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Let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous function. Suppose that $\displaystyle\int_a^b x^nf(x) \, dx=0$ for all $n\in\{ 0,1,2, \ldots \}$. Prove that $f=0$.

(Hint. Consider $\displaystyle\int_a^b f(x) \, f(x) \, dx$.)

Hello,

I have been trying to solve this question.

My attempt at the solution has involved looking at similar functions. Firstly, i know that if we take the absolute value of the integral above it is clear that the functions must be 0. To elaborate, if the function had a point x such that f(x) was nonzero it would also follow that there is a delta interval around the point f(x) so that the integral from x-delta to x+delta of the above would be positive and so this would be a contradiction. So, i thought i could apply a similar principal here. However, that doesn't seem to work. I feel that the hint is drawing upon the same idea using f(x)*f(x) as being always positive since we are working in R. Any ideas?

J. W. Perry
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greg
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  • Do you already have a theorem about approximability of continuous functions in a closed interval by polynomials? – André Nicolas Nov 02 '13 at 04:56
  • You can use the Stone-Weierstrass theorem to prove this. Comment if you need more hints. – Euler....IS_ALIVE Nov 02 '13 at 04:56
  • i have Weierstrass Approximation theorem and Stone-Weierstrass Theorem which is actually the last theorem in the notes, so im still trying to figure out what it says. However, if i am to use the Stone-Weierstrass theorem, i will have a go at understanding and using it and come back if i still dont get it. – greg Nov 02 '13 at 05:00
  • So, to confirm i must use the Stone-Weierstrass Theorem? – greg Nov 02 '13 at 05:06
  • Let $g$ is a polynomial which approximates $f$. Consider, $(f-g)^2=f^2-2fg+g^2$. – Worawit Tepsan Nov 02 '13 at 05:10
  • Well, you don't HAVE to use Stone-Weierstrass, but it is one way to prove it. – Euler....IS_ALIVE Nov 02 '13 at 05:14
  • Are there any good videos that i can watch that explains the Weierstrass and Stone-Weierstrass Theorem in depth? I can't seem to find anything to assist me in actually understanding the theorem – greg Nov 02 '13 at 05:19

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