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

Abstract

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

8 pages