Talk:Fourier transform

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Citation issue[edit]

FYI: I'm seeing a red warning message: Harv error: link to #CITEREFHewittRoss1971 doesn't point to any citation.

I dont have time to fix it now, but I thought Id post a notice. --Noleander (talk) 21:37, 24 April 2012 (UTC)

Yes check.svg Done Fixed: the in-text citation used the year 1971, but the reference itself at the bottom of the page was 1970. I changed the in-text reference to 1970, which seems to be the correct year. This was a book in a multi-volume set, so I checked that the reference at the bottom is the one actually being referenced in the text, and it is (since chapter 7 is in volume 2). Good spot! This has been wrong ever since the citation was added (in-text citation added, reference added). Quietbritishjim (talk) 12:49, 25 April 2012 (UTC)

Overemphasis on time as a variable[edit]

The article begins:

"expresses a mathematical function of time as a function of frequency, known as its frequency spectrum" [italics mine]

Throughout time is emphasized, although in fact the transform applies to a function of any variable whatsoever. Most noticeably, the Fourier transform applies to functions of distance, and allows expansions in terms of components characterized by wavevector.

The article does contain the section on space, but this minor consideration does not convey the generality of the method, which should be made apparent at the beginning. Brews ohare (talk) 15:15, 26 April 2012 (UTC)

A very general discussion is found here in terms of distributions and test functions. Brews ohare (talk) 15:47, 26 April 2012 (UTC)

It is not an absolute requirement that an article must start in the maximum possible generality. In fact, there are generally good reasons for not doing this, and I believe this is the case here. Sławomir Biały (talk) 16:18, 26 April 2012 (UTC)
It is well known that different minds react differently to material, with some responding best to development from the particular to the general, and others the reverse. However, it is not desirable to allow the initial impression that an introductory example is the entire subject, and that impression is easily avoided by a clear statement at the outset. Brews ohare (talk) 15:39, 18 May 2012 (UTC)
Agree with Brews. This should be first defined in the most general way (that can be any variable). After that, one can tell something like this: "For example, if defined as a function of time ..." and so on (and mostly keep the current text in introduction as not to cause anyone's objections). Moreover, it tells "Fourier's theorem guarantees that this can always be done". It would be better to tell: "Fourier's theorem guarantees that this can always be done for periodic functions". My very best wishes (talk) 01:38, 20 May 2012 (UTC)
So, per you, the Fourier transform should be defined in the lead as follows: "The Fourier transform is the decomposition of a tempered distribution on a locally compact group as an integral over the spectrum of the Hecke algebra of the group." Sławomir Biały (talk) 12:17, 20 May 2012 (UTC)
No, quite the opposite. It would be much easier for me just to fix the text instead of discussion reductio ad ridiculum, but I suggest that Brews should do it, just to look if there are any problems with his editing, or ths is something else. My very best wishes (talk) 13:13, 20 May 2012 (UTC)
Well, I disagree quite strongly with the assertion that the Fourier transform "should be first defined in the most general way". I don't have a problem mentioning in the lead of the article the extension to Euclidean spaces, or indeed to other locally compact groups, nor indeed including an entire paragraph about that. Sławomir Biały (talk) 13:24, 20 May 2012 (UTC)
And I do not even see any reason to mention time so many times in introduction. In fact, the introduction could be completely rewritten for brevity and clarity.My very best wishes (talk) 02:00, 20 May 2012 (UTC)
The lead is written to be read by someone with no prior background in the subject (WP:LEAD, WP:MTAA). It's true that it could be shortened substantially, thereby rendering it useless to such an individual. Sławomir Biały (talk) 12:22, 20 May 2012 (UTC)
I am extremely surprised that such an obvious matter (the Fourier_transform is not about time) becomes a matter of discussion. My very best wishes (talk) 13:13, 20 May 2012 (UTC)
To be sure. But the issue that I bring up is not about whether the Fourier transform is about time, but how to explain it to a lay person. Indeed, this is not an easy question! Sławomir Biały (talk) 13:25, 20 May 2012 (UTC)
An explanation is most likely to be understood when it starts with the familiar and particular and gently proceeds to the alien and abstract. For this reason it is good (when possible) to assume that time and frequency are the independent variables (so not for the multidimensional cases). Perhaps a second paragraph in the lede could say that the transform can be between the domains of any two reciprocal variables, but that for simplicity the article will assume that these are time and frequency - as is commonly the case. --catslash (talk) 15:16, 20 May 2012 (UTC)

