For a locally integrable $u$, the <change of variables formula> $y=Ax$ gives
$$
\int_{\mathbb R^n}u(Ax)\varphi(x)\,dx
=\frac1{|\det A|}
\int_{\mathbb R^n}u(y)\varphi(A^{-1}y)\,dy.
$$
This motivates
$$
\boxed{
\langle A^*u,\varphi\rangle
=\frac1{|\det A|}
\langle u,(A^{-1})^*\varphi\rangle}.
$$
Because pullback by an invertible linear map acts continuously on the <Schwartz space>, the right side is a continuous linear functional of $\varphi$. It therefore defines a <tempered distribution> and agrees with ordinary pullback when $u$ is a function.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 327 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-327/3/b/solution