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I've frequently heard that a Fourier transform is "just a change of basis".

However, I'm not sure whether that's correct, in terms of the terminology of "change of basis" versus "transformation" in linear algebra.

Is a Fourier transform of a function merely a change of basis (i.e. the identity transformation from the original vector space onto itself, with merely a change of basis vectors), or is indeed a linear transformation from the original vector space onto another vector space (the "Fourier" vector space, so to speak)?

Also: Does the same answer hold for other similar transforms (e.g. Laplace transforms)? Or is there a different terminology for those?

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up vote 18 down vote accepted

Calling the Fourier transformation "a change of basis" is misleading in the sense that the Fourier transformation is a unitary (linear) transformation between two different Hilbert spaces, namely $L^2(\mathbb R)$ and $L^2(\hat{\mathbb R})$.

Here $\hat{\mathbb R}$ is the dual group of $\mathbb R$. It turns out that $\hat{\mathbb R}\cong\mathbb R$, but there is no canonical isomorphism. So, only if you fix some arbitrary isomorphism $\hat{\mathbb R}\cong\mathbb R$, you can consider the Fourier transformation as a unitary transformation from some Hilbert space to itself, which really is essentially a change of basis.

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Whoops. Sorry. It's late. – Qiaochu Yuan Mar 23 '12 at 9:47
When is a morphism canonical? The construction of $\hat{\mathbb{R}}$ relies on $\text{Hom}(\mathbb{R},T)$, right? Does this miss any of the structure of $\mathbb{R}$? Don't both not even have a metric which can be matched etc.? – NikolajK Nov 3 '12 at 1:26

Rasmus' answer is probably what you want. But remember that the Fourier transform can also be defined in a discrete domain, and we have the DTFT and the DFT. The later, finite, maps a sequence of N complex numbers to other complex sequence of same length, via a linear unitary transform. In this case (and, perhaps, only in this case) we can confidently say that it's a "change of basis".

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+1 The context was certainly also related to DTFTs and DFTs/DFSs, so this answer points out something important; thanks. – Mehrdad Mar 24 '12 at 0:59

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