Some Characterizations of Wavelets with Composite Dilations

In order to overcome classical wavelets´ shortcoming in image processing problems, people developed many producing systems, which built up wavelet family. In this paper, the notion of AB-multiresolution analysis is generalized,and corresponding theory is developed. For an AB-multiresolution analysis associated with an expanding matrix, we deduce that there exists a singe scaling function in its reducing subspace. Under some conditions,wavelets with composite dilations can be gotten by AB multiresolution analysis, which permits the existence of fast implementation algorithm. At last, some wavelets with composite dilations which has nice geometrical properties are constructed.