Holomorphic extensions from open families of circles

Author:
Josip Globevnik

Journal:
Trans. Amer. Math. Soc. **355** (2003), 1921-1931

MSC (2000):
Primary 30E20

DOI:
https://doi.org/10.1090/S0002-9947-03-03241-0

Published electronically:
January 8, 2003

MathSciNet review:
1953532

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Abstract: For a circle write . A continuous function on extends holomorphically from (into the disc bounded by ) if and only if the function defined on has a bounded holomorphic extension into . In the paper we consider open connected families of circles , write , and assume that a continuous function on extends holomorphically from each . We show that this happens if and only if the function defined on has a bounded holomorphic extension into the domain for each open family compactly contained in . This allows us to use known facts from several complex variables. In particular, we use the edge of the wedge theorem to prove a theorem on real analyticity of such functions.

**[AG]**M. Agranovsky and J. Globevnik: Analyticity on circles for rational and real-analytic functions of two real variables. J. d'Analyse Math., to appear.**[E]**L. Ehrenpreis: Three problems at Mount Holyoke. Radon Transforms and Tomography. Contemp. Math. 278 (2001) 123-130. MR**2002d:44002****[G1]**J. Globevnik: Analyticity on rotation invariant families of curves. Trans. Amer. Math. Soc. 280 (1983) 247-254. MR**85f:30060****[G2]**J. Globevnik: Analyticity on families of curves. Talk at Bar Ilan University, November 1987**[G3]**J. Globevnik: Testing analyticity on rotation invariant families of curves. Trans. Amer. Math. Soc. 306 (1988) 401-410. MR**89g:30078****[G4]**J. Globevnik: Holomorphic extensions and rotation invariance. Complex Variables Theory Appl. 24 (1993) 49-51. MR**95g:30002****[R]**W. Rudin:*Lectures on the edge of the wedge theorem.*Conf. Ser. in Math. No. 6. American Mathematical Society, Providence, RI, 1971. MR**46:9389****[T]**A. Tumanov: A Morera type theorem in the strip. Preprint.

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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

DOI:
https://doi.org/10.1090/S0002-9947-03-03241-0

Received by editor(s):
July 24, 2002

Published electronically:
January 8, 2003

Dedicated:
Dedicated to Professor Ivan Vidav on the occasion of his eighty-fifth birthday

Article copyright:
© Copyright 2003
American Mathematical Society