A Research On Fredholm Frames And (p,Y)-Operator Frames

The theoty of frames is an important part of wavelet analysis. In 1952, Duffin and Schaeffer generalized Gabor' ideas and firstly introduced the concept of frames in a Hilbert space. From then on, more attention were paid to this subject. In this thesis, Fredholm frames are introduced and discussed, stabilities of frames for a Hilbert space are studied, some important properties and stabilities of (p, Y)-operator frames are obtained.This paper is divided into four chapters, the followings are the structure and the main contents of the paper:In Chapter 1, we introduce the history and development of the theory on frames and list some preliminaries which will be used in following chapters. Our main research work is summaried finally.In Chapter 2, Fredholm frames for a Hilbert space H are introduced by com-bining Fredholm operators and theory of frames, which are special frames between frames and Riesz bases; By using operator theory method, some important prop-erties and equivalent characterizations of Fredholm frames are obtained; Based On the theory of operator decomposition, Fredholm frames are decomposed as the unin of linear combination of two orthogonormal basis and a finite number of vectors; It is proved that the set of all Fredholm frames for a Hilbert space H is an open set in the Banach space consisting of all Bessel sequences in H, it is proved that Fredholm frames are stable under small and operator perturbations and inflations of frames and Riesz bases are also discussed.In Chapter 3, several inflation forms of frames are introduced, the stabilities of Bessel sequences, frames and tight frames are also discussed.In Chapter 4, we study the stability of (p, Y)-operator frames. We firstly discuss the relations between p-Bessel sequences(resp. p-frames) and (p, Y)-operator Bessel sequences(resp. (p, Y)-operator frames); Through defining a new union, we prove that adding some elements to a given (p, Y)-operator frame, the resulted sequence will be still a (p, Y)-operator frame, and we obtain a necessary and sufficient condi-tion for a sequence of compound operators to be a (p, Y)-operator frame; Lastly, we show that (p, Y)-operator frames for X are stable under some small perturbations.