  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 \[
M-\frac{1}{2}\le \alpha< M+\frac{1}{2}. \] We also show that such a system is
not a Riesz Basis for $L^2 (0,1)$. In particular, the weighted trigonometric
system $\{t^{\alpha}\cdot e^{2\pi i n t}\}_{n\in\mathbb{Z}\setminus A}$ 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 $L^2 (0,1)$.
