Incompleteness and Minimality of Exponential System

Abstract:

      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.