The Approximation Problem Of Random Function Systems In Weighted Banach Spaces

In this paper,we study the approximation problem of random function system｛eλn(ω)t} in the weighted Banach space Cα and also ｛eλn(ω)t} in C0(E).In the first part,under the hypothesis of infinity upper density of {λn} is allowed and change a weaker condition of the power function α(t)instead of being convex,by using the Khabibullin’s uniqueness theorem of entire functions,a new approach to obtain neces-sary and sufficient condition for the completeness of random exponential system in Ca is found.In another part,a method from random inverse problem helps to estimate the difference between an entire function depending on ｛λn(ω)} and its expect function.Thus we can use the known result about the completeness of ｛tλn} and ｛eλnlogmnt} in C0(E)to gain our goal in a probabilistic generalization.Cα and C0(E)are weighted Banach space of complex continuous functions f defined on R and E respectively with f(t)exp(-α(t))vanishing at infinity in the uniform norm,and where E = ∪=n1∞In,In =[an,bn],0<a1<b1<a2<b2<...<bn<...,limn→∞bn=∞.