Caratheodory type representation with unit weights and related approximation problems

TL;DR

This paper presents a new Carathéodory-type representation with unit weights for complex numbers, enabling improved approximation methods and harmonic extraction from trigonometric polynomials, based on estimates of uniform approximation rates.

Contribution

It introduces a modified Carathéodory representation with unit weights and applies it to approximation and harmonic extraction problems, extending classical results.

Findings

Representation with unit weights for complex numbers on the unit circle

Enhanced approximation rates for bounded analytic functions

Effective harmonic extraction from trigonometric polynomials

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 $|\lambda_k|=1$.