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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
DOI: https://doi.org/10.1090/S0002-9939-1973-0320326-3
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.


References [Enhancements On Off] (What's this?)

  • [1] C. H. Ching and C. K. Chui, Uniqueness theorems determined by function values at the roots of unity, J. Approximation Theory (to appear). MR 0367211 (51:3453)
  • [2] -, Asymptotic similarities of Fourier and Riemann coefficients, J. Approximation Theory (to appear). MR 0377392 (51:13564)
  • [3] -, Mean boundary value problems and Riemann series, J. Approximation Theory (to appear). MR 0382661 (52:3543)
  • [4] -, Analytic functions characterized by their means on an are, Trans. Amer. Math. Soc. (to appear).
  • [5] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 3rd ed., Clarendon Press, Oxford, 1954. MR 16, 673. MR 0067125 (16:673c)
  • [6] D. J. Patil, Recapturing $ {H^2}$ functions from boundary values on small sets, Notices Amer. Math. Soc. 19 (1972), A-307, Abstract #72T-B42 (Paper to appear). MR 0298017 (45:7069)
  • [7] -, Representation of $ {H^p}$ functions, Bull. Amer. Math. Soc. 78 (1972), 617-620. MR 0298017 (45:7069)
  • [8] D. Sarason, The $ {H^p}$ spaces of an annulus, Mem. Amer. Math. Soc. No. 56 (1965). MR 32 #6256. MR 0188824 (32:6256)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0320326-3
Keywords: Annulus, mean boundary values, Fourier coefficients, Riemann coefficients, Riemann series, Möbius function, holomorphic function
Article copyright: © Copyright 1973 American Mathematical Society

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