The Study Of The Properties And Extension Of Weaving Frames

It is a new research direction called frames theories,which studied in wavelet analysis in Hilbert space.As all know,every vector in Hilbert space can be represented by basis.Frames is an extension of basis which is linearly related and redundant in some sense.Because of these properties,frames play an important role in some areas such as signal transmission and image processing.At present,there are many new types of frame theories,such as g-frames,fusion frames,K-frames and so on.In recent years,the research on the properties and application of weaving frames has become a new direction in frame theories.Weaving frames are obtained by selecting a limited traditional frames which are defined in the same set I,dividing the set I and reorganizing the limited traditional frames from the divided subsets.Because of the particular construction of weaving frames,weaving frames can be combined with other frames,such as weaving g-frames,weaving fusion frames,weaving K-frames and so on.As we know,the research about weaving frames is few.In this paper,we will study some properties about weaving frames and its extension.The main contents are as follows:1.We studied some properties about weaving frames.The study of frames is mainly on operator and dual.We gave some expression of analysis operator and frames operator in weaving frames,according to the relationship between weaving frames and frames.And then,we studied the dual in weaving frames and gave the relationship between the dual of weaving frames and the weaving of dual frames.In additon,we gave the sufficient condition when the relationship is equivalent.Futhermore,by adding a turbulence to weaving frames,we proposed some conditions so that the frames are weaving frames under turbulence.2.We studied some properties about weaving g-frames.First,we got some properties in weaving frames.For example,we learned how to obtain weaving g-frames.Secondly,as we know,the property of weakly weaving is important in weaving frame.In weaving g-frames,weakly weaving is equal to weaving after deduction.Finally,we gave some new equalities and inequalities,which are more generalized.Based on these equalities and inequalities,we could be more easier to construct and propose some properties of signal transmission in weaving g-frames.