# Bochner's Theorem and Total Positivity

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?
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The matrix (finite or infinite) obtained by Fourier transform of a measure always has $a_{i,j}=a_{i+1,j+1}$ (circulant matrix), so you don't have a bijection, only an injection. Morever for a real measure, they are complex (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. – BS. Jul 19 '12 at 9:07
If I have read it correctly, that 1st sentence looks very very wrong... (as per BS's comment) – Yemon Choi Aug 15 '12 at 23:52

## 1 Answer

There is a survey paper http://www.math.lsa.umich.edu/~fomin/Papers/tptp-color.ps by Fomin and Zelevinsky containing various conditions for total positivity and further references.

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Unless that survey answers one of those two questions, this seems more appropriate as a comment than an answer. – Allen Knutson Apr 3 '12 at 9:51
This survey answers the first question though it is difficult to say whether those total positivity conditions are indeed "nice". – Anatoly Kochubei May 24 '12 at 4:33