────────────────────────────────────────────────────────────────────────────────────────────────────Catslash: Why not put your disclaimer the transform can be between the domains of any two reciprocal variables, but that for simplicity the article will assume that these are time and frequency - as is commonly the case at the outset, and then proceed. That would satisfy me. Brews ohare (talk) 16:18, 20 May 2012 (UTC)

I would prefer the case of n dimensions to be handled in a separate paragraph after the first paragraph. (I'm generally against disclaimers such as the one you suggest.) This paragraph can also mention the case of locally compact groups. Sławomir Biały (talk) 19:42, 20 May 2012 (UTC)
But please remember that it must be understandable for a lay person like myself. My very best wishes (talk) 02:09, 21 May 2012 (UTC)
I'm confused how n-dimensions entered this discussion. The point under discussion is that time and frequency are not the universe of applicability. A statement to this effect is not a disclaimer: it is simply pointing out that the example to follow is selected for the sake of keeping the discussion simple. The transform can be between the domains of any two reciprocal variables, but for simplicity the article will assume initially that these are time and frequency - as is commonly the case Brews ohare (talk) 03:51, 21 May 2012 (UTC)
First, disclaimer is your word, not mine. Second, if you're not objecting to focusing on the one-dimensional case in the lead, then I must admit that I'm quite baffled by your objections. I thought we were talking about using space as a variable instead? Sławomir Biały (talk) 12:26, 21 May 2012 (UTC)
Hi Sławomir. It really hadn't occurred to me to stress multidimensional Fourier transforms, although maybe that should be part of the thinking here. I am unsure just how to express the generality of the Fourier integral. But at the moment I am not alone in feeling the emphasis on time and frequency appears overstated. Some explanation at the outset that the use of time and frequency is only as a very common example would fix this impression. Can you propose some wording that you would accept? Brews ohare (talk) 15:03, 21 May 2012 (UTC)

Fourier's theorem[edit]

The article refers to Fourier's theorem as follows:

The Fourier transform is a mathematical operation with many applications in physics and engineering that expresses a mathematical function of time as a function of frequency, known as its frequency spectrum; Fourier's theorem guarantees that this can always be done.

This wording is incorrect. Fourier's theorem applies to periodic functions and Fourier series, and what is needed here is a theorem regarding arbitrary functions, not restricted to periodic functions. The more general theorem came later with Dirichlet and others. Brews ohare (talk) 17:26, 20 May 2012 (UTC)

That's bothered me as well. Should the reference to Fourier's theorem be removed? Sławomir Biały (talk) 19:44, 20 May 2012 (UTC)
We are not moving anywhere. Brews, could you please post here, on this talk page, your new version of the introduction. And please do not be shy, rewrite everything that needs to be rewritten. Thank you, My very best wishes (talk) 19:52, 20 May 2012 (UTC)
According to Sean A. Fulop (2011). Speech Spectrum Analysis. Springer. p. 22. ISBN 3642174779.  the equation:
s(t)=\frac{1}{2\pi}\int\limits_{-\infty}^\infty \left(\int\limits_{-\infty}^\infty \ s(x) e^{i\omega x }dx\right) e^{i\omega t} d\omega \ ,
was derived by Fourier in 1811 and is called the Fourier transform theorem. Apparently its examination and various conditions upon the functions involved were pursued into the 20th century. I guess that is what the article should refer to. Brews ohare (talk) 20:42, 20 May 2012 (UTC)
The source you found identifies this as the "Fourier integral theorem". I will change the lead to reflect that. Sławomir Biały (talk) 21:27, 20 May 2012 (UTC)

Sorry I don't have time to weigh in on this properly right now, but I just want to point out that the theorem in Brew ohare's comment is the Fourier inversion theorem (an article that desperately needs clarifying). I've never heard it called anything other than the Fourier inversion theorem before, which might be why you're having trouble finding references. The above statement is not correct: one of the exponentials should have a minus sign in the exponent (it doesn't matter which), otherwise the left hand side would be s(-t) instead of s(t). I'd be surprised if Fourier proved it in 1811 rigorously by today's standard, instead it seems more likely that he just gave a heuristic argument, but this is just my guess. Even if I'm wrong about that, he certainly wouldn't have proved the most general case (i.e. with the weakest assumptions on the function s); he would almost certainly have assumed that sL1 (i.e. it's absolutely integrable) and probably that it's also infinitely differentiable with compact support (the simplest case). Quietbritishjim (talk) 23:49, 20 May 2012 (UTC)

