sequences, their limit is $0$. That is:
\begin{equation*}
\sum_{n = 0}^\infty \mu(S_n) = \sum_{m = 0}^\infty
\Lim\bigl([m = n]\bigr) = 0.
\end{equation*}
While we have for their union:
\begin{equation*}
\mu\biggl(\bigcup_{n \in \N} S_n \biggr) = \mu\bigl([n \in \N]\bigr)
= 1.
\end{equation*}
Hence, $\mu$ is not countable additive.

Kisses, Cheers, So long and thanks for all the fish,

JT.

1. $[\dots]$ is the Iverson Bracket and is used to represent the indicator function.

# TU Delft Eduroam Android settings

Update (16 March 2013): The settings have been updated, I found these work better.

The Delft University of Technology has like many other universities a public eduroam network. For the iPad and iPhone there is a TUvisitor network which provides you with a certificate to connect to the Wi-Fi. The (pdf) manuals that explain how to do this are not available for most Android phones. Perhaps due to the fact that there are so many of them. Nevertheless, most people around will be able to connect without a manual using the correct settings.

# Hermite polynomials

Hermite polynomials are quite lovely creatures indeed. I use them a lot in my research as they relate well to normal distributed stochastic processes. The thing is, the Graduate School of the Delft University of Technology requires each PhD Candidate to write a ‘Proof of Competency’ which has to prove you are competent. As I adore our Hermitian friends, I wrote something about that.

Cut the… you say. Indeed, the document can be found here. I will update the document and fix errors as I come by. There is a definite need of combinatorial interpretations of these in this document. Will be done in the future!

The document should also be useful as a very brief introduction to special functions.

I believe that introductory university mathematics (say calculus) should be sufficient to be able to understand the document. Nevertheless, it probably does require experience in such matters before one can understand it fully. Do not conclude that you are too stupid too quickly. Many things can be the case, my style of writing is not your preferred style of reading. Perhaps I am just a lousy writer. Please do tell me how to improve. I like that. I like improvement.

Some words have been emphasized like this such that one has some pointers to know what to search for on the web!

Cheers,

JT

# How to be a famous mathematician

Wake up! There is no such thing as a famous mathematician. Hence, this post is about ‘how to be a mathematician’. Today I will tell you about the tools required to be a mathematician, or some of the most important ones. Alright. Let’s start!

First of all, what you do need is a pen. That is something like this:

A black pen

This is as the caption says: a black pen. Indeed it is. Why black? That does not really matter. What is important for mathematics is that you are consistent in what you do. It is required to keep the stereotype of the lonely genius going. I used to use pencils. First I used a refillable pencil, but I felt that such an ancient craft as the craft of mathematics is does not require mechanical pens. Next, I switched to old school pencils with a pencil sharpener and stuff. The idea was that I could erase my mistakes and start over again! Perhaps this was due the the idea that a lecturer once said that you can read intelligence from the fact that someone writes with a pencil. Of course, I liked the idea. What a fool that man must have been. Pencils are not up to the job! When you do mathematics you need to do it fast and mark the mistakes – do not erase the mistakes! -. Got it? Good.

Alright, now that we have a pen, we need to make sure we can do mathematics whenever possible. The brain of a mathematician cannot shut down. Never. If it does, he or she is dead and henceforth an ex-mathematician. So indeed, what we need is redudancy. This is to make sure that when an idea pop up you are able to write to down. And pop up it will, as I said before: the brain never sleeps. You do not want to run out of ink! My solution is the following:

The pens, but not in the chest pocket!

Indeed. The pens should be put there, not in the chest pocket. There is a big advantage to placing them there: they do not drop out when you bend whereas they do if you decide to put them in the chest pocket. I have tried many different carrying configurations, but I found this one to be optimal. As you see there are two red pens: I must have been very sleepy as I need a redundant black pen and a blue one as well. Indeed, there is also a stylus as I own an iPad to read papers and so on, but that is not so crucial: you can print them and carry them along.

Also this is something you don’t need:

A phone – you don’t need it. Mathematicians do not talk.

A phone with a 3G connection can be useful when figuring out which airplane to take to go to a conference, which is indeed very useful. Nevertheless, not crucial. Finally for today, I provide you with the most crucial of it all! The notebook. Next you’ll see one which I have bought recently.

The front side.

The back side

The front and the back side. Perhaps you can see this is a Moleskine notebook and indeed it is. I like them. However, the brand is not that important with respect to mathematics. I just find these very comfortable to work with. Also, the color is not that important. For instance, black:

Black is no problem.

When you open it up, you will see some personalia:

The first page!

There you write your name and so one, such that the finder of your Precious can return it back to you. Make sure you don’t lose it. It is forbidden. I also give my books names. This is Scramblings #1, but I also have Random Ramblings #1-#7, some other ramblings and so on. Got it? Good! Also, I have them in very different formats. You need to experiment a bit to know which methods works best for you. Some are for profound things, but of course, most are not. This is an example of a page:

A page of non-profoundness.

Okay kids. That’s it for today. If you want to have more serious advice on how to be a mathematician, check out Terence Tao’s blog: http://terrytao.wordpress.com/. He appears to be well-known (but not famous of course, as that would be a counterexample to my claim above.).

Kiss,

JT.