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The Fourier transform can be defined on $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$, and we can extend this to $X:=L^2(\mathbb{R}^n)$ by a density argument.

Now, by Plancherel we know that $\|\widehat{f}\|_2 = \|f\|_2$, so the Fourier transform is an isometry on this space.

My question now is, what is a theorem that guarantees that the Fourier transform has a fixed point on $L^2$? I know the Gaussian is a fixed point, but I'm also interested in other integral transforms, but I just take the Fourier transform as an example.

The Banach Fixed Point Theorem does not work here since we don't have a contraction (operator norm $< 1$). Can we apply the Tychonoff fixed point theorem? Then we would need to show that there exists a non-empty compact convex set $C \subset X$ such that the Fourier transform restricted to $C$ is a mapping from $C$ to $C$. Is this possible?

If we have a fixed point, what would be a way to show it is unique? By linearity we obviously have infinitely many fixed points of we have at least two of them.

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mathoverflow.net/questions/12045/… –  anonymous Sep 12 '10 at 13:52
    
There they directly compute it but I just want to know if we can use one of the fixed point theorems. –  Jonas Teuwen Sep 12 '10 at 14:29
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I don't think you can get the results you want with fixed point theorems. –  John D. Cook Sep 12 '10 at 22:29
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up vote 2 down vote accepted

My Functional Analysis Fu has gotten bit weak lately, but I think the following should work:

The Schauder fixed point theorem says, that a continuous function on a compact convex set in a topological vector space has a fixed point. Because of isometry, the Fourier transform maps the unit ball in $L^2$ to itself. Owing to the Banach Alaoglu theorem, the unit ball in $L^2$ is compact with respect to the weak topology. The Fourier transform is continuous in the weak topology, because if $( f_n, \phi ) \to (f, \phi)$ for all $\phi \in L^2$, then $$ (\hat{f}_n, \phi) = (f_n, \hat{\phi}) \to (f, \hat{\phi}) = (\hat{f}, \phi). $$

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True, but if say the continuous function is $f\mapsto -f$ then the only fixed point is zero, which I don't think Jonas is looking for. –  Robin Chapman Sep 13 '10 at 6:53
    
@Robin Chapman: Could you elaborate on that remark? What function do you mean? –  Jonas Teuwen Sep 13 '10 at 17:24
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@Jonas: Robin is quite rightly pointing out, that my approach just shows that there is a fixed point in the unit ball of L^2. However, since 0 is in the unit ball and it is trivially a fixed point of the Fourier transform, this does not tell us anything new. –  Michael Ulm Sep 14 '10 at 5:02
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Ah, right. Okay, then this is not the answer I'm looking for, sorry ;-). Maybe John D. Cook has a point that it might not work with fixed-point theorems? –  Jonas Teuwen Sep 14 '10 at 8:52
    
Since nobody else answers I will accept this as answer. The used technique can be of some value anyway. –  Jonas Teuwen Oct 11 '10 at 18:57
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