Study On Generalizations Of Fusion Frames In Hilbert Spaces

Fusion frames analyze signals by projecting them onto multidimensional subspaces in Hilbert spaces.Fusion frames are a generalization of frames,which are weighted and dis-tributed processing procedures that fuse together information in all subspaces of a Hilbert space,and also can handle some large systems which are impossible to handle efficiently by just a simple frame.In this thesis,we focus on the generalizations of fusion frames,the following two parts will be the main content of study.The first part is a systematic study of controlled g-fusion frames.Starting from g-fusion frames and operators with special properties,controlled g-fusion frames are constructed.Firstly,the definition of controlled g-fusion frame is introduced,several properties of con-trolled g-fusion Bessel sequences are discussed.Then,some sufficient conditions and some characterizations of controlled g-fusion frames are studied.Finally,the sum of operators of controlled g-fusion frames is discussed under positive constants and operators with special properties in Hilbert spaces.The study of weaving K-fusion frames is in the second part.The weaving K-fusion frames are studied by basic characteristics of weaving frames and K-fusion frames.Firstly,the notion of weaving K-fusion frames is proposed and several approaches for constructing weaving K-fusion frames are given.Secondly,some novel characterizations and properties about weaving K-fusion frames are given,and the relation between weaving K-frames and weaving K-fusion frames are discussed.Finally,an example is presented to prove that weav-ing K-fusion frames are not usually transitive,and then a sufficient condition of transitivity of weaving K-fusion frames is presented.