Detecting Fourier Subspaces

Abstract

Let G be a finite abelian group. We examine the discrepancy between subspaces of $$l^2(G)$$ which are diagonalized in the standard basis and subspaces which are diagonalized in the dual Fourier basis. The general principle is that a Fourier subspace whose dimension is small compared to $$|G| = \mathrm{dim}\left( l^2(G)\right) $$ tends to be far away from standard subspaces. In particular, the recent positive solution of the Kadison–Singer problem shows that from within any Fourier subspace whose dimension is small compared to |G| there is a standard subspace which is essentially indistinguishable from its orthogonal complement.