L_p Norm Or Quasi-norm Mean Size Of Subdivision Tree And Asymptotic Behavior Of Wavelet Packets

The purpose of this dissertation is to investigate the subdivision tree of the general vector refinement equation whose integer expanding matrix M satisfying lim_n→8 M^-n = 0. We establish the L_p norm or quasi-norm estimatesof the subdivision tree. On the basis of the L_p norm or quasinorm estimates of the subdivision tree, we obtain the L_p norm mean size formula of general (vector)wavelet packets. The L_p estimates is not only true for 1≤p≤∞, but also for 0 < p < 1. We remark here that there are very few discussions on L_p estimates for 0 < p < 1 in the literatures. We apply our conclusions established in above to multivariate quincunx biorthogonal wavelet packets and multiple biorthogonal wavelet packets in L_p. We obtain a asymptotic formula of the mean size for multivariate quincunx biorthogonalwavelet packets. Similarily, we also get a asymptotic mean size formula for multiple biorthogonal wavelet packets in L_p. These two concrete formulascontain important information for these two kinds of wavelet packets. Some results in [40,41,56] are special cases of our theorems. Furthermore, we generalize the L_p norm of the subdivision tree from Euclid space R^s to the noncommutative Heisenberg group H^s. We get the L_p norm estimates of the subdivision tree on Heisenberg group H^s, which extend the corresponding conclusions in [22,23].