无穷远处趋于零但非平方可积的实整函数

Non-square integrable real entire function which tending to zero at infinity

如果一个整函数限制在实轴上取实值,我们称这个整函数是实整函数.我们给出一类满足xl→im∞f(x)=0但在实轴上不是平方可积的实整函数f(x)的例子.特别地,我们的结果表明,xl→im∞f(x)=0是型不超过π的指数型整函数f(z)属于Paly-Wiener空间的必要而非充分条件.

An entire function is called real if it maps the real line into itself.In this paper,by using the classical theory of entire function,we give an explicit construction of real entire functions which tends to zero at infinity but not integral on the real line.Our results show that an exponential entire function tends to zero at infinity is not the sufficient condition for it belong to the Paly-Winer space.