Is it possible to use Fourier transform to evaluate the integral $$\int_{-\infty}^{+\infty}\left(\frac{\sin x}{x}\right)^3dx$$
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1See (37) here – K.defaoite Dec 08 '22 at 14:36
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https://math.stackexchange.com/questions/894649/closed-form-for-integral-of-integer-powers-of-sinc-function – Doug Dec 08 '22 at 15:13
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Hint: Let $$ f(x)=\bigg(\frac{\sin x}{x}\bigg)^2,g(x)=\frac{\sin x}{x} $$ and then $$ \hat f(\xi)=?, \hat g(\xi)=?$$ Next use the Parsevell Identity $$ \int_{-\infty }^{\infty }f(x)\bar{g(x)} dx=\int_{-\infty }^{\infty }\hat f(\xi) \bar{\hat g(\xi)}d\xi. $$
xpaul
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