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] 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)
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