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I have a vector valued function from $L^p(\mathbb{R}^d,\mathbb{R}^m)$. I'd like to take a Fourier transform of this and do further computations, but in all my further computations, the only derivatives of this function I use are first order partial derivatives and never ask any questions on higher order derivatives. So, can I conveniently take component-wise Fourier transform without having any issues? ($m$ Fourier transforms of $m$ real valued functions)

Rajesh D
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    What definition of the Fourier transform on vector functions are you using? This is trivially true if it is defined component wise, which is the only case that I have ever seen. – mathematician Jun 22 '17 at 06:34
  • @mathematician : Even I don't know. Thanks for the answer. I saw someone talking existence of some other definition, so I just wanted to be safe in my context. https://math.stackexchange.com/q/34485/2987 – Rajesh D Jun 22 '17 at 08:05
  • @RajeshDachiraju I have a similar problem which I've posted here: bit.ly/2JrFVkq

    So is it okey to do the Fourier transform component-wise on a vector field?

     – Ali May 10 '19 at 02:46
  • @Ali : I believe so. I dont think any other definiton exists or necessary. – Rajesh D May 10 '19 at 03:22

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