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For arbitrary $n$ complex numbers $a_{\nu-1}$, $\nu=1,\dots,n$, where $n$ issufficiently large, we get the representation in the form of power sums:$a_{\nu-1}=\lambda_1^\nu+\dots+\lambda_{2n+1}^\nu$, where $\lambda_k$ aredistinct points, such that $|\lambda_k|=1$. We study several applications tothe problem of approximation by exponential sums and by $h$-sums, to theproblem of extracting of harmonics from trigonometric polynomials. The resultis based on an estimate for the uniform approximation rate of bounded analyticin the unit disk functions by logarithmic derivatives of polynomials, all ofwhose zeros lie on the unit circle $C : |z| = 1$. Our result is a modificationof 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 $|\lambda_k|=1$.

Caratheodory type representation with unit weights and related approximation problems