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 \(F{\text{(}}t)\).