The Fourier transform \(\mathcal{F}\colon L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)\) coincides on \(L^2(\mathbb{R}^n)\cap L^1(\mathbb{R}^n)\) with the definition for \(L^1(\mathbb{R}^n)\). Otherwise, we can approximate it. To be more precise, we define for \(R>0\)
\[ g_R(\xi)=\frac{1}{(2\pi)^{n/2}} \int_{B_R} f(x)\e^{-x \xi} \; dx. \]Then for every \(f\in L^2(\mathbb{R}^n)\) we have
\[ g_R \xrightarrow{L^2} \mathcal{F}f \quad \text{as } R\to \infty. \]