On exact systems $\{t^{\alpha }\cdot e^{2\pi int}{\}}_{n\in \mathbb{Z}\setminus A}$ in $L^2(0,1)$ which are not Schauder Bases and their generalizations

Let $\{e^{i{\lambda }_{n}t}{\}}_{n\in \mathbb{Z}}$ be an exponential Schauder Basis for $L^2(0,1)$, for ${\lambda }_n\in \mathbb{R}$, and let $\{r_n(t){\}}_{n\in \mathbb{Z}}$ be its dual Schauder Basis. Let $A$ be a non-empty subset of the integers containing exactly $M$ elements. We prove that for $\alpha >0$ the weighted system $$\{t^{\alpha }\cdot r_n(t){\}}_{n\in \mathbb{Z}\setminus A}$$ is exact in the space $L^2(0,1)$, that is, it is complete and minimal in $L^2(0,1)$, if and only if $$M-\frac{1}{2}\le \alpha <M+\frac{1}{2}\mathrm{.}$$ We also show that such a system is not a Riesz Basis for $L^2(0,1)$. In particular, the weighted trigonometric system $\{t^{\alpha }\cdot e^{2\pi int}{\}}_{n\in \mathbb{Z}\setminus A}$ is exact in $L^2(0,1)$, if and only if $\alpha \in [M-\frac{1}{2},M+\frac{1}{2})$, but it is not a Schauder Basis for $L^2(0,1)$.