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

Using multi-index notation, the Schwartz space isIts topology is generated by the displayed seminorms. The tempered distribution ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ is its continuous dual, with weak convergence defined by convergence of every pairing with a Schwartz function.

For the conventiondifferentiation under the integral and integration by parts giveThese identities bound every Schwartz seminorm of $\hat{\phi }$ by finitely many seminorms of $\phi$, proving continuity. The Fourier inversion theorem gives the continuous inverseso the Fourier transform is a continuous isomorphism of $\mathcal{S}$. By duality,defines a continuous isomorphism of ${\mathcal{S}}^{\prime }$.

For $\omega \ne 0$, insert the Gaussian damping factor $e^{-\epsilon x^2/2}$ and use the Gaussian integral:Taking $\epsilon \downarrow 0$ in ${\mathcal{S}}^{\prime }$ with the continuous square-root branch gives the Fresnel integralFor $\omega =0$, the function is constant and its Fourier transform is $\boxed{2\pi {\delta }_0}$.

For a locally integrable $u$, the change of variables formula $y=Ax$ givesThis motivatesBecause pullback by an invertible linear map acts continuously on the Schwartz space, the right side is a continuous linear functional of $\phi$. It therefore defines a tempered distribution and agrees with ordinary pullback when $u$ is a function.

For every $\phi \in \mathcal{S}$, the change of variables formula gives the identityUsing this, the distributional Fourier transform, and the pullback formula from part b,Therefore

By the spectral theorem for real symmetric matrices, write $A=O^{t}DO$ with $O$ orthogonal and $D=\mathrm{diag}({\xi }_1,\dots ,{\xi }_n)$. Part a and the tensor-product property of the Fourier transform giveThusApplying the pullback rule from part c to the orthogonal change of variables, for which $|\det O|=1$, replaces $D$ by $A$ and $D^{-1}$ by $A^{-1}$. Since determinant and signature are invariant under orthogonal conjugation,