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][1]  

  [1]: http://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.