Generalization of a theorem of Carathéodory

Abstract

Carathéodory showed that $n$ complex numbers $c_1,...,c_n$ can uniquely be written in the form $c_p=\sum_{j=1}^m ρ_j {ε_j}^p$ with $p=1,...,n$, where the $ε_j$s are different unimodular complex numbers, the $ρ_j$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 $ρ_j$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 $ρ_j$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