On the completeness of the system of function \(\{e^{-\alpha \lambda_ nx}\sin \,\lambda_ nx\}^{\infty}_{n=1}\) (Q2640087)

The author finds the corresponding biorthogonal system of the system \(\{e^{-\alpha nx} \sin nx\}^{\infty}_{n=1}\) in an appropriate form which allows to obtain good estimates of its norm. Using the stability of the completeness property the completeness of the system \(\{e^{-\alpha \lambda_ nx} \sin \lambda_ nx\}^{\infty}_{n=1},\) if \(\lambda_ n=n+O(1/n^{\epsilon}),\quad \epsilon >0,\) in \(L^ p(0,\pi)\) \((1\leq p\leq 2,\quad \alpha >0)\) is proved. This solves a problem of A. G. Kostyuchenko.