Construction And Characterization For Multivariate Wavelet Frames

The notion of the frame is the generation of Riesz basis. Tight frames possesssome properties of orthonormal bases. The theory of frames has been developed into anew research field for nearly twenty years. Also it is a popular subject of waveletanalysis. It plays an important role in wavelet analysis and irregular sampling theory.The concept for the frame of Hilbert space was proposed by Duffin and Schaefer in1952when they studied non-harmonic Fourier analysis. Since wavelet frames havegood quality, now, it has become a focus of attention of many scholars. Wavelet frameshave been successfully used in signal processing, data compression, image processing,function approximation theory, sampling theory and other fields. By using framemultiresolution analysis method, time-frequency analysis method, operator theory, andmatrix theory and so on, we research into construction and characterization ofmultivariate wavelet frames in this paper. Several new results have been obtained.Firstly, the development course of wavelet analysis and the theory of waveletframe and the latest internal and external research results are outlined.The concepts andproperties of wavelet frames, p-frames in Banach spaces, WH-frames, exact frames, fu-sion-frame are summarized．Secondly, the concept of minimum energy bivariate wavelet frames is introduced,The equivalent characterization and sufficient and necessary condition for minimumenergy bivariate wavelet frames are established. Via making the polyphase decompo-sition that corresponds to the symbol functions for the scaling function and the waveletfunction, the decomposition and reconstruction formulas for minimum energy bivariatewavelet frames are deduced. And several numerical examples are provided.Thirdly, the existence for the wavelet frames of spaceL2(R s)is studied. For the real expansive matrixs A, B and the function h(x)∈L2(Rs), the necessary and sufficient condition of the function system{DAmTqBh}m∈Z,q∈Zs, which constitute wavelet frames in space L2(RS) is presented. The characteristics in the frequency domain of h(x)∈L2(RS) that become wavelet frame generator in space L2(RS) is characterized.Finally, the concept for matrix Fourier multipliers concerning Parseval multivariate multi-wavelet frames is introduced. A sufficient and necessary condition for a matrix-valued function M(t), which become Matrix Fourier multipliers for Parseval multivariate multi-wavelet frames is provided. Lots of constructive examples are also presented.