| Wavelet analysis is a research field developing rapidly in mathematics presently. It has profound theory and comprehensive application, the conctruction of scaling function and wavelet is very important to the theory and application of wavelet analysis, it has drawn more and more mathematical researcher's attentions.Firstly, the paper summarizes the characteristics of multiresolution analysis and elicits the essential characteristics of multiresolution analysis, and then we give the simplest definition of multiresolution analysis. At the same time, the conctruction of scaling function and wavelet is introduced. As an example, compactly supported orthogonal scaling functions and wavelets are constructed. It will help people understand the conctruction of scaling function and wavelet.Secondly, the author study two-scale matrixes and the conctruction of orthogonal wavelet, obtain a common method for constructing orthogonal wavelet bases by two-scale matrixes. In addition, through the study of the transfer functions of wavelets, a new kind of spline wavelets is conctructed. And the corresponding two-scale sequences are obtained easily, these will provide a new wavelet bases for wavelet theory and its applications. Lastly, this paper discusses detailedly the relationship between frame and Riesz basis in Hilbert space. Proves that if wavelet space {Wj}constructed through multiresolution analysis {Vj} satisfy W0⊥V0,the dual multiresolution analysis {(V|)j}of {Vj}and multiresolution analysis {Vj}is the same multiresolution analysis ,here the waveletψ( x)conctructed through multiresolution analysis is semi-orthogonal wavelet. If the waveletψ( x) conctructed through multiresolution analysis is a R-function, it exists necessarily only one dual waveletψ| so as toψandψ| satisfy the biorthogonal condition, and we obtain three methods for calculating the dual wavelet of the mother waveletψ( x) constructed through multiresolution analysis.
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