F_a-dual Frames In L~2(r_+)

The theory of Fa-frames is based on the Fa-inner product on L2(R+).Although the Fa-inner product has many properties similar to those of the usual inner product,it is essentially different from the usual inner product.The result of usual inner product is a number,but the result of the Fa-inner product is a function.Therefore,it is a natural problem whether the results on the usual frames can be established on the Fa-inner product-based L2(R+)-frames.The literature shows that this problem is not trivial.Given an Fa-frame for L2(R+),this thesis addresses the characterization and the expression of its Fa-dual frames.This thesis is organized as follows.Chapter 1 is the introduction of this thesis,it introduces the research background and the main results.Chapter 2 focuses on some auxiliary lemmas,using the Fa-analysis operator and the Fa-synthesis operator we characterize Fa-dual frames,and using a-factorable unitary operators we establish the connection between the Fa-Bessel sequences(Fa-complete sequences,Fa-frames,Fa-Riesz bases,Fa-orthonormal systems,Fa-orthonormal bases)in L2(Z x[1,a))and those in L2(R+).Chapter 3 is devoted to canonical Fa-dual frames.Given an Fa-frame,we proves that its Fa-frame operator is invertible,and using its inverse we obtain its canonical Fa-dual frame.And simultaneously,we show that the canonical Fa-dual frame corresponds to the minimal pointwise l2-norm representation of functions in L2(R+).Chapter 4 addresses the operator characterization of Fa-frames.Using a-factorable operators we establish the link between Fa-frames and Fa-orthonormal bases.Specifically,we prove that an Fa-Bessel sequence(Fa-frame,Fa-Riesz basis)is the image of an Fa-orthonormal basis under an a-factorable bounded(bounded surjective,bounded invertible)operator.Based on the results of Chapter 4,Chapter 5 presents an explicit expression of Fa-dual frames.