"Is it true that if a holomorphic function in the unit disk converges uniformly to the 0 function some connected arc of the unit circle, this function is globally null? If that is true, this would stand for a uniqueness fact and so how to recover some holomorphic function from its boundary value on an arc (if the uniform convergence still holds on the arc)"

Marley Meyers 2022-10-24 Answered
Is it true that if a holomorphic function in the unit disk converges uniformly to the 0 function some connected arc of the unit circle, this function is globally null?
If that is true, this would stand for a uniqueness fact and so how to recover some holomorphic function from its boundary value on an arc (if the uniform convergence still holds on the arc)
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Answers (1)

Jimena Torres
Answered 2022-10-25 Author has 20 answers
Let the (open) arc on which the boundary values of f vanish be A. Since the boundary values of f on A are real, by the Schwarz reflection principle we know that the function
g ( z ) = { f ( z ) , | z | < 1 0 , z A f ( 1 / z ¯ ) ¯ , | z | > 1
is holomorphic on the connected open set D A ( C D ¯ ). Since g vanishes on a non-discrete set, the identity theorem yields g 0, in particular f 0 follows.
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