Osgood-Hartogs type properties of power series and smooth functions

TL;DR

This paper investigates conditions under which formal power series and smooth functions are analytic based on their restrictions to certain curves, extending classical theorems in complex and real analysis.

Contribution

It generalizes theorems of Lelong and Bochnak-Siciak by establishing new criteria for analyticity from curve restrictions in Osgood-Hartogs-type problems.

Findings

Convergence of power series on specific curve families implies overall convergence.

Smooth functions are analytic if their restrictions to certain curves are analytic.

Extends classical results to broader classes of functions and curve families.

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^\infty $ 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 fall into the category of "Osgood-Hartogs-type" problems.