Introduction to Discontinuous Wavelets

Wavelets provide a tool for efficient representation of functions. This efficient representation has proven useful in the numerical solution of non-linear evolution equations. In this paper we provide a brief review of the use of wavelets for efficient representation of functions, and in particular we describe the piecewise-discontinuous basis of wavelets proposed by Alpert. We review the useful properties this basis has for the solution of PDE’s, and introduce an illustrative approach to the representation of boundary conditions. We also discuss the extension to higher dimensional problems.