Some Studies On M-Band Wavelets And Wavelet Frames

The typical wavelet theory is researched around two-band wavelets, but in order to acquire satisfactory outcoming with analyzing and processing the signal precisely, it is not enough to use only two-band wavelets, and M-band wavelets is needed. Besides, M-band wavelets have some good properties that two-band wavelets don't have, such as being orthogonal and liner phase simultaneously. So it has been a hotspot in information fields to research M-band wavelets'theory and construction. In this paper, we studied some properties of three-band orthogonal wavelets based on the classical two-band wavelets theory. Especially, we proved the relationship between discrete and continuous moments of the scaling functions in case three. Then an algebraic method and a few examples to construct the three-band orthogonal wavelets were given. This method converts the structure of scaling function to solving the constraint equations of the filter coefficients. It overcame that the existing algorithms were too complex or could not ensure the symmetry etc.Wavelet frames and tight wavelet frames become new research hotspots in recent years. Daubechies and Meyer advanced necessary and sufficient conditions for mother wavelet to generate frames in "Ten Lectures on Wavelets" and "Ondelettes et Operateurs". Characterizations of univariate tight wavelet frames are implicit in the works of Wang and Weiss. Then an explicit characterization of tight wavelet frames in the multivariate case was obtained by Bin Han. Independently of these, Ron and Shen gave a general characterization of all wavelet frames, and specialized this to the case of tight wavelet frames. Recently, Daubechies and Bin Han constructed MRA-based tight wavelet frames, and discussed the existence of dual wavelet frames and its construction method. Considering MRA plays a very important role during the construction of the wavelet frames, we presented and proved a sufficient condition for constructing the scaling functions of M-band dual wavelet frames based on tight wavelet frames and FMRA.