A Class Of Gabor Frames On The Half Real Line

During past more then thirty years,wavelet and Gabor analysis on L2(R)have seen great achievements,but the study of time-frequency on the half real line R+ has been rarely reported.It is because R is a locally compact Abelian group under addition and usual topology,while R+ is not.Motivated by Gabor analysis on locally compact groups,this thesis addresses the theory of Gabor frames on L2(R+,1/xdx).We introduce the concept of Gabor systems in L2(R+,1/xdx),and investigate its frame properties and characterization of dual frames.this thesis is organized as follows.Chapter 1 is an introduction,states the backgrounds and main rusults of this thesis.Chapter 2 focuses on some auxiliary lemmas.we present the commutative properties of some operates;present some orthonomal bases for special spaces;and obtain some basic properties of B-splines in L2(R+,1/xdx).Chapter 3 focuses on Gabor frames for L2(R+,1/xdx).some necessary and suf-ficient conditions and some examples for Gabor frames are given.Chapter 4 characterizes dual Gabor frames in L2(R+,1/xdx).Chapter 5 characterizes Gabor frames and their Gabor duals using the operator methods,and present duality principles for Gabor frames in L2(R+,1/xdx).