TITLE: find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that : $ f(f(x))=x^2-2 $
QUESTION [5 upvotes]: This is a very hard functional equation.
the problem is this :
find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that : $ f(f(x))=x^2-2 $
to solve it i have no idea! can we solve it with highschool olympiad education?
please help : )

REPLY [2 votes]: As shown by Gottfried Helms in a linked question, a solution over $[-1,1]$ is given by a function defined over $(-2,+\infty)$:
$$ 2\cdot T_{\sqrt{2}}(x/2) $$
where $T_n$ is a solution to the Chebyshev differential equation 
$$ (1-x^2)\frac{d^2 y}{dx^2}-x\frac{dy}{dx}+ n^2 y = 0.$$
The first terms of the Taylor series in zero are:
$$2 \cos\left(\frac{\pi }{\sqrt{2}}\right)+\sqrt{2}\,\sin\left(\frac{\pi }{\sqrt{2}}\right) x-\frac{1}{2} \cos\left(\frac{\pi }{\sqrt{2}}\right) x^2-\frac{1}{12\sqrt{2}}\sin\left(\frac{\pi }{\sqrt{2}}\right) x^3-\frac{1}{48} \cos\left(\frac{\pi }{\sqrt{2}}\right) x^4+\ldots$$