Caratheodory type representation with unit weights and related approximation problems

Abstract

For arbitrary $n$ complex numbers $a_{\nu-1}$, $\nu=1,\dots,n$, where $n$ is sufficiently large, we get the representation in the form of power sums: $a_{\nu-1}=\lambda_1^\nu+\dots+\lambda_{2n+1}^\nu$, where $\lambda_k$ are distinct points, such that $|\lambda_k|=1$. 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 Carath\'eodory representation $a_{\nu-1}=\sum_{k=1}^{n} X_k \lambda_k^\nu$, $\nu=1,2,\dots,n$, where weights $X_k\ge 0$, and $\lambda_k$ are distinct points, such that $|\lambda_k|=1$.