# Does there exist a function such that $\int_{\mathbb{R}_+^{\star} } t^nf(t)dt=0$? [closed]

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 call Hausdorff theorem.

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

In fact this exercice was posted in MSE and actually I don't understand how someone can construct a such example ? Can we find it by ourselves ? Is there exist some reference of this theorem (History perhaps..) ?

Reference

Thank you in advance for your time,

Julien.

-

## closed as off-topic by Andrés Caicedo, Lucia, Fernando Muro, Chris Godsil, Andrey RekaloMar 7 '14 at 6:01

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Andrés Caicedo, Lucia, Fernando Muro, Chris Godsil, Andrey Rekalo
If this question can be reworded to fit the rules in the help center, please edit the question.

See Fourier Analysis by T. W. Körner. – Andrés Caicedo Mar 6 '14 at 23:43
@AndresCaicedo Thank you but 80$for a book is a bit expensive .. – user46896 Mar 7 '14 at 0:04 @NeilHoffman What is your$k$? I did not really understand everything. Can you elaborate a little more please ? – user46896 Mar 7 '14 at 0:08 Perhaps I missing an implied assumption, but I think this works. Consider a function$g_0(t)=\{0 \mbox{ on } [0,1], 1/k \mbox{ on } [2k-1,2k], -1/k \mbox{ on } (2k,2k+1)\}$. Let$g_1(t)$be a continuous function that approximates$g_0(t)$such that$g_1(t)$also has the property that$\int_{x=2k} ^{2k+2} g_1(t)dt=0$. Now, let$f(t)=g_1(t)/t^n$if$t\ne 0$and$f(0)=0$. – Neil Hoffman Mar 7 '14 at 0:09 I just cleaned the comment up a little, so hopefully its more readable.$k\$ is the floor of (t+1)/2. – Neil Hoffman Mar 7 '14 at 0:12

## 1 Answer

The standard reference is N. I. Akhiezer, The Classical moment problem and some related questions of analysis, MR0184042.

There is also a paper: MR1627806 Simon, Barry The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137 (1998), no. 1, 82–203.

-
Available here. – Andrés Caicedo Mar 7 '14 at 2:11
I didn't know this reference. Thank you. – Andrés Caicedo Mar 7 '14 at 2:11
Andres, thanks for the reference:-) I did not know about the free English translation. Almost all Russian (Soviet) books are available free. – Alexandre Eremenko Mar 7 '14 at 2:15