Fourier coefficients of borelian measures on the circle

Abstract

Let $\mu$ be a complex borelian measure on the unit circle $\bf T$ and let $\hat{\mu}(n)$ be its Fourier coefficients. The sum of the series $\sum\limits_{n=-\infty}^\infty \frac{\hat{\mu}(n)}{n+z}$ is calculated. There are different criteria a sequence $c_n$, $n\in(-\infty,\infty)$, to be a sequence of Fourier coefficients of some complex borelian measure on $\bf T$. One more criterion of such type is proved.