Osiris Wavelets in Three Dimensions

Osiris wavelets and Set wavelets

“…A (1) , σ = (0,0), A (2) , σ = (1,0), A (6) , σ = (0,1), A (4) , σ = (1,1), A (5) , σ = (0,2), A (3) , σ = (2,0), 9) , σ = (0,1), A (4) , σ = (1,1), A (3) , σ = (0,2), A (5) , σ = (2,0), A (2) , σ = (1,2), A (6) , σ = (2,1), A (1) , σ = (2,2), we can rewrite this bound as…”

“…The decomposition of a scalar field into continuous piecewise-linear wavelets is the central approach in a new hierarchical modeling program [1,2,3,4,5] for studying the critical behavior of certain classical systems in equilibrium statistical mechanics (such as the classical dipole gas and the Ginzburg-Landau spin system). Naturally, such a wavelet decomposition must be based on a multiscale triangulation of space, and in two dimensions we have been generating it [4,5] from the triangulation of the next-to-minimum-scale square given by the diagram in Figure 1.1.…”

“…The notion of orthogonality between continuous piecewise-linear functions that is natural to the modeling is based on the Sobolev inner product f ,g = ∇ f • ∇g, (1.1) but the orthonormal set is not a basis. We measure this lack of completeness with the homogeneous mode density [1,4]. Indeed, consider the set of all such wavelets of scale greater than or equal to 1 supported in the region [0,2 N ] 2 .…”

“…The homogeneous mode density of the system of wavelets is defined as the infinite-volume limit of the ratio of the number of wavelets to the number of degrees of freedom-in this case, the N = ∞ limit of the ratio of the number of wavelets to the number of vertices. Thus In [1,2,3,4,5] we have referred to these wavelets as Osiris wavelets, and the lack of completeness may be regarded as a hierarchical cutoff on the scalar field. Notice that every wavelet in our system is supported by its localization block, and this is the property that the term "hierarchical" refers to in the context of statistical mechanics.…”