Quietbritishjim: Your comments strike me as accurate. I added a few links about this in the text. However, if these matters deserve more attention, perhaps a subsection about these matters is better? Brews ohare (talk) 03:22, 21 May 2012 (UTC)

Fourier integral theorem as an historical note[edit]

I moved the material on the Fourier integral theorem to a separate section Fourier transform#Historical note. It seems pertinent to me to state this theorem explicitly and to note its easy derivation using modern analysis. The links to other WP articles also helps the reader. Brews ohare (talk) 15:32, 21 May 2012 (UTC)

The article already does state the theorem in the Definition section, and includes the attribution to Fourier (with I think a more authoritative source). Sławomir Biały (talk) 15:34, 21 May 2012 (UTC)
Sławomir Biały: I see that you are quite determined to avoid the introduction of Fourier's integral theorem in the double integral form that naturally leads to the Dirac delta function. This may be a matter of aesthetics? To the uninitiated the use of the Dirac delta function and its representations is a very straightforward approach to much of the article, and the needed mathematical rigor that completely obscures the meaning can be relegated to the specialist articles on distributions.
In any event here are a few observations:
  1. Reference to Théorie analytique de la chaleur is not very helpful without a page reference where the theorem can be found. Even if that is done, Fourier's notation may defeat an attempt to connect his work to the article. The reference I provided may be less authoritative, but it is way more understandable.
  2. The article now refers to what is commonly called a Fourier transform pair as the "Fourier integral theorem". Although the pair obviously are connected by the theorem, they are not themselves the theorem. The common usage is the double integral form that results when one of two is substituted into the formula for the other. If you insist upon mentioning only the single integral formulas, the Fourier integral theorem consists of stating in words the result of the substitution of one into the other. See, e.g. this.
  3. Through some error, (Titchmarsh 1948, p. 1) is not provided in the citations.
I'd suggest that some changes in presentation would be a service to the community. However, as we seem to be at odds, I won't pursue these matters. Brews ohare (talk) 16:23, 21 May 2012 (UTC)
The definition section says that under appropriate conditions, the function can be recovered from its Fourier transform, and then gives the integral formula for the inverse transform. Later the article discusses some sufficient conditions under which the theorem is true. Sławomir Biały (talk) 16:46, 21 May 2012 (UTC)
Sławomir Biały: The issue here is clarity of exposition, not whether the thing is said somehow, somewhere. Maybe of interest: The presentation of Myint-U & Debnath suggests the exponential form of the Fourier theorem originates with Cauchy. Brews ohare (talk) 17:06, 21 May 2012 (UTC)
Well, it does say exactly the same theorem in the definition section as you would have it say. Actually what's there now is more technically correct than your version, which in no way alludes to there being any conditions on the function whatsoever. Sławomir Biały (talk) 17:59, 21 May 2012 (UTC)
You have not provided a page number to refer to Théorie analytique de la chaleur, but it appears that the formulation of Fourier is:
f(x)=\frac{1}{\pi}\int \limits_0^\infty\ \int \limits_{-\infty}^\infty \ f(z) \cos \left[ u(x-z)\right] \ dz\ du \ ,
which is of the double integral form, but not of exponential form. Brews ohare (talk) 17:44, 21 May 2012 (UTC)
Excellent. This seems to be progress. Sławomir Biały (talk) 17:59, 21 May 2012 (UTC)
See, for example, the English translation of Fourier. Brews ohare (talk) 18:10, 21 May 2012 (UTC)

Proposal[edit]

How about an historical section along these lines, with maybe more on the modern developments, and with the references properly formatted, of course?

