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,

  • $\begingroup$ 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? $\endgroup$ – Etienne Mar 2 '14 at 22:44
  • $\begingroup$ @Etienne Is exactly that except here we need $f:\mathbb{R}^+ \rightarrow \mathbb{C}$. In 'my' example $f$ value is not in $\mathbb{C}$ $\endgroup$ – user117932 Mar 3 '14 at 10:16
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    $\begingroup$ Usually, one would say that if $f$ is real-valued, it is in particular complex-valued. $\endgroup$ – Etienne Mar 3 '14 at 18:18
  • $\begingroup$ @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) $\endgroup$ – user117932 Mar 3 '14 at 18:23
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    $\begingroup$ Right now, I don't know. But this is an interesting question. $\endgroup$ – Etienne Mar 3 '14 at 18:24

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 $n$.


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