Compactly supported moments

The ensuing moment equation for the compactly supported moments is identical to that for the fip a,b) moments, except for the appearance of boundary terms. These boundary terms will take on the form ... 

Since the compactly supported moments are defined for any solution of the Schrodinger equation, we see that the original scalet equation is capable of generating solutions of the Schrodinger equation, for any E. 

There are several families of wavelets, proposed by different authors. Those developed by Daubechies [46] are extensively used in engineering applications. Wavelets from these families are orthogonal and compactly supported, they possess different degrees of smoothness and have the maximum number of vanishing moments for a given smoothness. In particular, a function f t) has e vanishing moments if... 

This chapter has briefly eluded to two wavelet families, the Haar and Daubechies wavelets. In fact when Nf = 2 the Daubechies wavelet is identical to the Haar wavelet. In this section we would like to discuss in greater detail more about these wavelet families and other wavelet families. We will also provide a brief comparison between the different properties possessed by these wavelets and other wavelet families. This is important because depending on your application, you may need to choose a wavelet that satisfies special properties. We first review the terms orthogonal and compact support. Following this, we will introduce some more properties, namely smoothness and symmetry of wavelets and also discuss the term vanishing moments. 

The decision to use a particular wavelet may extend to other properties besides orthogonality and compact support. Other properties include symmetry, smoothness, and vanishing moments. 

The number of tail-solvent interactions shows two regions as a function of n. Below n 45, f I6.3n which is consistent with the growth of compact, approximately spherical, clusters. Above n - 45, t" 5.1n which is consistent with the growth of cylindrical micelles, although the statistics become poor for these large clusters. These observations suggest a sphere to cylinder transition for the preferred cluster shape at n 45 and this hypothesis is supported by the moments of inertia data which show a transition from moments consistent with a sphere for consistent with a cylinder for n > 45 It... 

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