Polynomial approximations in a convex domain in $\mathbb C^n$ with the exponential decaying inside

Let $\Omega$ be convex domain in $\mathbb C^n$ satisfying some restrictions, $f$ be holomorphic in $\Omega$ and continuons in $\overline{\Omega}$, $f\in H^{r+\omega}(\overline{\Omega})$ with a modulus of continuity $\omega$. Then there are polynomials $P_N$, $\deg P_N\le N$, such that $ |f(z)-P_N(z)| \le cN^{-r}\omega(\frac{1}{N})$, $z \in \overline{\Omega}$, and $|f(z)-P_N(z)| \le c \exp(-c_0(K)N)$, $z\in K\subset \Omega$, where $K$ is any compact strictly inside $\Omega$.