The problem of understanding the Fourier-analytic structure of the cone of
positive functions on a group has a long history. In this article, we develop the first
quantitative spectral concentration results for such functions over arbitrary compact
groups. Specifically, we describe a family of finite, positive quadrature rules for the
Fourier coefficients of band-limited functions on compact groups. We apply these
quadrature rules to establish a spectral concentration result for positive functions:
given appropriately nested band limits A ⊂ B ⊂ G, we prove a lower bound on the
fraction of L2-mass that any B-band-limited positive function has in A. Our bounds
are explicit and depend only on elementary properties of A and B; they are the first
such bounds that apply to arbitrary compact groups. They apply to finite groups as
a special case, where the quadrature rule is given by the Fourier transform on the
smallest quotient whose dual contains the Fourier support of the function.
