Compressing H² matrices for translationally invariant kernels

H² matrices provide compressed representations of the matrices obtained when discretizing surface and volume integral equations. The memory costs associated with storing H² matrices for static and low-frequency applications are O(N). However, when the H² representation is constructed using sparse samples of the underlying matrix, the translation matrices in the H² representation do not preserve any translational invariance present in the underlying kernel. In some cases, this can result in an H² representation with relatively large memory requirements. This paper outlines a method to compress an existing H² matrix by constructing a translationally invariant H² matrix from it. Numerical examples demonstrate that the resulting representation can provide significant memory savings.