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Bonjour,

J'ai rencontré le problème suivant dans le livre "Real and Functional Analysis" de Lang, au chapitre $3$. J'explique d'abord le contexte, puis j'en viendrai à la question précise.

Il faut démontrer que les fonctions de la forme $e^{-x}p(x)$, où $p$ est un polynôme, sont denses dans l'ensemble des fonctions continues sur $[0, +\infty[$ qui tendent vers zéro en $+\infty$, muni de la norme sup.

Pour cela, on applique Stone-Weierstrass au complété $[0,+\infty]$, en considérant l'algèbre des fonctions de la forme $\sum_{n=1}^{N}{e^{-nx}p_n(x)}$ définies sur le complété (auxquelles on doit rajouter aussi des constantes). Ensuite, il faut montrer qu'on peut approcher uniformément les fonctions $e^{-nx}p(x)$ par des fonctions de la forme $e^{-x}q(x)$.

Lang suggère d'approcher d'abord $e^{-2x}$ par des fonctions $e^{-x}q(x)$ à l'aide de la formule de Taylor avec reste, puis $e^{-nx}p(x)$ en général. Je crois avoir réussi la première partie, et la deuxième partie en découle assez facilement pour $n \geq 3$. Par contre, je n'arrive pas à traiter le cas $n = 2$.

Alors ma question est la suivante: Pour $m \in \mathbf{N}$, peut-on approcher uniformément $x^m e^{-2x}$ sur $[0,+\infty[$ par des fonctions de la forme $e^{-x}q(x)$, où $q$ est un polynôme?

J'apprécierais beaucoup une référence ou une démonstration (ou les deux). (Vous pouvez répondre dans une autre langue.)


[Mod: attempt at translation below]

Hello,

I've encountered the following problem in the book "Real and Functional Analysis" by Lang, in chapter 3. I'll first explain the context, after which I'll pose my precise question.

We want to prove that the functions of the form $e^{-x}p(x)$, where $p$ is a polynomial, are dense in the set of continuous functions defined on $[0,\infty)$ which tends to 0 at $+\infty$, equipped with the sup norm.

For this, we apply Stone-Weierstrass on the completion $[0,+\infty]$, and consider the algebra generated from the functions of the form $\sum_{n=1}^N e^{-nx}p_n(x)$ (to which we add also the constant functions). Next, we need to show that we can uniformly approximate the functions $e^{-nx}p(x)$ by functions of the form $e^{-x}q(x)$.

Lang suggests to first approximate $e^{-2x}$ by functions of the form $e^{-x}q(x)$ using Taylor's theorem with remainders. Then consider $e^{-nx}p(x)$ in general. I think I know how to do the first step. For the second step, the $n\geq 3$ cases are easy. On the other hand, I don't know how to treat the case $n=2$.

So here's my question: can we approximate a function of the form $x^me^{-2x}$, where $m\in\mathbb{N}$, uniformly over $[0,\infty)$, by functions of the form $e^{-x}q(x)$, where $q$ is a polynomial?

I would appreciate a reference or a proof (or both). (You can post your answer in another language.)

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    Comme certainement tu as remarqué, ici on écrit en anglais! :)2011-02-08
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    Perhaps your browser has automatically translated this site into French for you, but this is actually an English-based site, so I think most people here will appreciate if you could translate your question into English.2011-02-08
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    @Mariano: Google Chrome is known to automatically translate pages into the user's preferred language, so it's possible that C.R. had not actually noticed that here we write in English! I've seen this issue raised on meta.SO, but I can't find it now.2011-02-08
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    A problem from Lang asks to show that the set $S$ of functions of the form $e^{-x} p(x)$, with $p$ a polynomial, is sup-norm-dense in $C_0[0,\infty)$: the set of continuous functions $f: [0,\infty) \to \mathbb{R}$ satisfying $\lim_{x \to \infty} f(x) = 0$. From an appropriate version of the Stone-Weierstrass theorem one knows that the subalgebra of $C_0[0,\infty)$ generated by $S$ is dense, so suffices to show that any function of the form $e^{-kx} p(x)$ is in the uniform closure of $S$, and it suffices to consider only $p$ a power of $x$. The OP is stuck on this case, with $k=2$.2011-02-08
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    J'ai vérifié dans le FAQ, et je n'ai rien vu au sujet de la langue. Je suis désolé si ça vous dérange, mais je ne voyais pas de mal à poser la question en français. Si vous ne comprenez pas la question, n'y répondez pas, et quelqu'un qui la comprend le fera.2011-02-08
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    I don't see any problems with users posting in any language they like. Indeed, if the poster's English is weak, questions will often be more comprehensible when written in his native language.2011-02-08
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    I apologize if this is the preferred practice in mathematical circles. I found that I was thinking of [the official policy on the StackOverflow](http://blog.stackoverflow.com/2009/07/non-english-question-policy/), but I understand if the people here want to follow a different policy.2011-02-08
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    (Continuing the translation of details) You can additionally assume known that $e^{-2x}$ is in the sup-norm-closure of $S$ (this follows from an argument involving Taylor's theorem which the OP knows). Lang's hint is that somehow this fact is of immediate use in showing that any $e^{-kx} q(x)$ is. The essential difficulty, as the OP states, is with the case $k=2$.2011-02-08
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    @C.R.: I edited in a more-or-less line-by-line translation of your original query. Hope you don't mind. @anon: thanks for the translation you gave in the comments. I double checked my translation against yours, so I think I've roughly captured everything the OP wanted to say. Do let me know if I messed up anything.2011-02-08
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    @C.R., le FAQ ne dit rien apropos de la langue... mais l'observation suggère sans doute que le français n'est pas fréquent, non?2011-02-08

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