Bernstein–Walsh Type Theorems for Real Analytic Functions in Several Variables

Abstract

The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions in the complex plane to real analytic functions in ℝN. In particular, we investigate the polynomial approximation behavior for functions F:L→ℂ, L={(Re z,Im z):z∈K}, of the structure \(F=g\overline{h}\), where g and h are holomorphic in a neighborhood of a compact set K⊂ℂN. To this end the maximal convergence number ρ(Sc,f) for continuous functions f defined on a compact set Sc⊂ℂN is connected to a maximal convergence number ρ(Sr,F) for continuous functions F defined on a compact set Sr⊂ℝN. We prove that ρ(L,F)=min {ρ(K,h)),ρ(K,g)} for functions \(F=g\overline{h}\) if K is either a closed Euclidean ball or a closed polydisc. Furthermore, we show that min {ρ(K,h)),ρ(K,g)}≤ρ(L,F) if K is regular in the sense of pluripotential theory and equality does not hold in general. Our results are based on the theory of the pluricomplex Green’s function with pole at infinity and Lundin’s formula for Siciak’s extremal function Φ. A properly chosen transformation of Joukowski type plays an important role.