Compressing $H^{2}$ Matrices for Translationally Invariant Kernels

$H^{2}$
matrices provide compressed representations of the matrices obtained when discretizing surface and volume integral equations. The memory costs associated with storing
$H^{2}$
matrices for static and low-frequency applications are
$O(N)$
. However, when the
$H^{2}$
representation is constructed using sparse samples of the underlying matrix, the translation matrices in the
$H^{2}$
representation do not preserve any translational invariance present in the underlying kernel. In some cases, this can result in an
$H^{2}$
representation with relatively large memory requirements. This paper outlines a method to compress an existing
$H^{2}$
matrix by constructing a translationally invariant
$H^{2}$
matrix from it. Numerical examples demonstrate that the resulting representation can provide significant memory savings.