Incompleteness and Minimality of Exponential System

DOI：10.3770/j.issn:1000-341X.2011.02.004

 作者 单位 柯思宇 北京师范大学数学科学学院, 北京 100875 邓冠铁 北京师范大学数学科学学院, 北京 100875

本文对指数函数系 $E(\Lambda, M)=\{z^l e^{\lambda_n z}:l=0,1,\cdots,m_n-1;\ n=1,2,\cdots\}$在半带形中某些解析函数组成的Banach空间 $E^2[\sigma]$ 中的不完备性和极小性给出了充分条件和必要条件, 指出了若$E(\Lambda, M)$ 在 $E^2[\sigma]$ 中不完备, 则它的线性组合的闭包中的任意函数都可以延拓为由 Taylor-Dirichlet 级数表示的解析函数. 并通过共形映射 $\zeta=\phi(z)=e^z$, 对于幂函数系$F(\Lambda,M)=\{(\log\zeta)^l \zeta^{\lambda_n}:l=0,1,\cdots,m_n-1;n=1,2,\cdots\}$ 在扇形中某些解析函数组成的Banach空间 $F^2[\sigma]$ 中的不完备性和极小性也得出了类似的结果.

Necessary and sufficient conditions are obtained for the incompleteness and the minimality of the exponential system $E(\Lambda, M)=\{z^l e^{\lambda_n z}: l=0,1,\ldots,m_n-1; n=1,2,\ldots\}$ in the Banach space $E^2[\sigma]$ consisting of some analytic functions in a half strip. If the incompleteness holds, each function in the closure of the linear span of exponential system $E(\Lambda, M)$ can be extended to an analytic function represented by a Taylor-Dirichlet series. Moreover, by the conformal mapping $\zeta=\phi(z)=e^z$, the similar results hold for the incompleteness and the minimality of the power function system $F(\Lambda,M)=\{(\log\zeta)^l \zeta^{\lambda_n}:l=0,1,\ldots,m_n-1; n=1,2,\ldots\}$ in the Banach space $F^2[\sigma]$ consisting of some analytic functions in a sector.