The following exercise is inspired in Daubechies - Lagarias: Two-Scale Difference Equations, Existence and Global Regularity of Solutions.
Define the following functions:
$$f_0(x)= \left\{ \begin{array}{lcc} 1+x & \text{if} & -1 \leq x \leq 0 \\ \\ 1-x & \text{if} & 0 \leq x \leq 1 \\ \\ 0 & \text{if} & \vert x \vert \geq 1 \end{array} \right. $$
Now we construct $f_1$ as follows: $f_1$ is linear in the restrictions to an interval of the form $[m/3,(m+1)/3]$ for all $z \in \mathbb{Z}$. The nodes of $f_1$ are given by $f_1(m)=f_0(m)$, $f_1(m+1/3)=f_0(m+2/3)$, $f_1(m+2/3) = f_0(m+1/3)$, for all $m\in\mathbb{Z}$. Then, do a linear interpolation between this nodes. The same procedure is repeated to obtain $f_{j+1}$ from $f_j$ for all $j \in \mathbb{N}$. The resulting $f_j$ are piecewise linear, with linear restrictions to the intervals $[m3^{-j},(m+1)3^{-j}]$, for all $z \in \mathbb{Z}$.
Now we want to check that:
$$f_{n+1} = Vf_n.$$
Where:
$$Vf(x) = f(3x)+\frac{1}{3}[f(3x+1)+f(3x-1)]+\frac{2}{3}[f(3x+2)+f(3x-2)]$$
The authors said that "it can be checked fairly easy...". I'm trying to show by induction, but I don't know how to start. I know that the formula of $V$ is an linear interpolation but I don't understand really well. I will be gratefull for any explanation or an attemp to demonstrate the formula. :(