A Systematic Study Of Gabor Frames

Since Gabor proposed that signals can be decomposed by basic functions,many scholars have devoted themselves to the research of basis and frame.A kind of frames in the frame theory is Gabor frame,which has further promoted the development of frame theory.In this paper,we mainly discuss the Gabor frame,which is a generalization of standard orthogonal basis and a set of vectors satisfying certain characteristics in Hilbert space.A wave of research upsurge followed one another,since Gabor frame theory was emphasized.In the 20th century,researchers also gave the concept of frame on direct and sum space.Firstly,this paper introduces algebraic theory and operator theory related to Gabor frame theory,especially some properties of Hilbert space and Lp space,basic theory and important properties of linear basis and lattice,and thus leads to the concept of frame.It can be seen from various theories that there is a close relationship among the frame tight frame normal tight frame and Riesz basis.Secondly,the Gabor frame is systematically discussed for vector(g,a,b),which conditions are satisfied to be Gabor frame and which conditions are ort-hogonal Gabor frame.The Gabor frame identity theorem and the characteriza?tion of Gabor frame are given.The condition of whether the vector satisfies Gabor frame is analyzed more clearly and simply.Finally,some properties of Gabor frame are given.Finally,we construct a super Gabor frame(especially when a=1 an orth-ogonal super Gabor frame,K,L is a lattice),under the definition of timefre-quency lattice K=aZ,L=bZ,and give the concept and necessary and sufficient conditions of matrix lattice transforme operator.Two super Gabor frames are lattice isomorphic because of matrix lattice transform operators.Based on our method of constructing super Gabor frames,matrix lattice transform operators can help us better understanding the basic theory of super Gabor frames.