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

Ching, Chin Hung; Chui, Charles K.

doi:10.1090/S0002-9939-1973-0320326-3

Recapturing a holomorphic function on an annulus from its mean boundary values
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by Chin Hung Ching and Charles K. Chui PDF

Proc. Amer. Math. Soc. 41 (1973), 120-126 Request permission

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.

References

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Chin Hung Ching and Charles K. Chui, Mean boundary value problems and Riemann series, J. Approximation Theory 10 (1974), 324–336. MR 382661, DOI 10.1016/0021-9045(74)90104-x

Analytic functions characterized by their means on an are

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