On exact systems {tα⋅e2πint}n∈Z∖A in L2(0,1) which are weighted lower semi frames but not Schauder bases, and their generalizations

Abstract

Let ${\left\{e^{i{\lambda }_{n}t}\right\}}_{n\in Z}$ be an exponential Schauder basis for $L^2(0,1)$, where ${\lambda }_n\in R$, and let ${\left\{r_n\right(t\left)\right\}}_{n\in Z}$ 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 ${\{t^{\alpha }\cdot r_n(t\left)\right\}}_{n\in Z\setminus A}$ 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 $\alpha \in [M-\frac{1}{2},M+\frac{1}{2})$. We also show that such a system is not a Riesz basis for $L^2(0,1)$.

In particular, the weighted trigonometric system ${\{t^{\alpha }\cdot e^{2\pi int}\}}_{n\in Z\setminus A}$ is exact in $L^2(0,1)$, if and only if $\alpha \in [M-\frac{1}{2},M+\frac{1}{2})$, but this system is not even a Schauder basis for $L^2(0,1)$. This result extends the ones obtained by Heil and Yoon (2012) who considered a similar problem when α is a positive integer.

The non basicity of ${\{t^{\alpha }\cdot e^{2\pi int}\}}_{n\in Z\setminus A}$ in combination with results of Heil et al. (2023), yields that for any $\alpha \ge 1/2$, the overcomplete system ${\{t^{\alpha }\cdot e^{2\pi int}\}}_{n\in Z}$ does not have a reproducing partner for $L^2(0,1)$. Nevertheless this overcomplete system is a weighted lower semi frame for $L^2(0,1)$. This follows from recent results of ours, where we proved that any exact system in a Hilbert space $H$ is a weighted lower semi frame for $H$. For the sake of completeness, we reprove here that result.

We point out that the invertibility of Vandermonde matrices plays a crucial role for the above systems to be exact as well as for their non-basicity.