  This paper develops the use of wavelets as a basis set for the solution of
physical problems exhibiting behavior over wide-ranges in length scale. In a
simple diagrammatic language, this article reviews both the mathematical
underpinnings of wavelet theory and the algorithms behind the fast wavelet
transform. This article underscores the fact that traditional wavelet bases are
fundamentally ill-suited for physical calculations and shows how to go beyond
these limitations by the introduction of the new concept of semicardinality,
which leads to the profound, new result that basic physical couplings may be
computed {\em without approximatation} from very sparse information, thereby
overcoming the limitations of traditional wavelet bases in the treatment of
physical problems. The paper then explores the convergence rate of conjugate
gradient solution of the Poisson equation in both semicardinal and lifted
wavelet bases and shows the first solution of the Kohn-Sham equations using a
novel variational principle.
