A Note on Wavelet Bases for Two-Dimensional Surfaces

Abstract

Recent work by Beylkin, Coifman and Rokhlin has demonstrated that integral equations for functions on $IR$ can be solved rapidly by expressing the integrands in a wavelet basis. Boundary element methods for solving partial differential equations in three dimension rely on integral equations for functions defined on surfaces embedded in $IR^{3}$. Accordingly, it is of interest to extend the wavelet work to functions defined on surfaces. In this report, we define a basis of piecewise constant functions on surfaces in $IR^{3}$ with properties akin to a wavelet basis. The basis we define is not useful for numerical computation because piecewise constant functions have poor approximation properties, but this work suggests an approach to define smoother wavelet bases for surfaces.