指数多项式在半带形中的闭包

Closure of Exponential Polynomials in a Half Strip

该文对一类复指数多项式组成的线性空间M(A)在Banach空间H_α中的不完备性给出了充分必要条件,其中H_α为在半带形I_α={z=x+iy:x≥0,|y|≤α}(α＞0)中连续,在I_α的内部解析且当x→∞时,f(x+iy)在I_α中关于y一致地趋向0的函数f(x+iy)全体,其范数为上确界范数．同时指出,如果M(A)在H_α中不完备,则它的闭包cl(M(A))中所有的函数都可以延拓为由Dirichlet级数表示的解析函数。

In this paper, the author obtains a necessary and sufficient condition for the incompleteness of complex exponential polynomials in the Banach space Hα consisting of all continuous functions f on the closed right half strip Iα={z=x＋iy：x≥0,｜y｜≤α}（α〉0）, analytic in its interior and such that f（z） vanishing uniformly at infinity. The author also proves that, if some conditions of incompleteness hold, then each function in the closure of the set of complex exponential polynomials can be extended to an analytic function represented by a Dirichlet series.