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Define $\hat{f}(\xi)\equiv F(f)(\xi):=\int_{\mathbb{R}}e^{ix.\xi}f(x)dx $

My question is: if we consider $x^{\alpha}$ as a distribution then what is $ F(x^{\alpha})(\xi)$ where $0<\alpha\in\mathbb{R}$. thanks

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    According to Maple, for positive integer $k$, the Fourier transform of $x^k$ is a certain constant multiple of the $k$th derivative of the Dirac delta function. (Start with the Fourier transform of the constant 1 as $2\pi$ times the delta function, then use the rule on how multiplying by $x$ corresponds to differentiating the transform.)2012-05-15

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