Construction And Algorithms Of Multi-scale Pseudoframes And Wavelet Frames

Frame theory has been developed into a new research field following waveletanalysis. At present, the theory of frames is a popular research subject for waveletanalysis. It plays an important role in wavelet analysis and irregular sampling theory. Ithas been successfully used in signal processing, image processing, data compression,sampling theory, Banach space theory investigation, optics theory study and so on. Inthis paper, the properties and construction of multiple multi-scale pseudoframes andmultiple wavelet tight frames are researched.Firstly, the origin and the development histories of frames are outlined. Gaborframes, wavelet frames and their properties as well as commonly used constructiveapproaches are briefly introduced.Nextly, the notion of multiple multi-scale pseudoframes for subspace is introduced.The features of multiple translation pseudoframes for subspace are characterized withthe help of operator theory and filter theory. Construction of the generalizedmultiresoution structure for the Paley-Wiener subspace is investigated. The pyramiddecomposition scheme of a generalized multiresolution structure is established, which issimilar to Mallat’s pyramid algorithm. A sufficient and necessary condition for theexistence of the pyramid decomposition scheme for a generalized multiresolutionstructure is presented. By virtue of the generalized multiresolution structure andapproximation theory, we can obtain multiple affine frame expansion of functions inspaceL2(R). And then the concept of minimum-energy multiwavelet tight frames isintroduced. The matrix symbols are derived by taking Fourier transform on the refinableequation. The conditions which are satisfied by multiple refinable function ofminimum-energy multiple wavelet frames are provided with the help of matrix symbols and matrix theory.A new multiple refinable function vector and a corresponding minimum-energymultiwavelet tight frame vector through implementing orthogonal transform on themultiple refinable function vector and the minimum-energy multiwavelet tight framevector are obtained. Finally, conditions which should be satisfied for several generatorsof symmetric tight wavelet frames are provided by constructing the mask functions ofthe refinable function.