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

Thank you in advance for your time,

Julien.

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

Thank you in advance for your time,

Julien.

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 found it by ourselves ? Is there exist some reference of this theorem (History perhaps..) ?

Reference

1. The exercise in Mathematics Stack Exchange

Thank you in advance for your time,

Julien.

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

Thank you in advance for your time,

Julien.