The Study Of The Properties And Extension Of K-g-frames

In the Internet era,information and communication technologies and their applications are developing rapidly.This has benefited from the unremitting exploration and contributions of many scholars,among which the study of frame theory is an important research direction in this field.Frames was discovered in the 1950 s and then extended to Banach space and Hilbert space.Since then,in order to adapt to complex and changeable practical problems,different types of frames have emerged one after another.For example,g-frames,weaving frames,K-frames,fusion frames,etc.This paper takes the properties of K-frames and K-g-frames as research topics,focusing on the properties of K-frames,the construction method of K-g-frames,and the dual K-g-Bessel sequences,stability,and related equalities and inequalities,in order to enrich the research results of the existing frame theory.The main research content of this article is divided into the following parts:(1)We investigate the relevant properties of K-frames.First,we introduce the definition,operators,equivalence conditions and dual K-Bessel sequences of K-frames.Secondly,we get the relationship between K-frames and the traditional frames,and when the sets of different types of K-frames have inclusion relationships,we give the relationship between the range of related operators.Finally,we obtain an inequality for a tight K-frame in a finite-dimensional Hilbert space.(2)We investigate the construction of K-g-frames and the duality on closed subspace.First,we get the method of constructing K-g-frames from g-frames or K-g-frames,give some sufficient conditions of the sum of K-g-frames and g-Bessel sequences is still K-gframes,and give a special form of the sum of K-g-frames.Next,we study the duality of K-g-frames in the closed subspace,and obtain the canonical dual K-g-Bessel sequences of K-g-frames in the closed subspace and all the dual K-g-Bessel sequences.Finally,we introduce the definition of the approximate dual K-g-Bessel sequences,and use it to get the method of constructing K-g-frames and its dual K-g-Bessel sequences.(3)We investigate the stability of K-g-frames and its qualities and inequalities.First of all,on the basis of the existing stability conclusions,we give four other sufficient conditions,so that K-g-frames under this condition is still K-g-frames after perturbation.Then,we generalize the relevant conclusions of g-frames qualities and inequalities,and obtain the qualities of K-g-frames and its dual K-g-Bessel sequences,as well as new and more general inequalities.