Historical background
Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:(see this)
f(x)=\frac{1}{\pi}\int \limits_{-\infty}^\infty\ \ d\alpha f(\alpha) \ \int \limits_0^\infty dp\ \cos  (px-p\alpha)\ .
which is tantamount to the introduction of the δ-function: See this.
\delta(x-\alpha)=\frac{1}{\pi} \int \limits_0^\infty dp\ \cos  (px-p\alpha) \ .
Later, Augustin Cauchy expressed the theorem using exponentials:Myint-U & Debnath Debnath & Bhatta
f(x)=\frac{1}{2\pi} \int \limits_{-\infty} ^ \infty \ e^{ipx}\left(\int \limits_{-\infty} ^ \infty e^{-ip\alpha }f(\alpha)\ d \alpha \right) \ dp \ .
Cauchy pointed out that in some circumstances the order of integration in this result was significant.Grattan-Guinness Des intégrales doubles qui se présentent sous une forme indéterminèe
Full justification of the exponential form and the various limitations upon the function f necessary for its application extended over several centuries, involving such mathematicians as Dirichlet, Plancherel and Wiener,some background, more background, and leading eventually to the theory of mathematical distributions and, in particular, the formal development of the Dirac delta function.
As justified using the theory of distributions, the Cauchy equation can be rearranged like Fourier's original formulation to expose the δ-function as:
f(x)=\frac{1}{2\pi} \int \limits_{-\infty} ^ \infty \ e^{ipx}\left(\int \limits_{-\infty} ^ \infty e^{-ip\alpha }f(\alpha)\ d \alpha \right) \ dp
       =\frac{1}{2\pi} \int \limits_{-\infty} ^ \infty \ \left(\int \limits_{-\infty} ^ \infty e^{ipx} e^{-ip\alpha } \ dp \right)f(\alpha)\ d \alpha \ =  \int \limits_{-\infty} ^ \infty \ \delta (x-\alpha) f(\alpha) \ d \alpha \ ,
where the δ-function is expressed as:
\delta(x-\alpha)=\frac{1}{2\pi} \int \limits_{-\infty} ^ \infty e^{ipx} e^{-ip\alpha } \ dp \ .

Brews ohare (talk) 18:49, 21 May 2012 (UTC)

Seems quite good, although really it's the theory of tempered distributions that are important in the development of the Fourier integral. More needs to be fleshed out in this later development, if you're up to it. Sławomir Biały (talk) 19:35, 21 May 2012 (UTC)
Here is a quote:
To this theory [the theory of Hilbert transforms] and even more to the developments resulting from it - it is of basic importance that one was able to generalize the Fourier integral, beginning with Plancherel's pathbreaking L2 theory (1910), continuing with Wiener's and Bochner's works (around 1930) and culminating with the amalgamation into L. Schwartz's theory of distributions (1945)...
and here is another:
"The greatest drawback of the classical Fourier transformation is a rather narrow class of functions (originals) for which it can be effectively computed. Namely, it is necessary that these functions decrease sufficiently rapidly to zero (in the neighborhood of infinity) in order to insure the existence of the Fourier integral. For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense. The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed and this removed many obstacles.
Query: do these quotes seem to you to cover the subject adequately for this historical discussion, or what else would you suggest? Brews ohare (talk) 20:05, 21 May 2012 (UTC)
The historical discussion looks good, but the last paragraph seems questionable or out of place. This is roughly how Cauchy proved the formula. (I have read Cauchy's account myself many moons ago.) The issue wasn't a lack of a notion of Delta function—Cauchy even had such a gadget—but a lack of appropriate function space on which the Fourier transform was defined. That is, it's the f in the formula that mathematicians subsequently worked so hard to clarify, not the δ. The emphasis on the Dirac delta seems misleading/wrong.
Also a general remark is that this content seems like it might ultimately be more suited to the Fourier inversion theorem article rather than here. At present, this article lacks any kind of history section, so it has to start with something I suppose. Sławomir Biały (talk) 18:50, 22 May 2012 (UTC)
I am happy you have read Cauchy on this topic. Perhaps you can supply a source?
The delta-function was used also by Fourier as noted in the proposed text. The point to be made is not the notion of a delta function, which is inevitable in any double-integral relation relating a function to itself, but an explication of the historical events that lead to its solid formulation as a distribution, among which are elucidation of exactly the points you raise: the correct function space and the role of distributions. Perhaps you might indicate what you consider to be the benchmark events?
Your last suggestion leaves me somewhat confused as to your recommendation. Are you saying Fourier transform needs a history section and maybe this proposal is a start that can be built upon? Brews ohare (talk) 14:58, 23 May 2012 (UTC)
I've elected to put this historical matter in the article Delta function. Brews ohare (talk) 19:54, 23 May 2012 (UTC)

