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

TL;DR

This paper extends the Clarkson-Erd"os-Schwartz theorem to sectors in the complex plane, providing conditions for the incompleteness and minimality of M"untz systems in specific function spaces, and explores analytic extension properties.

Contribution

It establishes a sector-specific version of the theorem, offering new criteria for M"untz system properties and analytic extension in complex sectors.

Findings

Conditions for M"untz system incompleteness in sectors

Characterization of minimality of M"untz systems

Analytic extension of functions in the span of M"untz systems

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 $I_\pi$.