Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $ One can show that $\text{span}\{f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right), k \in \mathbb{Z}, m>0\}$ is dense in $L_p(R)$.
Under which conditions does $\{f_{m,k}\}$ form a frame (or maybe a Riesz basis) for $L_p(R)$?
Thank you.