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Reconstructing holomorphic functions in a domain from their values on a part of its boundary. (English) Zbl 1230.30023

Agranovsky, Mark (ed.) et al., Complex analysis and dynamical systems III. Proceedings of the 3rd conference in honor of the retirement of Dov Aharonov, Lev Aizenberg, Samuel Krushkal, and Uri Srebro, Nahariya, Israel, January 2–6, 2006. Providence, RI: American Mathematical Society (AMS); Ramat Gan: Bar-Ilan University (ISBN 978-0-8218-4150-1/pbk). Contemporary Mathematics 455. Israel Mathematical Conference Proceedings, 393-410 (2008).
Summary: This survey article is about the class of holomorphic functions that are representable by integral Carleman formulas. Unlike Cauchy’s formula, these integral formulas represent a function holomorphic in a domain \(\mathbb D\) in terms of its values on a subset \(M\) of the boundary \(\partial\mathbb D\), if \(M\) has a positive Lebesgue measure satisfying \(0<\lambda(M)<\lambda(\partial\mathbb D)\). It is known that if \(f\in E^1(\mathbb D)\), then one can represent it for \(z\in\mathbb D\) as
\[ f(z)=\frac{1}{2\pi i}\lim_{m\to\infty}\int_M \frac{\phi^m(\zeta)}{\phi^m(z)}\frac{f(\zeta)}{\zeta-z}d\zeta, \]
where the quenching function \(\phi\in E^\infty(\mathbb D)\) satisfies \(|\phi(z)|=1\) a.e. on \(\partial\mathbb D\setminus M\) and \(|\phi(z)|>1\) for every \(z\in\mathbb D\). The problem, originally posed by L. A. Aizenberg, is the converse one: Suppose that \(f\) is holomorphic in \(\mathbb D\) and has angular boundary values denoted also by \(f\) on \(M\), such that \(f\in L^1(M)\), and assume that \(f\) satisfies the above formula (where the convergence is assumed to be just point-wise). What can be said about \(f\)? In other words, the problem is to describe the class of functions to which \(f\) belongs (this space is larger than the space \(E^1(\mathbb D)\)).
We have found the answer for a large class of examples in the case of one complex variable when the set \(M\) is an Ahlfors regular curve. Nothing is known if \(M\) is just a set of positive measure. Here, to illustrate the main ideas, we present a new reconstruction formula (Carleman formula), based on ideas of Aizenberg, for a holomorphic function \(f\) in the interior of a convex polygon from the values of \(f\) on one of its sides. In the case of several complex variables, for similarly formulated problems (but without the use of quenching functions), we know even less. Some known cases require quite explicit computations, obscuring a deeper understanding of the problem.
For the entire collection see [Zbl 1135.30001].

MSC:

30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
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