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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? –  Glen Wheeler Apr 22 '11 at 10:23
@Martin: What if you just do it componentwise? –  Jonas Teuwen Apr 22 '11 at 11:53
A more interesting question is if we replace $\mathbb R^m$ by a separable Banach space! –  Jonas Teuwen Apr 22 '11 at 13:36
It is not obvious that a component-wise approach is appropriate. Nice band-limited components can yield cusps, as in the deltoid. –  yasmar Apr 22 '11 at 20:14
One proposal is to use Clifford Algebras link, also on Citeseer. –  yasmar Apr 22 '11 at 20:15

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