Talk:Fourier transform/Archive 2

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Convergence Conditions Needed

Some one need to add a section on the convergence conditions for the fourier transform like Dirichlet's 3 conditions.

Definition

Any chance someone can put in a plain-English definition of Fourier transform? I consider myself pretty educated outside the field of math (PhD Immunology), but I can't make sense of the current article as it is... Fhayashi 13:31, 11 September 2007 (UTC)

I think the table and the definition is not consistent -- there are some $2\pi$ factors that don't tally. I won't fix it for fear of making it worse .... fiddly stuff. Lupin 15:16, 12 Jun 2004 (UTC)

It's a pain to keep things consistent through convention changes in the text, but nothing is jumping out at me right now. Which rules concern you? —Steven G. Johnson 15:48, Jun 12, 2004 (UTC)

There is at least an inconsistency with the Dirac $\delta$:

"Furthermore, the useful Dirac delta is a tempered distribution but not a function; its Fourier transform is the constant function 1."

Yet in the table the ${\frac {1}{\sqrt {2\pi }}}$ convention is used. Eldacan 20:57, 6 Jul 2004 (UTC)

The text is inconsistent with the FT definition used here; I'll correct it. —Steven G. Johnson 04:04, Jul 7, 2004 (UTC)

Convolution Theorem

Looks like I forgot about this page for a while :) Isn't the Fourier transform of a convolution the product of the fourier transforms, without any $2\pi$ factors? Unless you want your convolutions to have constant factors too... this appears to need fixage on the convolution page as well. Lupin 10:21, 7 Jul 2004 (UTC)

Whether you have constant factors in the convolution theorem depends upon your FT definition (and the convolution definition). With the FT and convolution definitions here, there are constant factors. Proof:

Let:

$h(t)=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )d\tau$

$f(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }F(\omega )e^{i\omega t}d\omega$

$g(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }G(\omega )e^{i\omega t}d\omega$

Then the Fourier transform of h is:

$H(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }h(t)e^{-i\omega t}dt$

$={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }dt\int _{-\infty }^{\infty }d\tau {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }d\omega 'F(\omega ')e^{i\omega '\tau }{\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }d\omega ''G(\omega '')e^{i\omega ''(t-\tau )}e^{-i\omega t}$

$={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }d\omega 'F(\omega ')\int _{-\infty }^{\infty }d\omega ''G(\omega '')\left({\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }e^{i(\omega ''-\omega )t}dt\right)\left({\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }e^{i(\omega '-\omega '')t}d\tau \right)$

$={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }d\omega 'F(\omega ')\int _{-\infty }^{\infty }d\omega ''G(\omega '')\left({\sqrt {2\pi }}\delta (\omega ''-\omega )\right)\left({\sqrt {2\pi }}\delta (\omega '-\omega '')\right)$

$={\sqrt {2\pi }}F(\omega )G(\omega )$

—Steven G. Johnson 17:11, Jul 7, 2004 (UTC)

Quite right, I dropped a constant in my rough calculation :-) By the way, there's a quicker proof where you don't need to use icky delta functions. The second line is obtained from the first with the change of variables $x'=x-t$.

$H(y)={\frac {1}{\sqrt {2\pi }}}\int \int f(t)g(x-t)\,dt\,e^{-ixy}\,dx$

$=\left({\frac {1}{\sqrt {2\pi }}}\int f(t)e^{-ity}\,dt\right)\left({\frac {1}{\sqrt {2\pi }}}\int g(x')\ e^{-ix'y}\,dx'\right)\cdot {\sqrt {2\pi }}$

$=F(y)G(y)\cdot {\sqrt {2\pi }}.$

Change normalization to simplify formulas?

Hello. I wonder if there's any support for changing the normalization convention to whatever makes the formulas come out simplest. For example, so that the convolution thm becomes F(f*g) = F(f) F(g). I believe that such a convention is commonly followed. Comments? Wile E. Heresiarch 02:02, 26 Oct 2004 (UTC)

main definition

I'm wondering why there is a ${\frac {1}{\sqrt {2\pi }}}$ in front of the transform and inverse integrals. Most definitions that I've seen are:

$X(\omega )=\int _{-\infty }^{\infty }x(t)e^{-i\omega t}dt$

and

$x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{i\omega t}d\omega$

Ok, thanks!

As noted in the article, the normalization is somewhat arbitrary and varies between authors; the only important thing is that the product of the two constants is $1/2\pi$. (The one here is common among physicists and mathematicians; see e.g. Mathematical Methods for Physicists by Arfken and Weber. It has the nice property that the transform is unitary without scaling, so that the forward and backwards transforms are conjugates.) —Steven G. Johnson 18:06, Nov 15, 2004 (UTC)

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I don't understand that: "The Fourier transform is close to a self-inverse mapping: if F(?) is defined as above, and f is sufficiently smooth, then..." if F(w) is defined as above - what does it refer to? ("as above") "f is sufficiently smooth" - what's the idea? f will be the result of the (inverse) transformation. How we can say before the transformation if f is sufficiently smooth? Jun 10, 2005 <btw, how to make a timestamp without registration?>

Typing four tildes (i.e, ~~~~) should do. Cburnett 05:17, Jun 11, 2005 (UTC)

---

You don't ask the writers' guild to rule on spelling; you ask the lexicographers.

It is certainly important to mention that there are various conventions for the factors on Fourier transforms. But it's mathematicians who invented the thing and who understand it forwards and backwards. This is a math article. The definition used by mathematicians is the one that should be central. (Which is the way this article is currently organized: it should remain this way.)

(If warranted, any pure math article could have a section of it discussing relevant applications; this would be where to emphasize the version of the Fourier transform favored by non-mathematicians.)

