The Properties And Perturbation Of K-g-frames And K-g-riesz Frames In Hilbert Space

Frame theory is an important component of wavelet analysis.Using frames can solve many problems in wavelet analysis,such as in the fields of compression sensing,image denoising,image processing,signal synthesis and restoration,which have extensive and effective applications.Therefore,designing a frame with good properties and structure is crucial.In this dissertation,we mainly study some properties of K-g-frames and K-g-Riesz frames in Hilbert space and their stability under perturbation.The research content mainly includes the following three aspects:(1)The properties and perturbation of K-g-frames in Hilbert space are studied.First,the original condition and the range of elements in the frame are changed,and a sufficient condition and a necessary condition of K-g-frame in Hilbert space are given,and then a sufficient and necessary condition of K-g-frame is obtained.Finally,the stability of the K-g-frame under the new perturbation is proved.(2)The properties and perturbation of K-g-Riesz frames in Hilbert space are studied.First,an equivalent condition for the K-g-Riesz basis is given,and it is pointed out that not all K-g-Riesz frames contain a K-g-Riesz basis.However,it can be inferred that each K-g-Riesz frame contains an exact K-g-frame.Secondly,the sufficient condition for the sum of a K-g-Riesz frame and a g-Bessel sequence to form a new K-g-Riesz frame are given.Finally,the stability of the K-g-Riesz frame under new perturbation is provided.(3)The properties of dual K-g-Riesz-Bessel sequence of K-g-Riesz frame in Hilbert space are studied.First,an equivalent condition for the K-g-Riesz frame is given,and then the concept of the dual K-g-Riesz Bessel sequence of the K-g-Riesz frame is proposed.Secondly,some properties of the dual K-g-Riesz-Bessel sequence of the K-g-Riesz frame are discussed.Finally,the canonical dual K-g-Riess Bessel sequences and all dual K-g-Riess Bessel sequences of the K-g-Riesz frame are presented.