Incompleteness and minimality of complex exponential system

A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(A,M)in C_α,where C_αis the weighted Banach space consisting of all complex continuous functions f on the real axis R with f(t)exp(-α(t))vanishing at infinity,in the uniform norm‖f‖_α=sup{|f(t)e~(-α(t))|:t∈R}with respect to the weightα(t).If the incompleteness holds, then the complex exponential system E(?)is minimal and each function in the closure of the linear span of complex exponential system E(?)can be extended to an entire function represented by a Taylor-Dirichlet series.

Incompleteness and minimality of complex exponential system

A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(Λ, M) in C _α , where C _α is the weighted Banach space consisting of all complex continuous functions f on the real axis ℝ with f(t) exp(−α(t)) vanishing at infinity, in the uniform norm ∥f∥ _α = sup{|f(t)e ^−α(t)| : t ∈ ℝ} with respect to the weight α(t). If the incompleteness holds, then the complex exponential system E(Λ, M) is minimal and each function in the closure of the linear span of complex exponential system E(Λ, M) can be extended to an entire function represented by a Taylor-Dirichlet series. This work was partially supported by the Research Foundation for Doctor Programme (Grant No. 20060027023) and the National Natural Science Foundation of China (Grant No. 10671022)