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

TL;DR

This paper characterizes when weighted systems formed by powers of t times dual bases are exact but not Schauder bases in L^2(0,1), revealing precise conditions on the weight exponent related to the size of missing index sets.

Contribution

It provides necessary and sufficient conditions for the exactness of weighted exponential systems in L^2(0,1) and shows these systems are not Riesz or Schauder bases, extending understanding of basis properties.

Findings

Weighted systems are exact iff alls within specific intervals related to the size of missing sets.

Such systems are not Riesz bases in L^2(0,1).

Weighted trigonometric systems are exact but not Schauder bases under certain conditions.

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 \[ \{t^{\alpha}\cdot r_n(t)\}_{n\in\mathbb{Z}\setminus A} \] 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 $L^2 (0,1)$.