The Characterization Of The Vector-valued Wave Packet Systems

Wave packet frames,as the fifth problem of Ole Christensen's article Six problems in frame theory in August 26,2013,is a hot research topic.The study of wave packet is mostly confined to unit-wavelet situation.Owing to the good properties:orthogonal-ity,symmetry and compact support,which all play an important role on application,multiwavelet theories and their applications catch more and more scholars' attention.Vector-valued wavelets are a class of special multiwavelets.Recently,Chen and Cheng introduced the notion of orthogonal vector-valued wavelets.Moreover,they investigated their existence and presented a constructive method of Vector-valued.However,compared with the conventional wavelet,Wavelet frames are redundant generally.This redundancy leads to robustness,r means that redundancy enables to obtain wavelet coefficients under low precision while can reply signal in a relatively high precision.The ground breaking work in this article is the combination of vector-valued and the wave packet system.The definition of vector-valued function F(x)=(f1(x),f2(x),...,fn(x)T is a vector-valued Parseval frame is given for the first time,meanwhile in L~2(Rd,Cn)defines the modula-tion operator,the translation operator and the dilation operator.Further,properties of the vector-valued wave packet systems ? in L~2(Rd,Cn)are obtained.However,some in-put,/output data and all filters are of discrete nature.Then combining the vector-valued and discrete space for enriching the theories of wavelet and frame is required.This paper is organized as follows:Chapter 1 introduces the background of wave packet frames and vector-valued wavelet-s and the structure and work of the thesis.Chapter 2 refers to relevant frames theories on L~2(Rd),defines the frames theories on L~2(Rd,Cn).Chapter 3 is one of the main contents.Firstly,defines the vector-valued wave packet orn L~2(Rd,Cn).Secondly,discusses the orthogonality property for vector-valued wave packet systems.Last but the most important,gives the necessary and sufficient conditions of the characterization of vector-valued wave Packet Systems to be the Parseval frames.Chapter 4 firstly defines the space form of l~2(Z,Cs).Then studies the fourier trans-form and convolution on l~2(Z,Cs),and deduces the properties of vector-valued wavelet on Z.Last,obtains the perfect reconstruction of discrete vector-valued framelet transform.