Systems of dilated functions: Completeness, minimality, basisness

The completeness, minimality, and basis property in $L^2[0,\pi]$ and $L^p[0,\pi]$, $p\neq 2$, are considered for systems of dilated functions $u_n(x)= S(nx)$, $n \in \mathbb{N}$, where $S$ is the trigonometric polynomial $S(x)=\sum_{k=0}^m a_k\sin(kx)$, $a_0 a_m \neq 0$. A series of results are presented and several unanswered questions are mentioned.