扇形区域中的Clarkson-Erd?s-Schwartz定理 献给余家荣教授100华诞

Clarkson-Erd?s-Schwartz theorem on a closed sector

本文得到区间中的Clarkson-Erdos-Schwartz定理在扇形区域中的类似结果,即得到Müntz函数系E(∧)={zλk}在空间Hα中不完备性和最小性的充分条件,以及在此条件下,Müntz函数系E(∧)线性生成的闭包span E(∧)中的每个元f可以解析开拓到扇形区域intIπ={z:|z|<1,|arg z|<π}中,且有形如∑akzλk的级数展开,其中Hα是所有在Iα={z:|z|≤1,|argz|≤α}(0≤α<π)中连续、在Iα的内部解析的函数f全体构成的Banach空间,其范数定义为‖f‖=max{|f(z)|:z∈Iα}.

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(A) in Ha and each element in the closure of the linear span of Müntz system E(A) can be extended analytically throughout int(Iπ)={z:|z| <1,|arg z| <π} with a series expansion of the form ∑akzλk,where Hα is a Banach space consisting of all complex continuous functions f on the closed sector Iα={z:|z|≤1,|arg z|≤a}(0≤α<π),analytic in the interior of Iα,and the norm is given by ‖f‖=max{|f(z)|:z ∈ Iα}.