how is this function a sum of these two functions?

Question

So chui's book on wavelets has this diagram:

I don't understand what he means by the highlight part here; aka I don't really see how the first two walsh basis functions are equal to the sums he's talking about.

Answer 1

Let $$\phi_H(t) = \begin{cases}1 & \text{, if } t \in [0,1)\\ 0 & \text{, otherwise} \end{cases}$$ and $$\psi_H(t) = \begin{cases}1 & \text{, if } t \in [0,\frac 12)\\ -1 & \text{, if } t \in [\frac 12,1)\\ 0 & \text{, otherwise} \end{cases}.$$ The top left graph of Fig. 1.4 corresponds to $$\phi_H(t)+\phi_H(t-1) = \begin{cases}1 & \text{, if } t \in [0,2)\\ 0 & \text{, otherwise} \end{cases}$$ periodically extended with a period of $2$.

Similarly the top right graph of Fig. 1.4 corresponds to the $2$-periodic extension of $$\psi_H(t)+\psi_H(t-1) = \begin{cases} 1 & \text{, if } t \in [0,\frac 12) \cup [1,\frac 32)\\ -1 & \text{, if } t \in [\frac 12,1) \cup [\frac 32,2)\\ 0 & \text{, otherwise} \end{cases}.$$