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)

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)

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)

You're right. That's a very dubious statement. Like you said, consider the direct analog for the Fourier series, then you have the symmetric partial sum, given by the Dirichlet kernel, which diverges in general. A more correct approach would be to stick a convolution approximate unit inside the integral, and try to obtain convergence in either in L^p or a.e. depending on the decay properties.

For a L^2 function f, the integral does converge to f in that sense. Mct mht (talk) 09:53, 7 August 2012 (UTC)

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)

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)

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)

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)

What? I am 12 and I can understand it. This is one of the clearest articles on all of Wikipedia. — Preceding unsigned comment added by 220.255.2.55 (talk) 03:37, 26 November 2011 (UTC)

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)

Yeah, Jim, you should really get to it. :)

I agree with all the comments and the proposed outline. Article is really amiss to not state the L¹ (or l¹l, even) case, and consequent implications such as continuity almost everywhere. Also, the current proof is unnecessarily cluttered. Any approximate unit in L¹ with sufficiently nice decay properties will do. In that sense, there's nothing special about the Gaussian kernel used. Mct mht (talk) 09:32, 7 August 2012 (UTC)

What is l¹l? Do you mean $L_{\mathrm {loc} }^{1}$? Or $\ell ^{1}$ for Fourier series? Quietbritishjim (talk) 11:37, 7 August 2012 (UTC)

By the way, although you were joking, I did actually start to rewrite this page, but I haven't had time to finish it yet. If you have any views on that it would be appreciated (bearing in mind that it's clearly not finished). Quietbritishjim (talk) 16:11, 7 August 2012 (UTC)

Yes, I meant the Fourier series case.

Your draft in progress looks good. I was going to leave some general comments/suggestions there but here seems a better place.

• The proof in the article right now proves the classical Fourier Inversion Formula: if both f and f^ lie in L^1, then f = f^~ almost everywhere (by ~ I mean the inverse transform; the proper symbols are not available to me for some reason.)
• The statement "This establishes that the Fourier transform is an invertible map of the Schwartz space to itself" is misleading. Bringing in the Schwartz class S is really unnecessary for classical inversion Formula. Bijectivity of Fourier transform follows from general inversion, and if you want to consider the distribution case, what you really need is homeomorphism, and that's a separate additional argument.
• There is also some superfluous comments on unitarity on L^2; again misleading and unnecessary.
• To repeat myself, in my opinion, the article should point out the underlying technique, rather than emphasizing the particulars, of the proof. Regardless of whether f^ lies in L^1 or not, the trick is to consider $\int _{\mathbb {R} ^{n}}\Phi _{t}(\xi )e^{2\pi ix\cdot \xi }\,({\mathcal {F}}f)(\xi )\,d\xi$ for an approximate unit $\Phi _{t}(\xi )$ of C_0 that also lies in L^1. For $\Phi _{t}(\xi )$ with sufficiently fast decay, the integral converges to f almost everywhere. When f^ is L^1, we then have inversion because of dominated convergence. The existence of such approximate units can be point out afterwards.
• The same trick applies to Fourier series, giving, for example, Cesaro and Abel sums. This connection should be pointed out. Mct mht (talk) 08:00, 8 August 2012 (UTC)

I've updated my draft of the article, hopefully to the point that it can replace the current version. I'll make a new section at the bottom to discuss it, but here are my responses to your specific comments.

• The draft you last read had a copy of the existing article at the bottom (to enable me to compare and copy/paste things easily) and I think you misread this as being a part of the proposed replacement. For example your second bullet point applies to the existing article, not my proposed replacement, and I think that's true of your first point too (outside of the conditions on the function section I assumed that f and f^ are integrable, plus I assumed that f is continuous because I thought it best to avoid using the term "almost everywhere").
• The draft you last read *did* have some comments on unitarity in L^2 in it. But these have now been moved to a section in the main Fourier transform article, apart from a passing mention in the applications section.
• To me, this seemed to be the part of the article with the least problems, so I didn't spend much time improving it. I don't object to your suggestion, I just didn't have time to do it myself. (The only changes I made were adjusting it so that it applies to the new conditions (f and f^ in L^1 and continuous), moving the observation that the inverse is two-sided to the statement section, and some minor formatting changes to hopefully make it clearer.)
• Connection with Fourier series: I agree that, beyond the "see also" link at the bottom, some discussion of the relationship between Fourier series and Fourier transform inversion would be nice, but I made no attempt to do this. Quietbritishjim (talk) 01:05, 31 December 2012 (UTC)

error?

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

$\lim _{\varepsilon \to 0}\int e^{2\pi ix\cdot \xi }{\widehat {f}}(\xi )e^{-\pi \varepsilon ^{2}|\xi |^{2}}d\xi$

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? —Preceding unsigned comment added by 79.129.223.145 (talk) 10:28, 29 July 2010 (UTC)

Right about the error, I fixed it. Compsonheir (talk) 03:46, 1 March 2011 (UTC)

Article rewrite - request for comments

I have mostly rewritten this article. I haven't tried to make the perfect article, only one that's better than the current one. The new version is currently over at User:Quietbritishjim/Fourier_inversion_theorem, but before I move it over here I'm giving fair warning so anyone interested can voice their comments / objections.

