text: # What's the Fourier transform of these functions?
  
  1. The Fourier transform of $|x|^{\alpha}$.
  
  This is the Fourier transform of a homogeneous function, and there are several cases of various $\alpha$: when $a\leq -n$, it's not a temperate distribution; when $-n<\alpha<0$,then the Fourier transform is $c_{n}|\xi|^{-n+\alpha}$, where $c_{n}$ is some constant; when $\alpha=2k$,a positive even number, then it's Fourier transformation is $(-\Delta)^{k}\delta_{0}$.
  
  My question is when $\alpha$ is any positive number (not the even case), then what's the Fourier transform of it ?
  
  2. The Fourier transform of $e^{it|x|}$ ?
  
  (the Fourier transforms I have mentioned here are in the sense of temperate distributions)
  
  -
  What is $n$? (need more characters) – Fabian Aug 16 '12 at 11:51
  n is the dimension of the space – sun Aug 16 '12 at 13:17
  
  ${1.}$ The corresponding operator is denoted $(-\Delta)^{\alpha/2}$ and is called fractional Laplacian. Basically the Fourier transform of $|x|^\alpha$ "should be" a homogeneous function $|\xi|^{-n-\alpha}$. Since it is not locally summable it is not a kernel of an integral operator. But it defines an elliptic linear integro-differential operator of order $\alpha$. For $0<\alpha<2$ $$(-\Delta)^{\alpha/2} u(x) = -c_{n,\alpha} \int_{\mathbb{R}^n}\frac{u(x-y)-2u(x)+u(x+y) }{|y|^{n+\alpha}}dy.$$
  One-dimensional analogue can be this. Function $|x|^{-1}$ is not locally summable and thus doesn't define a regular distribution. But it can be used to define a distribution as $$({\cal P}\frac1{|x|},\varphi)= \int_{|x|\le 1}\frac{\varphi(x)-\varphi(0)}{|x|}\,dx+ \int_{|x|> 1}\frac{\varphi(x)}{|x|}\,dx.$$
explanation: The document contains a series of questions and answers related to the Fourier transforms of specific functions. It starts with a question about the Fourier transform of \( |x|^\alpha \) for various values of \(\alpha\), and then provides detailed answers for different cases. It also includes a question about the Fourier transform of \( e^{it|x|} \) and provides an answer involving the fractional Laplacian and a specific operator.
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
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distilabel_metadata.content: # What's the Fourier transform of these functions?
  
  1. The Fourier transform of $|x|^{\alpha}$.
  
  This is the Fourier transform of a homogeneous function, and there are several cases of various $\alpha$: when $a\leq -n$, it's not a temperate distribution; when $-n<\alpha<0$,then the Fourier transform is $c_{n}|\xi|^{-n+\alpha}$, where $c_{n}$ is some constant; when $\alpha=2k$,a positive even number, then it's Fourier transformation is $(-\Delta)^{k}\delta_{0}$.
  
  My question is when $\alpha$ is any positive number (not the even case), then what's the Fourier transform of it ?
  
  2. The Fourier transform of $e^{it|x|}$ ?
  
  (the Fourier transforms I have mentioned here are in the sense of temperate distributions)
  
  -
  What is $n$? (need more characters) – Fabian Aug 16 '12 at 11:51
  n is the dimension of the space – sun Aug 16 '12 at 13:17
  
  ${1.}$ The corresponding operator is denoted $(-\Delta)^{\alpha/2}$ and is called fractional Laplacian. Basically the Fourier transform of $|x|^\alpha$ "should be" a homogeneous function $|\xi|^{-n-\alpha}$. Since it is not locally summable it is not a kernel of an integral operator. But it defines an elliptic linear integro-differential operator of order $\alpha$. For $0<\alpha<2$ $$(-\Delta)^{\alpha/2} u(x) = -c_{n,\alpha} \int_{\mathbb{R}^n}\frac{u(x-y)-2u(x)+u(x+y) }{|y|^{n+\alpha}}dy.$$
  One-dimensional analogue can be this. Function $|x|^{-1}$ is not locally summable and thus doesn't define a regular distribution. But it can be used to define a distribution as $$({\cal P}\frac1{|x|},\varphi)= \int_{|x|\le 1}\frac{\varphi(x)-\varphi(0)}{|x|}\,dx+ \int_{|x|> 1}\frac{\varphi(x)}{|x|}\,dx.$$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a series of questions and answers related to the Fourier transforms of specific functions. It starts with a question about the Fourier transform of \( |x|^\alpha \) for various values of \(\alpha\), and then provides detailed answers for different cases. It also includes a question about the Fourier transform of \( e^{it|x|} \) and provides an answer involving the fractional Laplacian and a specific operator.
  
  Label: multiple-question-and-answer