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Mathematics LibreTexts

5.E: Fourier Transform (Exercises)

Q5.1

Show

$$
\int_{{\mathbb{R}}^n}{\rm e}^{-|y|^2/2}\ dy=(2\pi)^{n/2}.
$$

Q5.2

Show that \(u\in \mathcal{S}(\mathbb{R}n^)\) implies \(\hat{u},\ \widetilde{u}\in\mathcal{S}(\mathbb{R}^n)\).

Q5.3

Give examples for functions \(p(x,\xi)\) which satisfy \(p(x,\xi)\in S^m\).

Q5.4

Find a formal solution of Cauchy's initial value problem  for the wave equation by using Fourier's transform.

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