Kronecker-product approximations for some function-related matrices

A new approximation tool such as sums of Kronecker products is recently found to provide a superb compression property on a series of numerical examples of quite a general nature. The purpose of the paper is explanation of this phenomenon in the form of “existence theorems” for matrix approximations of low Kronecker rank for some classes of function-related matrices including important specimens from potential theory. This lays the grounds for development of new approximation algorithms, for example, in the cases when a matrix is associated with a shift-invariant function on the Cartesian product of nonunform grids, which is of great practical interest in the solution of integral equations on plates or screens.