Clarkson-Erdös-Schwartz theorem on a closed sector

Abstract

The analogue results of the Clarkson-Erdos-Schwartz theorem on a closed sector are obtained, i.e., some sufficient conditions are obtained for the incompleteness and minimality of the Muntz system $E(\\Lambda)$ in $~H_\\alpha~$ and each element in the closure of the linear span of Muntz system $E(\\Lambda)$ can be extended analytically throughout $\\mbox{int}(I_{~\\pi~})=\\{z:~|z|~<1,$ $~|\\!\\arg~z~|<~\\pi\\}$ with a series expansion of the form $\\suma_kz^{\\lambda_k}$, where $~H_\\alpha~$ is a Banach space consisting of all complex continuous functions $~f~$ on the closed sector$I_{~\\alpha}=\\{z:~|z|~\\leq~1,~|\\!\\arg~z~|\\leq~\\alpha~\\}$ $~(~0\\leq\\alpha~<\\pi~)~$, analytic in the interior of $I_{~\\alpha}$, andthe norm is given by $~\\|f\\|=\\max~\\{~|f(z)|:~z~\\in~I_{~\\alpha~}\\}$.