Generalization of a theorem of Carath\'eodory

TL;DR

This paper generalizes Carathéodory's theorem by establishing conditions for representing complex numbers with real, positive coefficients linked to eigenvalues of Hermitian Toeplitz matrices, with applications in neutron scattering.

Contribution

It extends Carathéodory's theorem to real, non-zero coefficients and relates the number of sign choices to eigenvalues of a Hermitian Toeplitz matrix, providing new insights for inverse problems.

Findings

Conditions for real, positive coefficients in the theorem

Bound on sign choices based on eigenvalues

A lemma on unimodular roots of polynomials

Abstract

Carath\'eodory showed that $n$ complex numbers $c_1,...,c_n$ can uniquely be written in the form $c_p=\sum_{j=1}^m \rho_j {\epsilon_j}^p$ with $p=1,...,n$, where the $\epsilon_j$s are different unimodular complex numbers, the $\rho_j$s are strictly positive numbers and integer $m$ never exceeds $n$. We give the conditions to be obeyed for the former property to hold true if the $\rho_j$s are simply required to be real and different from zero. It turns out that the number of the possible choices of the signs of the $\rho_j$s are {at most} equal to the number of the different eigenvalues of the Hermitian Toeplitz matrix whose $i,j$-th entry is $c_{j-i}$, where $c_{-p}$ is equal to the complex conjugate of $c_{p}$ and $c_{0}=0$. This generalization is relevant for neutron scattering. Its proof is made possible by a lemma - which is an interesting side result - that establishes a necessary and sufficient condition for the unimodularity of the roots of a polynomial based only on the polynomial coefficients. Keywords: Toeplitz matrix factorization, unimodular roots, neutron scattering, signal theory, inverse problems. PACS: 61.12.Bt, 02.30.Zz, 89.70.+c, 02.10.Yn, 02.50.Ga