Analyticity on translates of a Jordan curve
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Abstract:
Let $\Omega$ be a domain in $\mathbb {C}$ which is symmetric with respect to the real axis and whose boundary is a real analytic simple closed curve. Translate $\overline \Omega$ vertically to get $K = \bigcup \{ \overline \Omega +it,\ -r\leq t\leq r\}$ where $r>0$ is such that $(\overline \Omega -ir)\cap (\overline \Omega +ir)= \emptyset$. We prove that if $f$ is a continuous function on $K$ such that for each $t,\ -r\leq t\leq r$, the function $f|(b\Omega +it)$ has a continuous extension to $\overline \Omega +it$ which is holomorphic on $\Omega +it$, then $f$ is holomorphic on $\textrm {Int}K$.References
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Additional Information
- Josip Globevnik
- Affiliation: Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Ljubljana, Slovenia
- Email: josip.globevnik@fmf.uni-lj.si
- Received by editor(s): June 15, 2005
- Received by editor(s) in revised form: December 1, 2005
- Published electronically: May 8, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 5555-5565
- MSC (2000): Primary 30E20
- DOI: https://doi.org/10.1090/S0002-9947-07-04264-X
- MathSciNet review: 2327042