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Prove that if $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ is a continuous function such that: $\int_{0}^{1}f(x)e^{nx}dx=0$ for all $n=0,1,2,...$. Prove that $f(x)=0$ for all $0\leqslant x\leqslant 1$ using two methods: 1/ change of variables and then apply Weiestrasss Theorem. 2/ Apply Stone Weierstrass Theorem .

I already know how to prove that $f(x)=0$ for all $0\leqslant x\leqslant 1$ if $\int_{0}^{1}f(x)x^{n}dx=0$. I did the following change of variable: $e^{x}=y$ and then I got $\int_{1}^{e}f(y)y^{n-1}dy$. since $\int_{0}^{1}f(y)x^{n-1}dy=0$, we are left with $\int_{1}^{e}f(y)x^{n-1}dy$ which I want to prove equal to zero. Any help with this?

For the second part, I don't have any idea how to use the Stone Weierstrass Theorem to prove it. I have never used this theorem before in solving problems, so I appreciate if someone helps with details for this part.

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"Webmistress"!? Was that OCR? – Dirk Dec 13 '11 at 16:45
Wouldn't you have to write $f(\log y)$ after the change of variable? – Dylan Moreland Dec 13 '11 at 16:46
Do you know the statement of Stone-Weierstrass? Do you know what "separating the points" mean? – Dirk Dec 13 '11 at 16:46
@Dirk: Yes, I know what separating the points means because it is one the conditions in Weierstrass Theorem. By the way, I made a mistake: instead of "Webmistress", change it to "Weierstrass" – M.Krov Dec 13 '11 at 17:01
@Dylan: You're right, it should be $f(logy)$ – M.Krov Dec 13 '11 at 17:02
up vote 2 down vote accepted

I think you are on the wrong track with the substitution. Do try to prove:

if $A$ is a dense subset of $C([0,1])$ (wrt uniform convergence, of course) and, for $f\in C([0,1]),$ $\int_0^1 f(x) g(x) dx = 0 \quad\forall g\in A$ then $ f=0$.

Weierstrass/Stone-Weierstrass give you dense subsets of $C([0,1])$. The Weierstrass part should be more or less obvious with the statement I asked you to prove. For the Stone-Weierstrass part try to find an algebra satisfying the hypothesis of the theorem by looking at the functions which you are given, $e^{nx}$.

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ok I kind of missed the fact you are supposed to use change of variable theorem. So my answer applies first of all to 2/ but also to your transformed integral. Sorry for not being precise here. – user20266 Dec 13 '11 at 18:41
Ok, two more hints: i) If $A$ is the dense subset approximate $f$ by functions $g_k \in A$ converging uniformly. Uniform convergence implies $$0=\int fg_k \rightarrow \int f^2 $$ ii) look at the algebra generated by the $e^{nx}$. – user20266 Dec 13 '11 at 18:56

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