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)
2026-04-13 08:22:20.1776068540
Fourier transform of vector valued function, would component wise approach suffice for this application?
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