Name of the theorem[edit]

To reiterate one of the things I said in my last comment the name of this theorem is the "Fourier inversion theorem", NOT the "Fourier integral theorem". This naming convention is essentially universal: I have seen myriad references to that name over many years, but I've never heard the name "Fourier integral theorem" before this discussion. What's more, as I said before, Fourier inversion theorem is already an article (admittedly one in need of some attention). Obviously it's a critical theorem relating to the Fourier transform, so perhaps it should have a short section in this article, with one of those notes at the top like "for more information see Fourier inversion formula". The history section above seems to be exclusively about that theorem, so it belongs in that article. Quietbritishjim (talk) 21:57, 22 May 2012 (UTC)

I'm not sure what makes you think that the naming convention is universal. Most of the classical literature in the subject uses "Fourier integral theorem" or some variant thereof to refer to the theorem, including the now referenced textbook by Titchmarsh—at one time required reading in the subject. I believe this convention remains in engineering and related areas. Google books bears this out: 34,000 hits for "Fourier integral theorem" versus 13,000 hits for "Fourier inversion theorem". Google scholar, which does not index older literature, gets about 1000 each. Sławomir Biały (talk) 22:19, 22 May 2012 (UTC)
Sorry about that. "... engineering and related areas" This seems to be the reason, I'm a pure mathematician, and looking at the top results in Google Books it seems Fourier inversion theorem is used in pure maths and Fourier integral theorem is used in science. The Google results aren't an accurate test because a lot of the Fourier integral theorem results (even in the top 20) are just results that have Fourier, integral and theorem anywhere in the name, but I agree that the name is in use. Quietbritishjim (talk) 23:23, 22 May 2012 (UTC)
Your Google results seem to differ from mine. The links I gave search (for me) for the exact phrase "Fourier integral theorem" and the exact phrase "Fourier inversion theorem". All of the top twenty links seem to be about the theorem we're discussing. Sławomir Biały (talk) 00:02, 23 May 2012 (UTC)
The term Fourier integral theorem is used in many authoritative works. A Google count is a poor way to find accepted usage because many, maybe even most, authoritative works are not searchable, and so do not appear in a Google search. A compromise position is that either name can be used, and both are widely understood. Brews ohare (talk) 17:48, 23 May 2012 (UTC)

ξ vs ν ?[edit]

Since it bothers me, maybe it bothers others as well. Instead of:

\hat{f}(\xi) = \int_{-\infty}^{\infty}f(x) e^{-i 2\pi \xi x}\, dx
\hat{f}(\nu) = \int_{-\infty}^{\infty}f(x) e^{-i \nu x}\, dx

I would prefer:

\hat{f}(\nu) = \int_{-\infty}^{\infty}f(x) e^{-i 2\pi \nu x}\, dx
\hat{f}(\xi) = \int_{-\infty}^{\infty}f(x) e^{-i \xi x}\, dx

Examples:

--Bob K (talk) 12:48, 7 June 2012 (UTC)

There are of course various conventions on where the 2π goes and what letters to use. The one here is that found in most harmonic analysis texts such as Stein and Weiss, and Grafakos. This is also the convention used, e.g., by Terrence Tao [1] in his entry to the The Princeton Companion to Mathematics. Your prefered version seems to be more common in the dispersive PDE community (e.g., Hormander). In any event, I don't really know if there is any good reason for preferring one convention over the other, besides individual familiarity and tastes. Sławomir Biały (talk) 15:56, 11 June 2012 (UTC)

Thanks. I don't have a familiarity preference, because I use f\, for hertz, myself. What it comes down to for me is that, for the sake of beginners, I don't like to make anything look any more intimidating that absolutely necessary. And I think this

\hat{f}(\nu) = \int_{-\infty}^{\infty}f(x) e^{-i 2\pi \nu x}\, dx   and  f(x) = \int_{-\infty}^\infty \hat f(\nu) e^{2 i \pi x \nu} \, d\nu

are a little friendlier looking than this

\hat{f}(\xi) = \int_{-\infty}^{\infty}f(x) e^{-i 2\pi \xi x}\, dx   and  f(x) = \int_{-\infty}^\infty \hat f(\xi) e^{2 i \pi x \xi} \, d\xi.

And indeed, as of 9/24/2008, the Hz convention was represented here by \nu\,.

