Systems of Dilated Functions: completeness, minimality, basisness

Abstract

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

4 pages