Can someome help me to proe the following result:
There exist two functions $\chi $ and $\phi$ valued in the interval $[0, 1],$ belonging respectively to $\mathcal{C}^{\infty}_0((0, \frac{1}{2}))$ and to $\mathcal{C}^{\infty}_0(\frac{1}{2},3),$ and such that \begin{align*} &\forall x\in [0,+\infty[,\quad\quad\chi^2(x) +\sum_{j\ge0}\phi^2(2^{-j} x) = 1 \end{align*}
This is what i tried to do:
I started by considering $\chi_0\in\mathcal{C}_0^{\infty}(\left[0,1\right[)$ such that $\chi_0=1$ in $\left[0,\frac{1}{2}\right]$ and $\chi_0=0$ when $x\ge\frac{3}{4}$ . And then i defined $\chi_1(x)=(1-\chi_0)(x)\chi_0(x/4)$.
But I don't know how to contitue. Please help me to construct such a partition. Thanks in advance