Systems of Dilated Functions: completeness, minimality, basisness

We discuss completeness, minimality, and basisness, in $L^2[0, π]$ and $L^p[0, π]$, $p \neq 2$, of dilated systems $u_n(x) = S(nx)$, $n \in \mathbb{N}$, where $S$ is a trigonometric polynomial $S(x) = \sum_{k = 0}^m a_k \sin(kx), \quad a_0 a_m \neq 0.$ We will present some results and mention a few unsolved questions.

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