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Let $f(x)$ be a real-valued function on $\mathbb{R}$ such that $x^nf(x), n=0,1,2,\ldots$ are Lebesgue integrable.

Suppose $$\int_{-\infty}^\infty x^n f(x) dx=0$$ for all $n=0,1,2,\ldots. $

Does it follow that $f(x)=0$ almost everywhere?

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    This is a duplicate question.2011-01-29
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    Here: http://math.stackexchange.com/questions/16831/a-nonzero-function-in-c0-1-for-which-int-01-fxxn-dx-0-forall-n-g2011-01-29
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    @TCL: Note also that the indefinite integral cannot equal $0$...2011-01-29
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    This is more apt I think: http://math.stackexchange.com/questions/17026/what-can-we-say-about-f-if-int-01-fxpxdx-0-for-all-polynomials-p2011-01-29
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    Neither of the questions linked in the comments above is a duplicate, but Hans Lundmark's answer on the first question shows that the answer to this one is no: http://math.stackexchange.com/questions/16831/a-nonzero-function-in-c0-1-for-which-int-01-fxxn-dx-0-forall-n-g/17047#170472011-01-30
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    I have voted to reopen, because this is not a duplicate of the linked question. It turns out that Hans Lundmark had posted an answer that also addresses this question as interesting related info, but it seems strange to not allow this question to be directly answered now that it has been asked.2011-01-30
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    @Jonas: The question was edited after it was closed! Of course, before the closure, the integral was not even definite. Given the edit, I have voted to reopen too.2011-01-30
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    @Moron: Thanks. The integral was definite from the beginning, but I suppose there was notational abuse. The combination of stating the domain, mentioning Lebesgue integrability, and having the right hand side $0$ led me to assume that what was intended is what is now more explicit.2011-01-30
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    @Jonas: I have voted to reopen it as well and to add to Moron's point when the question was posted the integral was indefinite causing some confusion.2011-01-30
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    @Jonas: Yes, there was an edit. Before, the integral was indefinite, it was not entirely clear what was meant. With the edit, this is a different question, so I'll withdraw my vote to close as a duplicate.2011-01-30
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    @Sivaram, @Arturo: Thanks. Yes, I know there was an edit. My first comment was nearly simultaneous with the closure, and my second comment was right afterward, and before TCL made the integral more explicit. What I hadn't realized at the time was that the integrals had actually been interpreted by many as indefinite integrals, and it's definitely a good thing that TCL cleared it up.2011-01-30

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