Perturbation And Equivalence Of Generalized Frames

Perturbation and Equivalence of Generalized FramesYao Xi-yanAbstract: Frames for a Hilbert space H were formally defined by R. J. Duffin and A. G. Schaeffer [1] in 1952. Generalized Frames for a Hilbert space H were first introduced by G.Kaiser [2] in 1995. Generalized Frames play an important role in the theory of frames for a Hilbert space H, it can be viewed as a "generalized discrete frames". This thesis consists of four chapters. We will investigate the properties of generalized frames and generalized frame operators, the stability of generalized Frames under perturbations, equivalence relations between two generalized Frames and uniformly continuous of the normalized windowed Fourier transform (NWFT) on L2(R). In chapter 1, we introduce a new concept, Bessel sets. We discuss the basic properties of orthonormal bases, frames and generalized frames, and give the notion and properties of tight generalized frames, dual generalized frames, independent frames and generalized frame operators, we extend frame expansion / = IC^/ni it is obtained that any element / E H to be written / 0 so that for all / e H, we have JM I < Mm - km > \2dn(m) < ?min(/M | < /,/im > |2d/i(m),/M | < f,km > \2dp(m)). In chapter 3, it is introduced that the concept of two generalized frames quadratic closeness and nearness in a Hilbert space H. It is studied that two generalized frames {hm} and {km} are Q equivalent, unitary equivalent if and only if their analysis operators have same range. It is proved that an equivalent characterization, which two generalized frames are Q equivalent if and only if they are near. In chapter 4, it is introduced the notion of the normalized windowed Fourier transform (NWFT) on L2(R). It is discussed that the properties of windowed function and the NWFT on L2(R). It is proved that the NWFT of a L2(R) function is a uniformly continuous bound function on L2(R). An inverse formula of the NWFT is proved.