  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 $|\lambda_k|=1$.
