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I should derive

$$\int_{R^{3}}e^{-2\pi ix \cdot \xi}\left(\frac{1}{4\pi}\int_{S^2}f(x-\gamma t)d\sigma(\gamma)\right)dx=\hat{f}(\xi)\frac{\sin(2\pi |\xi| t)}{2\pi |\xi|t}$$

I already calculate the useful lemma : $\frac{1}{4\pi}\int_{S^2}e^{-2\pi i \xi \cdot \gamma}d\sigma (\gamma)=\frac{\sin(2\pi |\xi|)}{2\pi |\xi|}$ .

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up vote 2 down vote accepted


You can start with the right hand side and show that it is equal to $$\frac{1}{4\pi} \int_{\mathbb{R}^3} \int_{S^2} e^{-2\pi i \xi \cdot (x+t\gamma)} f(x) dx d\sigma(\gamma), $$ Then use substitution to show it is the same as the left hand side.

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