Frequency localization properties of the density matrix and its resulting hypersparsity in a wavelet representation

$O\left(N\right)\mathrm{}$ methods are based on the localization properties of the density matrix in real space, an effect refered to as nearsightedness. We show that, in addition to this real-space localization there is also a localization 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 that exhibits additional sparseness properties compared to the scenario where one has a basis set with real-space localization only. Taking advantage of this hypersparsity, 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. We expect that hypersparsity will pave the way for highly accurate $O\left(N\right)\mathrm{}$ calculations of large systems requiring many basis functions per atom.