Osiris Wavelets in Three Dimensions

Abstract

We extend to three dimensions an unusual wavelet construction that we first introduced in a two-dimensional context (in “Wavelet Transforms and Time–Frequency Signal Analysis” (L. Debnath, Ed.), Birkhäuser, Basel, in press). This is part of an on-going program to analyze critical behavior in classical equilibrium statistical mechanics. Following Golner's general idea of using an incomplete multiscale set of functions (1973, Phys. Rev. B8, 339) to obtain more realistic modeling that is still hierarchical, we introduce a wavelet set whose mother wavelets are continuous, piecewise-linear functions supported in the unit cube. Such a wavelet set is necessarily incomplete, but in three dimensions we have packed seven mother wavelets into the unit cube—four based on the 8 sub-cubes, and three based on the 12 octahedra that intersect adjacent sub-cubes. The generated wavelet set is not Sobolev-orthogonal, but we derive a positive lower bound on the multi-scale Sobolev overlap matrix.