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The question that has been asked here for several times in various forms is somewhat like this (cf. the following links):

Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$

Also,

Integral of a continuous function

Again,

Prove that $f(x) = 0$ on the interval $[0,1]$

And so on!!

Now just being curious I am asking for similar result (BUT on unbounded domain) if at all true, in the form of following problem:

Suppose $v \in C([0,\infty))$ and $v$ vanishes at infinity. If $v$ is assumed to be square integrable and $lim_{x \to \infty} \int_{0}^{x}(x-y)^{k} v(y) dy = 0 \quad \forall k \in \mathbb{N}\cup{0}$, can we prove or disprove that $v$ is identically equal to 0 ??

Any form of solution/discussion is much appreciated!!

user92360
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  • I'm sorry, I deleted my previous answer because in the end I couldn't solve it after the mistake you pointed out in my attempted solution. It seems like this is a hard problem. In case you find a solution I'd be glad if you posted the solution here. – Lorenzo Q May 11 '18 at 14:52
  • Hello Lorenzo and anybody who is interested... but I suggest you to google a bit about "Ghost matrices and a characterization of symmetric Sobolev bilinear forms" by Kwon et al. See equation 2.7 and 2.8. I HOPE this is our sought after counterexample. Cheers. – user92360 May 12 '18 at 17:24

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