关于满足自相似方程的紧支连续函数对称支集等价条件的证明问询

Hey everyone, I'm tackling a problem involving a continuous compactly supported function $\varphi \in \mathcal{C}^0(\mathbb{R})$ that satisfies the following self-similarity equation:

$$\varphi(x)=\sum_{k \in \mathbb{Z}}a_k\varphi(mx-k+\tau), ;; x \in \mathbb{R},$$

Here are the given constraints:

• $m \in \mathbb{N}\setminus{1}$ (so $m$ is a natural number greater than 1)
• $\tau \in [0,1) \cap \mathbb{Q}$ (a rational number in the half-open interval $[0,1)$)
• The sequence $(a_k)_{k \in \mathbb{Z}} \subset \mathbb{R}$ is compactly supported (only finitely many non-zero terms)

I'm trying to prove two related statements here:

1. First, if $\varphi$ is compactly supported, we can reindex the sequence and adjust the parameter $\tau$ such that we can assume $\mathrm{supp}(\varphi)=[-s,s]$ for some $s>0$.
2. More specifically, let $k_l=\mathrm{min}{k \in \mathbb{Z} , : , a_k \neq 0}$ (the smallest index with non-zero coefficient) and $k_r=\mathrm{max}{k \in \mathbb{Z} , : , a_k \neq 0}$ (the largest index with non-zero coefficient). I need to show that $\varphi$ has symmetric support if and only if one of the following holds:
   • (a) $\tau=0$ and $k_l=-k_r$
   • (b) $\tau=1/2$ and $k_l=1-k_r$

I'd appreciate any hints, proofs, or relevant insights to help me work through this!

备注：内容来源于stack exchange，提问作者AndreaBaleani