Osgood–Hartogs-type properties of power series and smooth functions

Buma L. Fridman and Daowei Ma

Vol. 251 (2011), No. 1, 67–79

DOI: 10.2140/pjm.2011.251.67

Abstract

We study the convergence of a formal power series of two variables if its restrictions on curves belonging to a certain family are convergent. Also analyticity of a given C^∞ function f is proved when the restriction of f on analytic curves belonging to some family is analytic. Our results generalize two known statements: a theorem of P. Lelong and the Bochnak–Siciak theorem. The questions we study can be regarded as problems of Osgood–Hartogs type.

Keywords

formal power series, analytic functions, capacity

Mathematical Subject Classification 2000

Primary: 26E05, 30C85, 40A05

Milestones

Received: 5 May 2010

Revised: 24 August 2010

Accepted: 14 December 2010

Published: 22 April 2011

Proposed: Paul Yang