--Bob K (talk) 20:40, 12 June 2012 (UTC)
Most of the references use ξ and not ν. This is consistent with the book of Stein and Weiss (which is a canonical textbook in modern Fourier analysis) and Hormander (a canonical textbook in dispersive PDE). Sławomir Biały (talk) 22:19, 19 June 2012 (UTC)

Confused section on LCH spaces[edit]

That short section is confused; it simply contains some trivial statements. The context Gelfand-Pontryagin-Fourier transform is unitary representations of topological groups. No group structure, no transform.

The Gelfand transform is the same as the Pontryagin transform, in this context: take a locally compact topological group G, one forms the convolution algebra L^1(G). This algebra comes with a natural involution, given by taking inverses of group elements. Its enveloping C*-algebra is the group C*-algebra C*(G). The one dimensional representations of C*(G) can be identified with the Pontryagin dual G^, i.e. one dimensional representations of G. The Gelfand transform is then an isomorphism from C*(G) to C_0(G^).

In the special case G = the real line R, the Gelfand transform is exactly the Fourier transform, but extended by continuity to all of C*(R). It says C*(R) is isomorphic to C_0(R), not the vacuous statement currently in the section. Mct mht (talk) 19:13, 28 June 2012 (UTC)


Legibility of example images in Introduction[edit]

I have some comments that may help improve legibility of the images in the introduction (http://en.wikipedia.org/wiki/Fourier_transform#Example).

  1. Make the graphs shorter (by a factor of 2). The plots just look like a bunch of lines as they are and take up too much vertical space.
  2. Move the vertical text to the captions under the figures. Vertical text is hard to read.
  3. Label the figures (Fig. 1, Fig. 2, etc.) and center the caption text. The figure numbers should be referred to in the text above to assist the reader.
  4. In the figure "The Fourier transform of f(t)" increase the font size of the boxed text. It is too hard to read.

Putnam.lance (talk) 08:58, 4 November 2012 (UTC)

Until a very recent edit these graphs were so small that the text you talk about was just incoherent squiggles. I'm not sure that that was such a bad thing, since the captions (outside of the images) and the text that refers to them seems to be descriptive enough. So maybe we should just shrink them back, assuming no one can be bothered to make versions with the text removed entirely. I agree with your first and third points though. Quietbritishjim (talk) 11:55, 4 November 2012 (UTC)

I have no objection to the Oct 26 version. The "thumbnails" are not meant to be legible. They are links to the large size versions.
--Bob K (talk) 13:54, 4 November 2012 (UTC)

Suspect wrong equations in section 'Square-integrable functions'[edit]

The equations in the form below section 'Square-integrable functions' should be taken from reference: The Fourier transforms in this table may be found in (Campbell & Foster 1948), (Erdélyi 1954), or the appendix of (Kammler 2000). I would suggest someone have these to double check these equations (equations 201 to 204).

Here's the suspect problem. There're three columns in the table: unitary ordinary frequency, unitary angular frequency, and non-unitory angular frequency. I think the Fourier transform for the last two columns should not have PI in their denominators. While there should be PI for the first column results. Because there're no 2*PI in the index of last two types of Fourier transforms, doing the integral, there's no way to generate a PI coefficient in denominator. I did calculation for the rectangular function, which showed the equations are wrong. I actually also corrected the equations in page of 'Rectangular function' under section 'Fourier transform of rectangular function'. But I'm not a math student and don't have enough resources, I'm not confident to make changes here. Anyone familiar with Fourier transforms please take a look at these equations.


--Allenleeshining (talk) 05:29, 16 November 2012 (UTC)

You don't say which transform you're talking about but it sounds like you're talking about the Gaussian (at least mostly that). Maybe this fact will help:
\int_{\mathbb{R}^{d}}e^{-\kappa|t|^{2}}e^{-it\cdot\eta}\,dt=\left(\frac{\pi}{\kappa}\right)^{d/2}e^{-|\eta|^{2}/4\kappa}
so that might explain where the pi comes from. Quietbritishjim (talk) 21:26, 30 December 2012 (UTC)
Our normalization of the sinc function differs from the one in those references, and this explains the discrepancy. Sławomir Biały (talk) 22:04, 30 December 2012 (UTC)

I'm sorry I didn't make it clear. I was talking about equantions from 201 to 204 about the pi in denominator. Sławomir Biały could you explain more if the normalization is the problem. Thanks. Allenleeshining (talk) 17:30, 4 January 2013 (UTC)

