Talk:Fourier inversion theorem

(talk before TOC)

There is a bunch of broken math code on this page, but I don't know how to fix it. Someone needs to do this! --Shoofle

Looks OK to me now. Try again. Or maybe try on a different browser? Maybe just a temporary glitch? Michael Hardy 18:43, 28 August 2006 (UTC)[reply]

Hrm, I see the same thing. The error reads: "Failed to parse (Can't write to or create math output directory): ..." Funny thing is, when I look through the history, even the current version doesn't give the same error. Edit this page/preview also 'fails' to show an error! Not sure how or why this is borfing like this. Btw, I'm getting the same error in Safari and Opera, but this doesn't seem like a problem is happening client-side. — gogobera (talk) 23:13, 17 May 2007 (UTC)[reply]

Reference?

When discussing the Fourier inversion for L¹ functions we have the statement

In such a case, the integral in the Fourier inversion theorem above must be taken to be an improper integral (Cauchy principal value)

\lim _{b\rightarrow \infty }{\frac {1}{2\pi }}\int _{-b}^{b}({\mathcal {F}}f)(t)e^{itx}\,dt

rather than a Lebesgue integral.

And I am concerned about the content here. I don't believe that the above limit exists for general L¹ function. (My reasons being that the corresponding statement is not true for Fourier series, as shown by Kolmogorov, and the Hilbert transform is not bounded on L¹). Is there a reference for this? Thenub314 (talk) 14:59, 8 October 2008 (UTC)[reply]

article needs explanation of the general case for a locally compact group

this is necessary e.g. to read Tate's thesis. i do not have the expertise to do it. —Preceding unsigned comment added by 76.182.61.207 (talk) 18:09, 14 January 2009 (UTC)[reply]

proof

I don't know if a proof would be appropriate here, but can anyone provide a source that actually includes a proof? Its probably online somewhere, but it has eluded me so far. 146.6.200.213 (talk) 22:27, 11 May 2009 (UTC)[reply]

There is a proof in the book "Introduction to Partial Differential Equations" by Gerald Folland, I added a proof here roughly along those lines. He places the

2\pi

in a different place, which I followed, I like it better his way. Let me know if anything is unclear or should be changed. Compsonheir (talk) 12:47, 25 June 2009 (UTC)[reply]

The article is completely useless for any non-mathematician.

Which means that the only people who can read it, are those who don’t need it anymore.
Well done. Really. Impressive job! Way to go!

The article just throws formulas in your face, and doesn’t even care to explain the history or examples/problem that it started with. It just vomits symbols and relations without meaning in your face.
Maybe you mathematician-types understand this better: For any real human, this article is not is the set of understandable articles, for any time or space location in all of space-time!
Read up on this article that explains how you can make a concept understandable for actual real humans!

Conclusion: You massively, epically and horribly fail! At remaining an actual real human, and at making useful articles.
— 94.220.250.151 (talk) 23:34, 16 October 2009 (UTC)[reply]

Improvements

Despite the crass way that someone said this above, this article could actually do with significant improvement (but despite what that guy said, it's fantastic start). Here are problems that I see, and suggestions to fix them:

The lead section could do with an intuitive argument. I have a good one, based on the L¹ proof, that I'll add when I get time.
The article uses the usual, somewhat indirect, proof on the Schwartz space. The usual proof for L¹, while not so elegant, is a lot more direct and I think easier for the beginner to understand. What's more it will link up with the intuitive argument I plan on adding.
The lead section says (about not clearly declaring what function space you're using) that "this way of stating a Fourier inversion theorem sweeps some more subtle issues under the carpet". While I agree that the space that we take the Fourier transform on is important, I don't think it's such a critical thing that we need to spend half the lead section talking about it, which should concentrate more on the idea with maybe a couple of sentences on this point.
On the other hand a new section that clearly breaks down the four most common cases (Schwartz space, L¹, L², tempered distributions) and the dependencies between their proofs would be useful.

Quietbritishjim (talk) 17:43, 2 March 2010 (UTC)[reply]

error?

I believe the integral in the last part of the proof of the theorem should read:

<math>\lim_{\varepsilon\to 0}\int e^{2\pi i x\cdot\xi}\widehat{f}(\xi)e^{-\pi\varepsilon^2|\xi|^2}d\xi

Nice article, and I think there is a good point for its existence. Why there is no reference to it by the other Fourier-related articles?