On Approximation Problems In Complex Banach Spaces

In this thesis,we consider the completeness of some function systems in several weighted Banach spaces.We study the completeness of function system with infinite multiplicities in the Banach space investigated.The main contents are as follows:In the second chapter of this thesis,we suppose that B is a Non-Carathéodory region which consists of simply connected region Bl(l=1,2,...,s),i.e.,B= ?l=1sBl.Furthermore,let ?(z)be a non-negative continuous function defined on R.And ?={?n:n=1,2,...} is a sequence of distinct complex numbers.Denote by ?1={?n,sn}n=1? where Sn?0,1,...,mn-1 are the multiplicity of ?n.The completeness of {z?n logsn z} in Lp(B)is investigated,wherever the multiplicity sn is allowed to tend to infinity as n tends to infinity.In the third chapter of this thesis,let C{e-?(z)]be the Banach space defined on the measurable set E,and the measurable set E defined on an unbounded curve C.Furthermore,let a(z)be a non-negative continuous function defined on R.And ?={?n:n=1,2,...} is a sequence of distinct complex numbers.Denote by ?1={?n,sn}n=1? where sn = 0,1,...,mn-1 are the multiplicity of ?n,The completeness of {z?n logsn z} in Banach space C[e-?(z)]which defined on the measurable set E is investigated,wherever the multiplicity sn is allowed to tend to infinity as n tends to infinity.In the fourth chapter of this thesis,we discuss that when the multiplicity sn is al-lowed to tend to infinity,the completeness of multivariate exponential system in Banach space Ca is investigated.