You have a function $\varphi \in \mathcal{C}^0(\mathbb{R})$ that satisfies an equation of the type $$\varphi(x)=\sum_{k \in \mathbb{Z}}a_k\varphi(mx-k+\tau), \;\; x \in \mathbb{R},$$ where $m \in \mathbb{N}\setminus\{1\}$, $\tau \in [0,1) \cap \mathbb{Q}$ and $(a_k)_{k \in \mathbb{Z}} \subset \mathbb{R}$ is a compactly supported sequence. I would like to prove the following:
- If $\varphi$ is a compactly supported function we can consider it to have $\mathrm{supp}(\varphi)=[-s,s], \;(s>0)$ by changing the indexing and the parameter $\tau$.
- In particular, denoting $k_l=\mathrm{min}\{k \in \mathbb{Z} \, : \, a_k \neq 0\}$ and $k_r=\mathrm{max}\{k \in \mathbb{Z} \, : \, a_k \neq 0\}$, we can prove that $\varphi$ has symmetric support if and only if one of the following is true:
$$(a) \;\; \tau=0 \;\; \mathrm{and} \;\; k_l=-k_r$$
$$(b) \; \; \tau=1/2 \;\; \mathrm{and} \;\; k_l=1-k_r$$