The Research Of Construction And Properties For Higher-Dimensional Wavelet Frames

The notion of the frame is the generalization of Riesz basis. Frames possess so-me properties of orthonormal bases. Frames have fine and rich structure. The theory offrames has been developed into a new research field for nearly twenty years. It is a partof wavelet theory. Frame theory has been widely applied in mathematics, physics andengineering technology, especially in signal processing, image processing, data compres-sion, sampling theory, earthquake detection and communication engineering and so on.By using time-frequency analysis method, operator theory, and matrix theory and so on,we research into construction and properties of wavelet frames in this paper.In the first part of this paper, the research background of frame theory, the itsdevelopment histories and its current research situation are overviewed. The conceptsand properties of frames are outlined. A brief introduction on several special frames ismade.And then, the unitary extension principle for constructing three-dimensional wav-elet frames is proposed. By introducing periodization operator, unitary operator andother tools and using approximation idea, the proof of the unitary extension principle isprovided. A construction formula is derived and a corresponding constructive exampleis presented.The constructive methods and properties for higher-dimensional wavelet framesare investigated by means of filters theory, time-frequency analysis method, operatortheory. A sufficient condition for three-dimensional refinable function (x)to generate a Parseval wavelet frames is presentd. In the presence of given refinable function (x),an explicit construction formula for corresponding Parseval wavelet frames is provided.According to the unitary extension principle of spaceL2(R3),the biorthogonality ofmulti-wavelet frames is investigated. The biorthogonality and periodicity of filter banksassociated with multi-wavelet frames are discussed as well. The reconstruction algori-thm for an arbitrary three-dimensional function be proposed by using the biorthogonal-ity of multi-wavelet frames. The formula for constructing multi-wavelet frames areestablished by means of the biorthogonality of multi-wavelet frames.