Clarkson-Erdös-Schwartz Theorem on a Sector

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 $E(Λ)={z^{λ_n}:n=0,1,...}$ in the space $H_α$, where $H_α=A(I_α)$, $I_α={z\in\mathbb{C}:|\arg (z)|\leq α\text{and} |z|\leq 1}$ 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 $\textbf{span}[E(Λ)]$ is not dense in $H_α$ then all functions $f\in \bar{\textbf{span}}[E(Λ)]$ can be analytically extended to the interior of the sector $I_π$.

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