The Runge approximation theorem for generalized polynomial hulls

الأصل · EN

It is known from the Runge approximation theorem that every function which is holomorphic in a neighborhood of a compact polynomially convex set K⊂ ⁿ can be approximated uniformly on K by analytic polynomials. We shall here prove the same result when the role of the polynomially convex hull K is played by the generalized polynomial hull hq(K) introduced by Basener and which can be defined, for each integer q∈ 0,...,n-1, by hq(K)=∈ [z₁,...,zₙ] Aₚ where Aₚ={z∈ ⁿ: |P(z)|≤ δₖ(P,z)}, and where δₖ(P,z) denotes the lowest value of ||P||ₖ∩ f⁻¹₍₀₎ when f ranges in the set of holomorphic polynomial maps ⁿ→ vanishing at z.