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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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Praise the lord Wolfram!!^5 – yohBS Feb 13 '12 at 7:16
@steven how do you define $x^{\pi}$ for $x<0$? – Andrew Feb 13 '12 at 7:56
The distribution defined by $x^\alpha$ (supported on $\mathbb R_{>0}$) is an example of a homogeneous distribution of degree $\alpha$. It is pretty clear (if you use proper notation) that the Fourier transform of a homogeneous distribution of degree $\alpha$ on $\mathbb R^n$ is homogeneous of degree $-\alpha-n$. To get everything exactly correct requires some work. This stuff is well known and should be in most/many books on distribution theory. E.g., see Vol I of Hormander's book "The Analysis of Linear Partial Diff. Ops", Example 7.1.17. – B R Apr 14 '12 at 1:37
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.) – GEdgar May 15 '12 at 2:53

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