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?
