Caratheodory type representation with unit weights and related approximation problems

2018/07/17 by Komarov, Mikhail A.
#41A20 #41A21 #41A25 #41A29 #41A30 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.06499

Abstract

For arbitrary n complex numbers a_ν-1, ν=1,…,n, where n is sufficiently large, we get the representation in the form of power sums: a_ν-1=λ₁^ν+…+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=∑_k=1ⁿ X_k λ_k^ν, ν=1,2,…,n, where weights X_k≥ 0, and λ_k are distinct points, such that |λ_k|=1.

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