Bochner's Theorem for LCA groups applied to the case of $G = U(1)$ and $G^{\vee} = \mathbb{Z}$ tells us that through the Fourier transform, probability measures on the circle are in bijection with infinite positive semidefinite matrices with $1$s along the main diagonal. In the case of finite $N \times N$ matrices, we know from convex analysis that these are *correlation matrices*. Indeed, this corresponds to the case $G = \mathbb{Z} / N \mathbb{Z} \hookrightarrow U(1)$ of $N$th roots of unity with its dual group $\mathbb{Z} \twoheadrightarrow \mathbb{Z} / N \mathbb{Z} = G^{\vee}$.

Concretely, every *principal* minor of a positive semidefinite matrix has non-negative determinant. If a matrix satisfies the stronger condition that *every* minor has non-negative determinant, we call it a *totally positive* matrix.

- Is there some nice condition on a positive semidefinite matrix which guarantees it is totally positive?
- Which probability measures on the circle correspond to infinite totally positive matrices with $1$s on the main diagonal?

bijection, only an injection. Morever for a real measure, they arecomplex(hermitian), unless the measure is symmetric (invariant under $g\mapsto g^{-1}$). Finally shifted principal minors correspond to principal minors of the FT of the orig measure times a characters and this is a positive measure for all characters only if the original measure is a Dirac at the origin. $\endgroup$ – BS. Jul 19 '12 at 9:07