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Let $f\in C([a,b],\mathbb{R})$ such that $\displaystyle\int_{a}^{b} t^nf(t)dt=0$ for all integer n. We know that $f\equiv 0$. It's "easy" to prove with Weierstrass theorem or with How to prove that $\,\,f\equiv 0,$ without using Weierstrass theorem?

  • This theorem is wrong on $\mathbb{R^+}$, we can choose : $$f(x)=\exp(-x^{\frac{1}{4}})\sin(x^\frac{1}{4})$$

Let $$ I_n=\displaystyle\int_{0}^{+\infty}t^n\exp(-\omega t)dt=\frac{n!}{\omega^{n+1}},\quad n\in \mathbb{N}, \quad \omega=exp(\frac{i\pi}{4}) $$ Proof. $$ |t^n\exp(-\omega t)|=t^n\exp \bigr(\frac{-t \sqrt{2}}{2}\bigl)\in L^1(\mathbb{R}) $$ by integration by parts we get $$ I_n=\frac{n}{\omega}I_{n-1} $$ Thus,

$$ I_n=\frac{n!}{\omega^{n+1}} $$

Plus for $n\geq 1$, $\quad \omega^{4(n+1)}=-(1)^{n+1}$

Then, $$ I_{4n+3}\in \mathbb{R} $$

Therefore, $$ 0=\Im(I_{4n+3})=\displaystyle\int_{0}^{+\infty}x^nf(x)dx $$

  • Let $f:\mathbb{R}_+ \longrightarrow \mathbb{C}$,

I would like to prove the existence of a function $f$ such that $\int_{0}^{+\infty}t^n f(t)dt=0$,

In fact this example it's not mine (I have already read it in a book) and the question is to find $f:\mathbb{R}_+ \longrightarrow \mathbb{C}$. So I would like to know if we can proove the existence more generally or just how can I construct a such function.

Thank you in advance,

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  • Maybe I don't understand your question: you have an example but you are not happy with it because you don't see where it is coming from? So you would like to know how you could have found it by yourself? – Etienne Mar 02 '14 at 22:44
  • @Etienne Is exactly that except here we need $f:\mathbb{R}^+ \rightarrow \mathbb{C}$. In 'my' example $f$ value is not in $\mathbb{C}$ –  Mar 03 '14 at 10:16
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    Usually, one would say that if $f$ is real-valued, it is in particular complex-valued. – Etienne Mar 03 '14 at 18:18
  • @Etienne I know that but perhaps we can found f with complex-valued, anyway do you have an idea to found it by myself ? (real-valued, complex-valued , it doesn't matter) –  Mar 03 '14 at 18:23
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    Right now, I don't know. But this is an interesting question. – Etienne Mar 03 '14 at 18:24

1 Answers1

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Start with your example, take the imaginary part, just user every fourth step with variable $t=s^{1/4}$. We get: $$ F(s) = \exp(-s^{1/4})\sin(s^{1/4}), $$ for which $$ \int_0^\infty s^n F(s)\;ds = 0 $$ for all nonnegative integers $n$.

GEdgar
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