I can not entirely follow the proof from section 3.1.1 from the book "A primer on Wavelets" by Walker. After the first part (listed below), I can grasp the rest so if you could help I would greatly appreciate it.
Given a level 1 Daub4 Wavelet: \begin{equation}W_m^1 = (0, 0, \dots, 0, \beta_1, \beta_2, \beta_3, \beta_4, 0, \dots, 0)\end{equation} with the non-zero entries starting at position $2(m-1)$, we have the following properties: \begin{equation}\beta_1 + \beta_2 + \beta_3 + \beta_4 = 0\end{equation} \begin{equation}0\beta_1 + 1\beta_2 + 2\beta_3 + 3\beta_4 = 0\end{equation} and given a signal $f$ sampled at a constant step-size \begin{equation}h = t_{n+1} - t_n, \forall n\end{equation} from an analog signal $g$ that is known to have a continuous second derivative over the support of $W_m^1$ such that \begin{equation}f_n=g(t_n)\end{equation} the author says we can write: \begin{equation}g(t_{2m-1+k}) = g(t_{2m-1}) + g(t_{2m-1})'(kh) + {\cal O}(h^2)\end{equation} where ${\cal O}(h^2)$ is a quantity that is a bounded multiple of $h^2$.
Why? How does $g(t_{2m-1+k})$ link that way to $g(t_{2m-1})$? Is there a property of $g(t_{2m-1} + t_k)$ that I'm missing and that leads to the above expansion?