A class of multiwavelet bases for L² 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 K whose kernel is smooth except along a finite number of singular bands has a sparse representation. In addition, the inverse operator (I - K)- 1 appearing in the solution of a second-kind integral equation involving K is often sparse in the new bases. The result is an order O(nlog ² n) algorithm for numerical solution of a large class of second-kind integral equations.
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Bradley K. Alpert (1993) studied this question.