Representability of Analytic Functions in Terms of Their Boundary Values

Abstract

Suppose that $\nu$ is an arbitrary finite complex Borel measure on the interval $[0;{\text{2}}\pi {\text{, }}u{\text{(}}re^{i\varphi } )$ is its Poisson integral, $u{\text{(}}re^{i\varphi } )$ and $v{\text{(}}re^{i\varphi } )$ are the conjugate harmonics of $F{\text{(}}z) = v{\text{(}}z) + iv{\text{(}}z),{\text{ }}z = re^{i\varphi }$, and $F{\text{(}}t)$ is the nontangential limiting value of the analytic function $F{\text{(}}z)$ as $z \to t = e^{i\theta }$. In this paper, we consider the problem of representing the analytic function $F{\text{(}}z)$ in terms of its boundary values $F{\text{(}}t)$.