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How to compute $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{[-1,1]^n}\exp[2\pi i(\theta_1 v_1.x+\theta_2v_2.x)]d^nx d\theta_1d\theta_2$
Let $\mathbf{v}_1, \mathbf{v}_2$ be two vectors in $\mathbb{R}^n$. I would like to compute the following singular integral:
$$\int_{-\infty}^{ \infty} \int_{-\infty}^{\infty}
\int_{[-1,1]^n}
e(\...
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How to prove the equality on the Fourier transformation of measure? [closed]
I cannot prove the following equality on the Fourier transformation of measure:
let $\mu$ be a finite Borel measure on $R^d,$ then
$$\lim\limits_{T\to \infty}\frac{1}{(2T)^d}\int_{[-T,T]^d}|\widehat{\...
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