Osgood-Hartogs-type properties of power series and smooth functions (Q539020)

The authors extend two theorems: one of Lelong, and another of Bochnak and Siciak about convergence of formal power series of two variables. In the first theorem it is shown that, if a power series is convergent when restricted to curves of a certain family, then it is convergent all together. Then in the next theorem the authors prove the real analyticity near the origin of a smooth function \(f\), when the restriction of \(f\) to certain analytic curves is real analytic. In the end they give examples showing that the conditions in the theorems are necessary in order to obtain convergence of the power series.