The system of weighted exponentials with power weights

The system of weighted exponentials $\left(\mid x\mid ^{\alpha }\exp^{i\lambda _{n}x}\right)_{n\in \mathbb{Z}}$ with power weights is considered in this paper, where $\lambda_{n}\in \mathbb{R}$. We find conditions on power indicator $\alpha$ which provide such properties of the system as to be a frame, Bessel system, Schauder basis in the space $L^{2}[-\pi ,\pi ]$.