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Consider $L^2(\mathbb R^n, \mathbb R^m)$. There should be a Fourier transform for these functions, like in the case $L^2( \mathbb R^n, \mathbb R )$. I wonder how these can be defined.

The application I have in mind is defining a Fourier transform for differential forms on $\mathbb R^n$.

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    Have you tried the obvious?2011-04-22
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    @Martin: What if you just do it componentwise?2011-04-22
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    A more interesting question is if we replace $\mathbb R^m$ by a separable Banach space!2011-04-22
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    Removing the differential geometry tag, because in the actual geometric setting, even the Fourier transform of scalar functions is problematic.2011-04-22
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    Of course you can try the obvious, but maybe there something you overlook and which appears only in higher dimensions.2011-04-22
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    It is not obvious that a component-wise approach is appropriate. Nice band-limited components can yield cusps, as in the [deltoid](http://en.wikipedia.org/wiki/Deltoid_curve).2011-04-22
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    One proposal is to use Clifford Algebras [link](http://portal.acm.org/citation.cfm?id=1070749), also on [Citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.93.2487).2011-04-22
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    @WillieWong : As long as the only derivatives (of this function), that we are dealing, are first order partial derivatives, then can we say that using the obvious (component-wise) would suffice without any issues? I am not trying to answer the OP question, but this is a question of mine.2017-06-22
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    I am posting it as a separate question, as I am not interested in the differential forms unlike what OP is asking.2017-06-22
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    Related : https://math.stackexchange.com/q/2332005/29872017-06-22

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