A Class of Bases in $L^2$ for the Sparse Representation of Integral Operators

Abstract

A class of multiwavelet bases for $L^2$ is constructed with the property that a variety of integral operators is represented in these bases as sparse matrices, to high precision. In particular, an integral operator $\mathcal{K}$ whose kernel is smooth except along a finite number of singular bands has a sparse representation. In addition, the inverse operator $(I - \mathcal {K})^{ - 1} $ appearing in the solution of a second-kind integral equation involving $\mathcal{K}$ is often sparse in the new bases. The result is an order $O(n\log ^2 n)$ algorithm for numerical solution of a large class of second-kind integral equations.