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. Drbnath. Ed.). Birkhauser. Basel. in press). This is part of an on-going program to analyze critical behaviour in classical equilibrium statistical mechanics. Following Golner`s general idea of using an incomplete multiscale set of functions (1993, Phys. Rev. B 8. 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. (C) 2000 Academic Press. [References: 7]
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机译：我们将不寻常的小波构造扩展到三个维度，这是我们在二维上下文中首次引入的（在“小波变换和时频信号分析”中（L. Drbnath。Ed。），Birkhauser。Basel。出版）。这是正在进行的程序的一部分，该程序用于分析经典平衡统计力学中的关键行为。遵循Golner使用不完整的多尺度函数集的一般想法（1993，Phys。Rev. B 8. 339）来获得更逼真的建模效果。我们介绍一个小波集，其母小波是连续的。单位立方体中支持的分段线性函数。这样的小波集必然是不完整的。但在三个维度上，我们基于8个子多维数据集将七个子波打包到了四个单位多维数据集中。和三个基于与相邻子立方体相交的12个八面体。生成的小波集不是Sobolev正交的。但是我们得出了多尺度Sobolev重叠矩阵的正下界。 （C）2000年学术出版社。 [参考：7]
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