  The wavelet transform, a family of orthonormal bases, is introduced as a
technique for performing multiresolution analysis in statistical mechanics. The
wavelet transform is a hierarchical technique designed to separate data sets
into sets representing local averages and local differences. Although
one-to-one transformations of data sets are possible, the advantage of the
wavelet transform is as an approximation scheme for the efficient calculation
of thermodynamic and ensemble properties. Even under the most drastic of
approximations, the resulting errors in the values obtained for average
absolute magnetization, free energy, and heat capacity are on the order of 10%,
with a corresponding computational efficiency gain of two orders of magnitude
for a system such as a $4\times 4$ Ising lattice. In addition, the errors in
the results tend toward zero in the neighborhood of fixed points, as determined
by renormalization group theory.
