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**


1. [The exercise in Mathematics Stack Exchange](https://math.stackexchange.com/questions/689218/does-there-exist-a-function-such-that-int-mathbbr-star-tnftdt-0)


Thank you in advance for your time,


Julien.



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


