Let g be integrable, even function. Let $\Lambda$ be uniformly discrete sequaence (i.e. $\inf|\lambda_i-\lambda_j|>0, \lambda_i, \lambda_j \in \Lambda$).
We say that system $\Psi=\{\psi_n\}$ is $(g, \Lambda)$-localized if $|\psi_n(t)|<cg(t-\lambda_n)$, where $c>0$ some constant.
Let function $f$ be continuous on $R$ with compact support and with exactly one maximum. We form the followinf system $\{f_{m,k}\}=\left\{f^m\left(t-\frac{k}{2^m}\right)\right\}, k\in Z, m>0.$
I would like to understand if the system $\{f_{m,k}\}$ is $(g, \Lambda)$-localized?
Thank you for your help.