The Study Of Several Problems For Periodic Wavelet Frames

Frames are a family of vectors in a linear space. They are a kind of overcomplete bases. The framework reflects the microstructure of linear space. Periodic wavelet frames are of good time-frequency localization characteristics, translation invariance and more design degrees of freedom. They are widely used in biomedical engineering, signal denoising and signal reconstruction, sampling theory, seismic exploration data, image fusion, image retrieval, quantum mechanics and many other fields. By using frame multiresolution analysis method, operator theory, unitary extension principle and the mixed extension principle, we research into construction methods for periodic tight wavelet frames and dual periodic wavelet frames, and the express structure of bivariate multis-cale tight wavelet frames. Several new results have been obtained.First of all, the development progress for frame theory is outlined. Some basic concepts, properties of the frame- work and the research significance for wavelet frames are briefly introduced. Secondly, [ ]2L -q,q is the space of square integrable periodic functions, and the transl-ation operatorkTtis defined in [ ]2L -q,q. According to the unitary extension principle, the time-frequency analysis method and frame multiresolution analysis, we put forward two kinds of methods to construct tight wavelet frames with any positive real number as a cycle, and get the condition of satisfying the filter of periodic tight wavelet frames. Next, a constructive method for periodic wavelet frames is studied. Bivariate tight wavelet frames are constructed by the unitary extension principle. Moreover, the compactly supported periodic tight bivariate wavelet frames are obtained. Also on the basis of the mixed extension principle,starting from the compactly supported scaling function filters, a pair of dual wavelet frames is constructed. The periodization of the dual wavelet frames is still a pair of dual compactly supported periodic wavelet frames.Finally, according to the frame multiresolution analysis and the symbol function0Q(w) associated with the scaling function F(x) which satisfies the inequality for222 20 00101Q() Q() Q() 1Mw w m mwm-+ + + +L+ + ￡, the sufficient condition for the existence of multi-scale tight bivariate wavelet frames is provided. The filters for tight bivariate wavelet frames are constructed. A coresponding numerical example is given.