Here is a summary of the changes (and things that haven't changed):

• Almost all mention over what function space is being acted on is moved to a separate section (and especially removed from the lead). Outside that section it is assumed that the function is continuous, integrable, and has integrable Fourier transform. This may not be the best choice, but any choice would be a compromise.
• I restated the theorem in the main body of the article, since every article is meant to make sense without its lead.
• The section on square-integrable periodic functions belongs in convergence of Fourier series, if anywhere (it might be too technical even for that article), so I removed that and I'll put it in the talk page of that article for anyone who wants to add it there. It is in the collapsed box below.

Extended content

Fourier transforms of square-integrable functions Plancherel theorem allows the Fourier transform to be extended to a unitary operator on the Hilbert space of all square-integrable functions, i.e., all functions satisfying $\int _{-\infty }^{\infty }\left|f(x)\right|^{2}\,dx<\infty .$ Therefore it is invertible on L2. In case f is a square-integrable periodic function on the interval $[-\pi ,\pi ]$, it has a Fourier series whose coefficients are ${\widehat {f}}(n)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)\,e^{-inx}\,dx.$ The Fourier inversion theorem might then say that $\sum _{n=-\infty }^{\infty }{\widehat {f}}(n)\,e^{inx}=f(x).$ What kind of convergence is right? "Convergence in mean square" can be proved fairly easily: $\lim _{N\rightarrow \infty }\int _{-\pi }^{\pi }\left|f(x)-\sum _{n=-N}^{N}{\widehat {f}}(n)\,e^{inx}\right|^{2}\,dx=0.$ What about convergence almost everywhere? That would say that if f is square-integrable, then for "almost every" value of x between 0 and 2π we have $f(x)=\lim _{N\rightarrow \infty }\sum _{n=-N}^{N}{\widehat {f}}(n)\,e^{inx}.$ This was not proved until 1966 in (Carleson, 1966). For strictly finitary discrete Fourier transforms, these delicate questions of convergence are avoided. Lennart Carleson (1966). On the convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157.

• I briefly discussed applications and properties of the inverse transform. I think these sections are not very satisfactory, but IMO they're better than nothing.
• Apart from very minor formatting fixes, and noting that it works with the conditions stated above, the proof is pretty much unmodified.
• I have not added any references, so as before the article has only Folland's book Introduction to PDEs. The article really needs more references, especially inline ones.
• As User:Mct mht suggested in another section on this talk page, it would be good to discuss the connection between this theorem and Fourier series, but I haven't done that (aside from keeping the current link in the "see also" section at the bottom).

So I think my proposed new article is a bit flawed but an improvement on the current situation. Quietbritishjim (talk) 01:10, 31 December 2012 (UTC)

That's a clear improvement. One small thing I would suggest is moving the section on conditions so it immediately follows the section stating the theorem. It seems strange to have it at the very end. Sławomir Biały (talk) 12:04, 31 December 2012 (UTC)

These revisions amount to a huge improvement; you should put them in the live article ASAP. My only suggestions are fairly minor. Although you refer to the "density argument" used to define the Fourier transform on L² functions by directing the user to the Fourier transform page, I think it bears some repeating within the article itself. Also, I'd move the section for L² functions before the section for absolutely integrable functions in 1D as it strikes me as being more important. Finally, there are also some minor typographical things that I should have done differently when I wrote the proof in the first place, but I can toy with those later once your revision goes live. Compsonheir (talk) 19:49, 31 December 2012 (UTC)

I don't think it makes sense to treat the L^2 case before the L^1 case. Sławomir Biały (talk) 01:17, 1 January 2013 (UTC)

Thanks for your comments. I updated it a bit:

• I added a section stating that the theorem is the analogue for Fourier transforms of the convergence of Fourier series.
• I moved the conditions section up like Sławomir Biały suggested.
• I explicitly mentioned an L^2 approximation sequence like Compsonheir suggested. I also gave more details in the tempered distribution case.
• I updated the notation and formatting of the proof to match the rest of the article (although if there are still things you'd prefer changed, of course go ahead Compsonheir).

Since a couple of people have looked at the draft and there hasn't been huge uproar, I'll go ahead and move it to the live article. Quietbritishjim (talk) 20:25, 1 January 2013 (UTC)

Error in the proof

for the dominated convergence theorem, should'nt it be $f\in L^{1}(\mathbb {R} ^{n})$ instead of ${\mathcal {F}}f\in L^{1}(\mathbb {R} ^{n})$?

For $\varepsilon \leq 1$ we have

$\left|e^{-\pi \varepsilon ^{2}|\xi |^{2}+2\pi ix\cdot \xi }({\mathcal {F}}f)(\xi )\right|=e^{-\pi \varepsilon ^{2}|\xi |^{2}}|({\mathcal {F}}f)(\xi )|\leq e^{-\pi |\xi |^{2}}|({\mathcal {F}}f)(\xi )|=:g(\xi )$,

so if $g\in L^{1}(\mathbb {R} ^{n})$, this would work as our dominating function. This is true, since

$\int _{\mathbb {R} ^{n}}g(\xi )\,d\xi \leq \|{\mathcal {F}}f\|_{L^{\infty }(\mathbb {R} ^{n})}\int _{\mathbb {R} ^{n}}e^{-\pi |\xi |^{2}}\,d\xi \leq \|f\|_{L^{1}(\mathbb {R} ^{n})}<\infty$.

Headfime (talk) 15:34, 16 December 2017 (UTC)