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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Recapturing a holomorphic function on an annulus from its mean boundary values

Authors: Chin Hung Ching and Charles K. Chui
Journal: Proc. Amer. Math. Soc. 41 (1973), 120-126
MSC: Primary 30A72
MathSciNet review: 0320326
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Abstract: Let $ D$ be an annulus in the complex plane with closure $ \bar D$ and boundary $ \partial D$. We prove that a function $ f$, holomorphic in $ D$ with $ {C^{1 + \varepsilon }}(\partial D)$ boundary data for some $ \varepsilon > 0$, is uniquely determined by its arithmetic means $ {s_n}(f)$ and $ {s_{0n}}(f)$ over equally spaced points on $ \partial D$. We also give an explicit formula for recapturing $ f$ from its means $ {s_n}(f)$ and $ {s_{0n}}(f)$. Furthermore, we derive the relations between $ {s_n}(f)$ and $ {s_{0n}}(f)$ which are necessary and sufficient for the analytic continuability of $ f$ from $ D$ to the whole disc.

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Keywords: Annulus, mean boundary values, Fourier coefficients, Riemann coefficients, Riemann series, Möbius function, holomorphic function
Article copyright: © Copyright 1973 American Mathematical Society