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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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]**Chin Hung Ching and Charles K. Chui,*Uniqueness theorems determined by function values at the roots of unity*, J. Approximation Theory**9**(1973), 267–271. MR**0367211****[2]**Chin Hung Ching and Charles K. Chui,*Asymptotic similarities of Fourier and Riemann coefficients*, J. Approximation Theory**10**(1974), 295–300. MR**0377392****[3]**Chin Hung Ching and Charles K. Chui,*Mean boundary value problems and Riemann series*, J. Approximation Theory**10**(1974), 324–336. MR**0382661****[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*, Oxford, at the Clarendon Press, 1954. 3rd ed. MR**0067125****[6]**D. J. Patil,*Representation of 𝐻^{𝑝}-functions*, Bull. Amer. Math. Soc.**78**(1972), 617–620. MR**0298017**, https://doi.org/10.1090/S0002-9904-1972-13031-3**[7]**D. J. Patil,*Representation of 𝐻^{𝑝}-functions*, Bull. Amer. Math. Soc.**78**(1972), 617–620. MR**0298017**, https://doi.org/10.1090/S0002-9904-1972-13031-3**[8]**Donald Sarason,*The 𝐻^{𝑝} spaces of an annulus*, Mem. Amer. Math. Soc. No.**56**(1965), 78. MR**0188824**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
30A72

Retrieve articles in all journals with MSC: 30A72

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