Define for \(f\in L^1(\mathbb{R}^n)\) and \(\xi\in \mathbb{R}^n\) the Fourier transform of \(f\)
\begin{equation*} (\mathcal{F}f)(\xi)=\frac{1}{(\sqrt{2\pi})^{n}}\int_{\mathbb{R}^n} f(x)e^{-ix \xi}\;dx. \end{equation*}Some authors denote the Fourier transform of \(f\) with \(\hat{f}\).
Remarks
Fourier transform defined on different domains Link to heading
- \(\mathcal{F}\colon L^1(\mathbb{R}^n)\to C_0(\mathbb{R}^n)\)
- \(\mathcal{F}\colon \mathcal{S}(\mathbb{R}^n)\to \mathcal{S}(\mathbb{R}^n)\)
- \(\mathcal{F}\colon L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)\)
- \(\mathcal{F}\colon L^p(\mathbb{R}^n)\to L^q(\mathbb{R}^n)\) with \(1\le p\le 2\) and \(\frac{1}{p}+\frac{1}{q}=1\)
- \(\mathcal{F}\colon \mathcal{S}'(\mathbb{R}^n)\to \mathcal{S}'(\mathbb{R}^n)\)