| Wavelet has the time-frequency localization character which is not owned by other analyzing tools and it can deal with odd signals effectively .So it has been the hotspot of theory research since it was presented. Many wavelets and multi-wavelets and M-band wavelets have been constructed, which have nice properties. These theory fruits have been using in many fields of science and production more and more widely. So it is significant to study the construction of scaling function and wavelet, which are the two vial composed elements of wavelet system, it is also significant to study wavelet packet. Based on deep research to the statuses of the wavelet theory at home and abroad, we do some research on the construction of compactly supported quarternary wavelet, orthogonal multiple vector-valued wavelet packet, general cardinal multiwavelet with dilation M and 3-dimensional multiwavelet.Firstly, we give a new algorithm for constuction compactly supported quarternary orthogonal saling function , the correspond wavelets are also given.Secondly, the property and the construction method of a class of orthogonal multiple vector-valued wavelet packet with dilation factor 6 is discussed.Thirdly, by relationship between balanced order and approximation order in time domain, we investigate interpolatory multiwavelet with dilation M and show its balanced order is equal to its approximation order. As for general cardinal orthogonal multiwavelet, the existent condition of its cardinal parameters are studied. Its balanced order is not equal its approximation order in a general way.Lastly, by starting from the 3-dimensional tensor product, several scaling functions wavelets for the 3-dimensional space are defined. The necessary and sufficient conditions for the orthogonality of scaling functions in 3-dimensional space and the wavelets in 3-dimensional space are presented.
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