Smooth dyadic decompositions of unity are classical and very applicable tools in analysis. This is usually an identity of the form $$ \sum_{j \in \mathbb{Z}} \varphi(t/2^j) = 1 $$ for all $t \in \mathbb{R}$ (except possibly when $t=0$) and some $\varphi$ smooth compactly supported on (say) $[2^{-1}, 2]$ and with bounded derivatives. Thus $\varphi(t/2^j)$ is supported on $[2^{j-1}, 2^{j+1}]$.
I wonder how one could go about generalizing this to the case when $2$ is replaced with some large $q$. Thus something of the form $$ \sum_{j \in \mathbb{Z}}\varphi(t/q^j) = 1. $$ However I want that each $\varphi(t/q^j)$ has derivatives decaying proportionately with the length of its support. If one were to follow the dyadic example above, one should expect $\varphi(t/q^j)$ to be supported on an interval of length $\asymp q^{j+1}$ but its first derivative is $\ll 1/q^j$ (if I'm not mistaken). Can we construct $\varphi$ in such a way that we also get $\|(\varphi(\cdot/q^j))^{(k)}\|_\infty \ll_k |\text{supp}(\varphi(\cdot/q^j))|^{-k}$ for each $k \geq 0$. Here $|\text{supp}(\varphi(\cdot/q^j))| $ denotes the length of the support of $t \mapsto \varphi(t/q^j)$. Thanks.