Independent local-scaling transformations

Following the projection property, the realization m[b,a]Wgr] t) in the time/scale domain is in general not a wavelet transformation and thus realizations s t) in the time domain depend - usually weakly - on the chosen wavelet g t) and the reconstruction wavelet h(t) respectively. To ensure at least asymptotic independence of the (reconstruction) wavelet, we define the processes such that for smaller and smaller scales more and more reproducing kernels fit into local structures of the process ... 

If one succeeds in transforming not only the occupied but also the virtual MOs to a set of well-localized MOs, such that one can associate q (occupied and virtual) MOs with each atom, then one can argue that for the description of intraatomic correlation only excitations in this g-dimensional (and hence n-independent) subspace need to be considered. For interatomic correlations between a pair of neighboring atoms excitations with the 2q dimensional space of the MOs of the two atoms are necessary, and so forth. Correlations beyond next-nearest neighbors may be regarded as unimportant. The number of pairs of atoms to be considered scales with n, so the overall computational demands should scale with n as well, provided that also takes advantage of fast multipole expansion [190, 191] for the Coulomb interaction. 

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