Properties And Constructions Of Fusion Frames

The concept of frames in Hilbert spaces was first introduced by Duffin and Schaeffer when they studied some problems of nonharmonic Fourier series in 1952 and gave some properties of frames.The concept of frames was represented in 1986 by Daubechies,Grossmann and Meyer.After their work,the theory of frames began to be studied widely and deeply.Fusion frames(frames of subspaces),as a generalization frame,have a profound background in practical application.The concept of fusion frames was raised by Casazza and Kutyniok and the relative theory was gradually developed and perfected by later scholars.But there are some unsolved problems,such as some problems of stability of fusion frames under the operator perturbation.Studying the condition of the stability of the fusion frames under operator perturbation is currently a hot topic.Many important results about fusion frames have been obtained by many authors.We may get more fusion frames by applying the method of the operator perturbation,which is an adequate preparation for the construction of the fusion frames.Some scholars gave many methods of the construction of fusion frames,such as Spectrum Tetris method,orthogonal method,and the method to compile algorithm.These methods play an important role in fusion frames theory because they are extremely useful for applications.Therefore in this paper we focus on fusion frame in Hilbert spaces.Firstly,we introduce the basic concept of frames theory and fusion frames theory.Next we recall some results of the frames and fusion frames in Hilbert spaces.Secondly,some problems of the stability of the fusion frames under the operator perturbation are studied in this paper.Our results enrich the famous results of Casazza.We also give some properties of the fusion-Riesz frame and the exact fusion frames under the operator perturbation.Finally,we discuss several problems of construction of fusion frames,some properties of fusion frames and frames and obtain their equivalent characterizations.We discuss a problem how to construct a new frame by a frame and a fusion frame.