Some Problems Of Weak Gabor Dual Frames On Periodic Sets

The theory of frames is an important branch of wavelet analysis.It is one of core issues in the frame theory of function spaces to construct dual frame pais with beautiful structure and high efficiency in computation.In the last thirty years,the theory of wavelet and Gabor frames on the whole space L~2(R)has seen great achievements,and the one on subspaces also has made some exciting progress.The present dissertation introduces the notions of special weak Gabor dual frames under the setting of different subspaces,addresses their characterizations,con-struction and uniqueness,etc..It includes the following two aspects:weak Gabor dual frame pairs on discrete periodic subsets;weak Gabor dual frame pairs on periodic subsets of the real line.Chapter 2 introduces the notion of weak Gabor dual frame pairs of type I in the square-summable sequence spaces defined on discrete periodic subsets,and presents a Zak transform domain characterization and a time domain characteri-zation of weak Gabor dual frames of type I.Chapter 3 introduces the following three notions in the vector-valued square-summable sequence spaces defined on discrete periodic subsets:weak Gabor dual frame pairs of type I,II,and weak oblique Gabor dual frame pairs.We obtain a Zak transform matrix-based characterization of these weak Gabor dual frames.Given a Gabor system,we characterize the uniqueness of its weak Gabor duals,and give an explicit expression of its weak Gabor duals of type I and II.Chapter 4 introduces the notion of weak Gabor dual frame pairs of type I in the vector-valued square integrable function spaces defined on periodic subsets of the real line,and presents its one Zak transform matrix-based characterization.Simultaneously,a density theorem for weak Gabor dual frame pairs of type I is obtained.Chapter 5 introduces the notion of weak Gabor dual frame pairs of type II in the vector-valued square integrable function spaces defined on periodic subsets of the real line.Using Zak transform matrix method,we characterize weak Gabor dual frame pairs of type II,and characterize the uniqueness of its weak Gabor dual frames of type II for an arbitrarily given Gabor system.Chapter 6 presents a time domain characterization of weak Gabor dual frame pairs of type I in Chapter 4.Zak transform matrices throughout this dissertation are all function matri-ces of finite order.Therefore,all Zak transform matrix-based characterization theorems,dual expression theorems and uniqueness theorems reduce to designing function matrices of finite order.We include examples in each chapter to illustrate the operability of the theorems.