Caratheodory type representation with unit weights and related approximation problems

Abstract

For arbitrary $n$ complex numbers $a_{ν-1}$, $ν=1,\dots,n$, where $n$ is sufficiently large, we get the representation in the form of power sums: $a_{ν-1}=λ_1^ν+\dots+λ_{2n+1}^ν$, where $λ_k$ are distinct points, such that $|λ_k|=1$. We study several applications to the problem of approximation by exponential sums and by $h$-sums, to the problem of extracting of harmonics from trigonometric polynomials. The result is based on an estimate for the uniform approximation rate of bounded analytic in the unit disk functions by logarithmic derivatives of polynomials, all of whose zeros lie on the unit circle $C : |z| = 1$. Our result is a modification of classical Carathéodory representation $a_{ν-1}=\sum_{k=1}^{n} X_k λ_k^ν$, $ν=1,2,\dots,n$, where weights $X_k\ge 0$, and $λ_k$ are distinct points, such that $|λ_k|=1$.