Beyond Wavelets: Exactness Theorems for Physical Calculations

Abstract

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 dVAgrammatic 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 allows basic physical couplings to be computed exactly from Iery sparse information, thereby oIercoming 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 varVAtional principle.