Our convention for the sinc is
\operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x}
This is mentioned in the "Comment" column of the table in the article. For more information, please consult the article sinc function. The convention in the links you gave ([2], [3]) is:
\operatorname{sinc}(x) = \frac{\sin \frac{x}{2}}{\frac{x}{2}}.
Obviously there's going to be an extra 2π to account for if you use our conventions. Sławomir Biały (talk) 18:52, 4 January 2013 (UTC)

Rewrite of Fourier inversion theorem article - request for comments[edit]

Since people who have this article on their watchlist are likely to also be interested in the Fourier inversion theorem article, this is just a note to let everyone here know that I'm proposing a rewrite of that article. Please leave any comments you have over on that talk page so that they're kept together. Thanks! Quietbritishjim (talk) 01:26, 31 December 2012 (UTC)

Plancherel theorem equivalent to Parseval's theorem?[edit]

According to the article, Plancherel theorem and Parseval's theorem are equivalent. So how do you go from Plancherel theorem to Parseval's theorem?

Maybe there should be a proof of some kind in the article that they are in fact equivalent, because to me it's not obvious. —Kri (talk) 11:44, 3 February 2013 (UTC)

Both assert that the Fourier transform is unitary, and it's well known that these are equivalent characterizations of unitarity. To get from one to the other, apply Plancherel's theorem to h(x)=f(x)+tg(x) with t a complex parameter. Sławomir Biały (talk) 13:41, 3 February 2013 (UTC)
If I substitute f(x)+t\,g(x) for h in \int_{-\infty}^\infty \left| h(x) \right|^2\,dx = \int_{-\infty}^\infty \left| \hat{h}(\xi) \right|^2\,d\xi, I eventually reach the expression
\int_{-\infty}^\infty \left(f(x)\,\overline{t\,g(x)} + \overline{f(x)}\,t\,g(x)\right)\,dx = \int_{-\infty}^\infty \left(\hat{f}(\xi)\,\overline{t\,\hat{g}(\xi)} + \overline{\hat{f}(\xi)}\,t\,\hat{g}(\xi)\right)\,d\xi.
How do I get from there to Parseval's? —Kri (talk) 12:56, 4 February 2013 (UTC)
Differentiate with respect to t. (Alternatively, take t=\int f\overline g - \hat{f}\overline{\hat{g}}.) Sławomir Biały (talk) 13:05, 4 February 2013 (UTC)

¨

If I differenciate with respect to t (which is the same in this case as just taking t = 1) I just get
\int_{-\infty}^\infty \left(f(x)\,\overline{g(x)} + \overline{f(x)}\,g(x)\right)\,dx = \int_{-\infty}^\infty \left(\hat{f}(\xi)\,\overline{\hat{g}(\xi)} + \overline{\hat{f}(\xi)}\,\hat{g}(\xi)\right)\,d\xi,
which is not Parseval's. The other aterlnative seems very complicated. Could you go ahead and show me what you mean by taking t=\int f\overline g - \hat{f}\overline{\hat{g}}? —Kri (talk) 13:51, 4 February 2013 (UTC)
You haven't taken the derivative correctly. See complex derivative for how to take the derivative with respect to a complex parameter. For the alternative suggestion, collect t and \overline t to obtain
t\int \left(\overline f g - \overline \hat f \hat g\right) + \overline t\int\left(f \overline g -  \hat f \overline\hat g\right)=0
This is true for all t, and in particular it is true when t=\int f\overline g - \hat{f}\overline{\hat{g}}:
2\left|\int \left(\overline f g - \overline \hat f \hat g\right)\right|^2=0
or
\int \left(\overline f g - \overline \hat f \hat g\right)=0
as required. Sławomir Biały (talk) 14:06, 4 February 2013 (UTC)
Ah, okay. I forgot that the derivative becomes a bit more complicated when the expression containes a complex conjugate involving the active parameter. This was actually a very nice proof. Thank you very much. —Kri (talk) 14:57, 4 February 2013 (UTC)

Proposal: Add complex conjugation to "Tables of important Fourier transforms" under "Functional relationships"[edit]

Even though it might not be in Erdélyi (1954), the following Fourier transform schould be added as a generalization of relationship 110:

Function Fourier transform
unitary, ordinary frequency
Fourier transform
unitary, angular frequency
Fourier transform
non-unitary, angular frequency
Remarks
\overline{f(x)} \overline{\hat{f}(-\xi)} \overline{\hat{f}(-\omega)} \overline{\hat{f}(-\nu)} Complex conjugation, generalization of 110
I dont have a problem with adding it. It's almost certain to be in Erdelyi (or at least one of the books listed). But someone should check just to be sure. Sławomir Biały (talk) 16:55, 9 August 2013 (UTC)

Yes check.svg Done Just checked it. It's in Erdélyi (1954) on page 117, number (2). I'll add it as 113 at the end of the list.

