Multiresolution separated representations of singular and weakly singular operators

For a finite but arbitrary precision, we construct efficient low separation rank representations for the Poisson kernel and for the projector on the divergence free functions in the dimension d = 3. Our construction requires computing only one-dimensional integrals. We use scaling functions of multiwavelet bases, thus making these representations available for a variety of multiresolution algorithms. Besides having many applications, these two operators serve as examples of weakly singular and singular operators for which our approach is applicable. Our approach provides a practical implementation of separated representations of a class of weakly singular and singular operators in dimensions d ≥ 2.