Problem about the relationship of 2 periodic functions

Question

If $f,\varphi$ are two even periodic functions with $T=2$. Suppose $$f(x) = x(2-x),\quad \varphi(x) = x,\quad x\in [0,1]$$ prove that: $$f(x)=\sum\limits_{n=0}^\infty\frac{1}{2^{2n}}\varphi(2^nx),\quad\forall x\in\mathbb{R}.$$ I wonder if I need to make use of the fourier expansion of $f$ and $\varphi$, can anyone offer me some help?