{scope}

The angular-frequency <Fourier transform> and its inverse on $\mathcal S(\mathbb R^n)$ are
$$
\widehat\varphi(\lambda)=\int_{\mathbb R^n}e^{-i\lambda\cdot x}\varphi(x)\,dx,
\qquad
\varphi(x)=\frac1{(2\pi)^n}\int_{\mathbb R^n}e^{i\lambda\cdot x}\widehat\varphi(\lambda)\,d\lambda.
$$
It extends to a <tempered distribution> by duality:
$$
\langle\widehat u,\varphi\rangle=\langle u,\widehat\varphi\rangle.
$$

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 327 / 1 / b

 b

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-327/1/b