Ok, good work. Sławomir Biały (talk) 23:44, 9 August 2013 (UTC)

Example appears to be incorrect[edit]

The example appears to be incorrect. The result of ƒ̂(ξ) shows maxima of around -3 and 3. The result of ƒ̂(ξ) according to WolframAlpha (which is likely to be correct) shows maxima of around -20 and 20. Source. Could someone confirm/refute this? If I'm correct then new images should be made. Jdp407 (talk) 20:31, 21 August 2013 (UTC)

Careful. Wolfram Alpha uses conventions that differ from those in the article. (In fact, the Fourier transform convention used by Wolfram is extremely unusual: apart from normalization they have a +i where most have a −i.) Sławomir Biały (talk) 22:52, 21 August 2013 (UTC)
Ah ok. Thanks for clearing that up! Jdp407 (talk) 17:05, 28 August 2013 (UTC)
The peaks in [our example] are at f = ± 3 hertz. The peak at [wolframalpha] is at ω = 2π f = 18.8 radians/sec. It's just the frequency units that differ.
The definition at [wolframalpha], after clicking on "more Details", is this one from our table:
\displaystyle \hat{f}_1(\xi)\ \stackrel{\mathrm{def}}{=}\ \int_{\mathbf{R}^1} f(x) e^{-2 \pi i x\cdot\xi}\, dx = \hat{f}_2(2 \pi \xi)=(2 \pi)^{1/2}\hat{f}_3(2 \pi \xi),       including the "-i" convention.
But the height of the peak is lower than our example by the factor \scriptstyle 1/\sqrt{2\pi}.  So they are apparently using this one, also from our table:
\displaystyle \hat{f}_3(\omega) \ \stackrel{\mathrm{def}}{=}\  \frac{1}{(2 \pi)^{1/2}} \int_{\mathbf{R}^1} f(x) \ e^{-i \omega\cdot x}\, dx = \frac{1}{(2 \pi)^{1/2}} \hat{f}_1\left(\frac{\omega}{2 \pi} \right) = \frac{1}{(2 \pi)^{1/2}} \hat{f}_2(\omega)
--Bob K (talk) 22:17, 28 August 2013 (UTC)

Fourier transform of an integral[edit]

The Fourier transform of the derivative of f(t) is listed, but not the fourier transform of the definite integral of f(t). I think that would be helpful as well, especially since naively putting -1 into the "n'th derivative" formula to find it would overlook a few subtleties. --AndreRD (talk) 16:23, 21 August 2014 (UTC)

Animated example not representative of transform[edit]

The animation in the Introduction section creates confusion between the Fourier Transform and Fourier Series. It shows a short section of the partial sum of the Fourier Series for a square wave, but then uses the \hat{f} notation for the transform to label a diagram that is essentially a visualization of the series coefficients. The Fourier Transform of the section of function shown is a continuous function that spans the entire real line, which can be thought of as the linear combination of the transforms of each sinusoidal component, each of which is a scaled, translated version of the sinc function. The transform would be shaped more like this (ignore units, as none are given in the example graphic): [4] The expression shown a_{n}cos(nx) + b_{n}sin(nx) is also very general and does not add anything to the example, especially since the function shown would normally be represented as the sum of sines (or cosines, but not both) and only with odd values for n and coefficients in the form of \frac{1}{n}. Adamsmith (talk) 16:19, 2 September 2014 (UTC)

The Fourier transform of a periodic function (shown) is a linear combination of delta functions (also shown). What's wrong with this visualization? (Incidentally, I don't know what went wrong with the image at the link you gave, but as the Fourier transform of a periodic function, it is very clearly not correct.) Sławomir Biały (talk) 23:56, 2 September 2014 (UTC)