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 be an annulus in the complex plane with closure and boundary . We prove that a function , holomorphic in with boundary data for some , is uniquely determined by its arithmetic means and over equally spaced points on . We also give an explicit formula for recapturing from its means and . Furthermore, we derive the relations between and which are necessary and sufficient for the analytic continuability of from to the whole disc.

**[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 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 functions*, Bull. Amer. Math. Soc.**78**(1972), 617-620. MR**0298017 (45:7069)****[8]**D. Sarason,*The 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