Since \( f, g \in \mathcal{S}(\mathbb{R}^n) \), the inverse \(\mathcal{F}^{-1}f\) is also in \(\mathcal{S}(\mathbb{R}^n)\) due to (0x67d92de3) . With (0x67ece8ef) we deduce that \( \mathcal{F}^{-1}f \in L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n) \) and similarly \( \mathcal{F}^{-1}g \in L^2(\mathbb{R}^n) \).

The convolution theorem implies

\[ \mathcal{F}(\mathcal{F}^{-1}f * \mathcal{F}^{-1}g) = (\sqrt{2\pi})^n \mathcal{F}(\mathcal{F}^{-1}f) \cdot \mathcal{F}(\mathcal{F}^{-1}g) = (\sqrt{2\pi})^n \hat{f} \cdot \hat{g} \]