  O(N) methods are based on the decay properties of the density matrix in real
space, an effect sometimes refered to as near-sightedness. We show, that in
addition to this near-sightedness in real space there is also a
near-sightedness in Fourier space. Using a basis set with good localization
properties in both real and Fourier space such as wavelets, one can exploit
both localization properties to obtain a density matrix which exhibits
additional sparseness properties compared to the scenario where one has a basis
set with real space localization only. We will call this additional sparsity
hyper-sparsity. Taking advantage of this hyper-sparsity, it is possible to
represent very large quantum mechanical systems in a highly compact way. This
can be done both for insulating and metallic systems and for arbitrarily
accurate basis sets. We expect that hyper-sparsity will pave the way for O(N)
calculations of large systems requiring many basis functions per atom, such as
Density Functional calculations.
