On subsystems of a trigonometric system which are complete on the set of positive measure (Q1921494)

An orthonormal system \(\{\phi_n(x)\}\) on \([a,b]\) is said to be \(L_2\)-series complete on a measurable set \(E\subset[a, b]\) of positive measure if for every function \(f\in L_2(E)\) there exists a series \[ \sum^\infty_{n= 1}a_n\phi_n(x),\quad \sum| a_n|^2< \infty, \] which converges to \(f\) in the metric of \(L_2(E)\). The following result is proved: For any increasing sequence \(\{k_n\}^\infty_{n= 1}\) of natural numbers such that \(\limsup_{n\to\infty} (k_{n+ 1}- k_n)= \infty\), the system \[ \{\sin n_jx\}^\infty_{j= 1}:= \{\sin nx\}^\infty_{n= 1}\backslash\{\sin 2^{k_n}x\}^\infty_{n= 1} \] is \(L_2\)-series complete on any set \([0,\pi]\backslash(a, b)\), where \((a,b)\subset [0,\pi]\) and \(b-a<\pi\).