A Bernstein–Walsh Type Inequality and Applications

Abstract

AbstractA Bernstein–Walsh type inequality forC∞functions of several variables is derived, which then is applied to obtain analogs and generalizations of the following classical theorems: (1) Bochnak– Siciak theorem: aC∞function on ℝnthat is real analytic on every line is real analytic; (2) Zorn–Lelong theorem: if a double power seriesF(x,y) converges on a set of lines of positive capacity thenF(x,y) is convergent; (3) Abhyankar–Moh–Sathaye theorem: the transfinite diameter of the convergence set of a divergent series is zero.