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