The Fourier transform \(\mathcal{F}\colon \mathcal{S}(\mathbb{R}^d)\to \mathcal{S}(\mathbb{R}^d)\) defined on the Schwartz space is a linear isomorphism . The inverse \(\mathcal{F}^{-1}\) is the inverse Fourier transform .
This extends on \(\mathcal{F}\colon L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)\).
Remarks
- It is isometric in \(L^2\) (see Plancherel theorem ).