Polynomial Approximations in a Convex Domain in ℂn with Exponential Decay Inside

Let Ω be a convex domain in ℂn satisfying certain restrictions, and let f function holomorphic continuous $$ \overline{\Omega},\kern0.5em f\kern0.5em \in \kern0.5em {H}^{r+\omega}\left(\overline{\Omega}\right) for an appropriate modulus of continuity ω. Then there exist polynomials PN, deg PN ≤ N, such that \left|f(z)-{P}_N(z)\right|\le {cN}^{-r}\omega \left(\frac{1}{N}\right),z\kern0.5em \overline{\Omega}, |f(z) − PN(z)| c exp(−c0(K)N), z ∈ K ⊂ Ω, where is any compact set strictly inside Ω.