Clarkson-Erd\"os-Schwartz Theorem on a Sector

Abstract

We prove a Clarkson-Erd\"os-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 M\"untz system $E(\Lambda)={z^{\lambda_n}:n=0,1,...}$ in the space $H_\alpha$, where $H_\alpha=A(I_\alpha)$, $I_\alpha={z\in\mathbb{C}:|\arg (z)|\leq \alpha \text{and} |z|\leq 1}$ and $A(K)=C(K)\cap H(\textbf{Int}[K])$ denotes the space of continuous functions on the compact set $K$ which are analytic in the interior of $K$. Furthermore, we prove that, if $\textbf{span}[E(\Lambda)]$ is not dense in $H_\alpha$ then all functions $f\in \bar{\textbf{span}}[E(\Lambda)]$ can be analytically extended to the interior of the sector $I_\pi$.