Convergence and Holomorphicity of Series in Reinhardt Domains within Complex Analysis

Let $D \in \mathbb{C^n}$ be a Reinhardt domain and $f \in \mathcal{O}(D)$. For every $\alpha \in \mathbb{Z}^n$ define \[f_\alpha : D \to \mathbb{C}, z \mapsto \left(\frac{1}{2\pi i}\right)^{n}\int\limits_{\mathbb{T}^n}\frac{f(\zeta \cdot z)}{\zeta^{\alpha +1}}d\zeta   (1.9) \] where $\zeta \cdot z = (\zeta_1z_1, ..., \zeta_nz_n).$ Then $f_\alpha \in \mathcal{O}(D)$ and the series $\sum\limits_{\alpha \in \mathbb{Z}^n}f_{\alpha}(z)$ converges compactly on $D$ towards $f$.

Prove the following.