Some Results On Generalized Shift Function Systems

Frames were first introduced in 1952 by Duffin and Schaeffer in the study of non-harmonic Fourier series,reintroduced in 1986 by Daubechies,Grossman and Meyer,and popularized from then on.Frame theory involves wavelet frames and Gabor frames.Be-cause the wavelet frames provide poor frequency localization in applications,however,could do nothing about high frequency.To overcome this disadvantage,the concept of wavelet frames must be generalized to include a library of wavelet frames,called framelet packets or wavelet frame packets.According to the character of the signals,we can not only choose different wavelet frames using framelet packets,but also analyze and pro-cess the signal with different frequency bands.First,we exploit splitting trick given by Daubechies in the thesis,and present a method with which we can construct the high dimensions framelet packets with a general dilation matrix from the unitary extension principles given by Ron and Shen,and give the decomposition and reconstruction algo-rithms for one dimension framelet packets.Because Sobolev spaces Hs(Rd)is a special and important class of Hilbert spaces,secondly,a sufficient condition or a sufficient and necessary condition for a generalized shift function system to be a Bessel sequence or even a frame for Hs(Rd)is established in the thesis.Furthermore,the results for a wave packet system to be a frame for Hs(Rd)are obtained.Finally,some frame properties associated with generalized shift function systems on locally compact abelian groups in Sobolev spaces H?s(G)are discussed A sufficient condition for a generalized shift function system to be a frame for the Sobolev spaces H?s(G)is established.Thus,on the one hand,in this thesis,we study the construction scheme and algo-rithms of framelet packets.On the other hand,we introduce frame properties of gen-eralized shift function systems and wave packet systems in Sobolev spaces.The thesis consists of five chapters.Chapter 1 introduces the background of the thesis.Chapter 2 lists the concept of frames in a Hilbert space and some basic facts of locally compact abelian groups throughout the thesis.Chapter 3 is one of the main contents.we exploit splitting trick given by Daubechies in the thesis,and present a method with which we can construct the high dimensions framelet packets with a general dilation matrix from the unitary extension principles given by Ron and Shen,and give the decomposition and reconstruction algorithms for one dimension framelet packets.Chapter 4 is the second work.A sufficient condition or a sufficient and necessary condition for a generalized shift function system to be a Bessel sequence or even a frame for the Sobolev spaces Hs(Rd)is established.Finally,the results on the wave packet system to be a frame for the Sobolev spaces Hs(Rd)axe obtained,and using the eigenvalues of the matrix theory,this thesis proves that if the Fourier transform of a function g on a certain open ball is greater than some positive number,then the wave packet system which is generated by it,cannot to be a frame for Hs(Rd).Chapter 5 is the third work.Some frame properties associated with generalized shift function systems on locally compact abelian groups in Sobolev spaces H?s(G)are dis-cussed.A sufficient condition for a generalized shift function system to be a frame for the Sobolev spaces H?s(G)is established.