I think this is asked as a standard exercise in books about wavelets (e.g. exercise 7.2 in Mallat's book), but I couldn't find a proof. Let $\phi$ be a scaling function (see definition below). I would like to learn why
$$\sum_{k\in\mathbb Z} \phi(x-k) = 1 $$
almost everywhere.
Definition. A sequence of subspaces $\{V_j: j\in \mathbb{Z}\}$ of $L^2(\mathbb R)$ is called a multiresolution analysis if it satisfies the following:
- $V_j \subset V_{j+1}$
- $\bigcap_{j}V_j = \{0\}$
- $\overline{\bigcup_jV_j} = L^2(\mathbb R)$
- $f(x)\in V_j$ if and only if $f(2x) \in V_{j+1}$
- There exists a function $\phi \in V_0$ such that $\{\phi(x-k)\}_{k\in\mathbb Z}$ is an orthogonal basis for $V_0$
The function $\phi$ here is called as a scaling function.