Prove that there exists $\varphi \in \mathscr{S}(\mathbb{R}^n)$(Schwartz space), satisfying $0\le \varphi \le 1$, $\text{supp}(\varphi)=\{x\in\mathbb{R}^n|\frac{1}{2}\le |x|\le 2\}$, such that $$ \sum_{k=-\infty}^{\infty} \varphi(2^{-k}x)=1, x\ne 0. $$
This question is relevant to partition of unity, I think there are only finite terms in above equality.
Thanks for your help!
Construction of $\varphi$
Take $\rho \in C_c^\infty(\mathbb R)$ such that $\operatorname{supp}\rho = [0, 1],$ $0 \leq \rho \leq 1$ and $\int \rho(t) \, dt = 1.$
Let $\sigma(t) = \int_{-\infty}^{t} \rho(s) \, ds.$ Then $\sigma \in C^\infty(\mathbb R),$ $\sigma(t)=0$ for $t \leq 0$ and $\sigma(t)=1$ for $t \geq 1.$ Furthermore, $\sigma^{(k)}(0) = 0 = \sigma^{(k)}(1)$ for $k=1,2,3,\ldots$
Define $\psi \in C^\infty(\mathbb{R})$ by $$ \psi(t) = \begin{cases} 0, & t<-1 \\ \sigma(t+1), & -1 \leq t < 0 \\ 1-\sigma(t), & 0 \leq t < 1 \\ 0, & t \geq 1 \end{cases} $$ Then $\psi \in C^\infty(\mathbb R),$ $0 \leq \psi \leq 1$, $\operatorname{supp}\psi = [-1, 1]$ and $\sum_{k\in\mathbb Z} \psi(t-k) = 1.$
Finally, set $\varphi(x) = \psi(\log_2|x|).$ Then $\varphi$ fulfills the requested conditions.