Completeness and basicity for a certain system of functions (Q1087771)

The author establishes necessary and sufficient conditions for the Riesz basis property with finite defect in \(L^ 2(0,\pi)\) for the following system of functions in \(L^ 2(0,\pi):\mu \sin nx+\sum^{N}_{k=1}c_ ke^{-a}k^{nx}\) \((n=1,2,...)\), and \(\mu e^{- 2inx}+\sum^{N}_{k=1}c_ ke^{-a}k^{nx}\) \((n=1,2,...)\), \(\mu e^{2inx}+\sum^{N}_{k=1}d_ ke^{-b}k^{nx}\) \((n=0,1,...)\), where Re \(a_ k>0\), Re \(b_ k>0\); \(c_ k\), \(d_ k\in C.\) After establishing the completeness of the system with finite defect in \(L^ 2(0,\pi)\) for any \(\mu\neq 0\), the author applies these results to solve the basis problem of certain quadratic bundles of differential operators.