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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Analytic functions characterized by their means on an arc


Authors: Chin Hung Ching and Charles K. Chui
Journal: Trans. Amer. Math. Soc. 184 (1973), 175-183
MSC: Primary 30A72
MathSciNet review: 0338375
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Abstract: It is known that a function f, holomorphic in the open unit disc U with $ {C^{1 + \varepsilon }}$ boundary data for some $ \varepsilon > 0$, is uniquely determined by its arithmetic means over equally spaced points on $ \partial U$. By using different techniques, we weaken the hypothesis $ {C^{1 + \varepsilon }}(\partial U)$ to functions with $ {L^p}$ derivatives, $ 1 < p \leq \infty $. We also prove that a function is determined by its averages over an arc K if f is holomorphic in a neighborhood of $ \bar U$, and that this result is false for some functions f in $ A \cap {C^\infty }(\bar U)$. On the other hand, we can capture a $ A \cap {C^2}(\bar U)$ function from its means and shifted means on K.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0338375-2
PII: S 0002-9947(1973)0338375-2
Keywords: Arithmetic means on an arc, Möbius function, holomorphic functions, uniqueness and nonuniqueness, representation formulas
Article copyright: © Copyright 1973 American Mathematical Society