Some Equalities And Inequalities Of Continuous G-bessel Sequences

The concept of frames in Hilbert spaces was formally introduced by Duffin and Scha-effer in 1952 for studying some deep problems in nonharmonic Fourier series.However,it didn't attract the attention of many scholars at that time.Until more than 30 years later,Daubechies,Grossmann and Meyer recalled the concept of frames,the frame theory was widely promoted and applied.A frame in a Hilbert space has some properties similar to an orthonormal basis for expanding arbitrary elements in a stable manner,but rep-resenting method is not the only(redundancy).Therefore,a frame don't have a unique characteristic of a basis when they represent an element.But the redundancy offered by frames has spurred their application in a variety of fields,such as signal and image processing,neural network,sampling theory and so on.With the further study of the frame theory,some scholars generalize the frames to various forms,such as continuous frame,generalized frame,K-frame and so on.Espe-cially,in 2008,Dehghan and Hasankhani Fard put forward the concept of continuous generalized frames for the first time on the basis of continuous frame and generalized frame.Then,there were a lot of studies on continuous generalized frame,scholars have tried to make some results of the frames extended to the continuous generalized frames,including some properties of the continuous generalized frame operators,equalities and inequalities of continuous generalized frames.It is well known that a frame must be a Bessel sequence.Therefore,some scholars focus on Bessel sequences.Based on some equalities and inequalities of frames,we establish equalities and inequalities for contin-uous generalized Bessel sequences in Hilbert spaces,and give some new equalities and inequalities for continuous generalized frames in this thesis.Further,the continuous gen-eralized frame's inequalities are extended to a more general form,which contains weight T.Finally,the Aldroubi criterion of continuous frames is discussed based on the existing Aldroubi criterion of frames.The thesis consists of five chapters.Chapter 1 states the development history of frames's inequalities.And then it s-ketches the structure and works of the thesis.Chapter 2 lists the definitions of continuous frames and generalized frames,and reviews some basic properties of the continuous generalized frames.Chapter 3 is one of the main contents.Using the bounded linear operator theory,we establish some equalities and inequalities of continuous generalized Bessel sequences in Hilbert spaces.Chapter 4 is also the main contents of the thesis.We try to make some equalities and inequalities of frames extended to the continuous generalized frames,and give some new equalities and inequalities for continuous generalized frames.Finally,we obtain some weight inequalities with weight ?.If ? = 1/2,these inequalities are the existing conclusions.Chapter 5 is the third work.It studies the Aldroubi criterion of continuous frames.