Clarkson-Erdös-Schwartz Theorem on a Sector

Abstract

We prove a Clarkson-Erdös-Schwartz type theorem for the case of a closed sector in the plane. Concretely, we get some sufficient conditions for the incompleteness and minimality of a Müntz system $E(\Lambda )=z^{{\lambda }_n}:n=0,1,...$ in the space $H_{\alpha }$, where $H_{\alpha }=A(I_{\alpha })$, $I_{\alpha }=z\in \mathbb{C}:|\arg (z)|\le \alpha \text{and}|z|\le 1$ and $A(K)=C(K)\cap H(\text{Int}[K])$ denotes the space of continuous functions on the compact set $K$ which are analytic in the interior of $K$. Furthermore, we prove that, if $\text{span}[E(\Lambda )]$ is not dense in $H_{\alpha }$ then all functions $f\in \overline{\text{span}}[E(\Lambda )]$ can be analytically extended to the interior of the sector $I_{\pi }$.

11 pages, 0 figures

References in corpus (1)

• Muntz type Theorems I