The reason mathematicians use 1/sqrt(2 pi) for each factor (as is mentioned in at least one Fourier transform article) is that this puts the transform and its inverse in the group of unitary transformations, ones that preserve distance in the Hilbert space L^2 of functions (or tempered distributions) on the reals. This permits all kinds of additional machinery to be brought to bear in the study of these transforms.

Also: it should be mentioned that iterating the Fourier transform twice takes f(t) to f(-t) (which explains the mysterious comment that the transform is almost the same as its inverse; having similar formulas certainly does not suffice for making two thing "almost the same"). And it should be mentioned that (therefore) iterating the transform four times gives the original function back; this operator iterated four times is the identity operator.Daqu 19:59, 12 December 2005 (UTC)

P.S. Meant to include that this beautiful property of the Fourier transform -- that iterating it four times is equal to the identity transform -- is but one example of a property that holds only when the 1/sqrt(2π) constants are used. (It will not hold for any other choice of real positive constant.)

So it's not really the case that "the only important thing is that the product of the constants before the integrals be equal to 1/2π." This is true, I'm sure, for certain applications, but not true for others.Daqu 04:57, 13 December 2005 (UTC)

Generalization

I changed the wording under generalization, because the forward transform has been defined as the one resulting in F(ω). PAR 3 July 2005 02:28 (UTC)

Yep, no problem - feel free to do so anytime. I overlooked that it was previously defined earlier in the article. --HappyCamper 3 July 2005 02:53 (UTC)

Hello - I removed the properties list because they are all mentioned and defined in the "Table of important Fourier tranforms" section. All except "duality". What is "duality" in this context? PAR 3 July 2005 14:53 (UTC)

I put the list up earlier, because I think the article does not emphasize the property of linearity early enough. "Duality" refers to the application of the forward transform twice. Actually, I would like to participate in changing the organization of the 4 Fourier transform articles, where is this being done? --HappyCamper 3 July 2005 15:57 (UTC)

Right here on the talk pages is probably best. What kind of reorganization did you have in mind? I agree linearity is fundamentally important and should be more than just mentioned in the table, maybe some of the other properties as well, but other fundamental properties like orthogonality and Plancherels theorem should not be sidelined. PAR 3 July 2005 16:47 (UTC)

Reorganization

There are a few things that I have in mind, what do you think about them? (These are in no particular order)

1. Emphasize in the beginning the historical use and development of continuous Fourier transforms. Although the article Fourier transforms mentions this, it should be reiterated here. It would be worthwhile to mention the historical difficulty in placing Fourier transforms on a firm mathematical grounding. It is also worthwhile to mention its tremendous impact on engineering applications, harmonic analysis and functional analysis. The article needs to mention that in some contexts, Fourier transforms are used with the implicit assumption that most practical signals are "Fourier transformable"
2. The sufficient conditions for convergence and existence of the Fourier transform should be mentioned
3. The article should be rewritten somewhat so that it conveys a more introductory tone in the beginning, and then goes progressively deeper into the theory. For example, I don't think Lebesgue integrable functions and Hilbert spaces should be mentioned so early.
4. Mention all the basic properties of the Fourier transform earlier
   • These include linearity, time shifting, and frequency shifting
5. Move all the deeper mathematical properties and definitions to later into the article.
6. Make a table of properties which contains
   • Transform pairs, with the most common (a,b) definition in use all written out. At minimum, I feel that the definition where (a,b) = (1,1) needs to be included. Even though this material might be redundant (since all the transform pairs are proportional to each other), the article as it stands is nontheless difficult to use for someone who has grown akin to using a particular definition of the Fourier transform.
   • Table of all properties which are common to all 4 Fourier transform types. Although I understand the motivation to split the Fourier transform articles into 4 subarticles, I feel that the deep interrelations between the 4 are being obscured somewhat. We need to find a better balance for them.
7. The interpretation of Parseval's theorm in terms of energy needs to be stated. --HappyCamper 3 July 2005 18:51 (UTC)

To respond:

1. (Emphasize in the beginning the historical use and development of continuous Fourier transforms.) I agree - but only a paragraph or so. People looking for a detailed history should be looking elsewhere.
2. (The sufficient conditions for convergence and existence of the Fourier transform should be mentioned) I agree - this should go in the "completeness" section, right?
3. (The article should be rewritten somewhat so that it conveys a more introductory tone in the beginning, and then goes progressively deeper into the theory. For example, I don't think Lebesgue integrable functions and Hilbert spaces should be mentioned so early.) - I agree with that.
4. (Mention all the basic properties of the Fourier transform earlier. These include linearity, time shifting, and frequency shifting) I agree.
5. (Move all the deeper mathematical properties and definitions to later into the article) Well, we need to figure out what is "deep".
6. Make a table of properties which contains
   • (Transform pairs, with the most common (a,b) definition in use all written out) We need to keep one normalization throughout, and that has been decided (after some heated argument) to be the (0,1). This was not my favorite but it is now, since its both common and unitary. With regard to inclusion of other normalizations in the table only, I'm doubtful it wont be cluttered. I would like to see the unitary normalization for all Fourier articles, but there is resistance to that in the DFT article.
   • (Table of all properties which are common to all 4 Fourier transform types.) Agree - although I don't know the best place for it. The articles need to be kept separate, or else there will be one huge confusing article. Maybe in the Fourier transform page?
7. (The interpretation of Parseval's theorm in terms of energy needs to be stated) Strongly disagree - energy is a physics concept and the relationship needs to be drawn in the appropriate physics article, not here. Perhaps a mention, but no more.

PAR 3 July 2005 19:43 (UTC)

Thanks for your response...I'll follow suit just like the way you have it set up...

1. (I agree - but only a paragraph or so. People looking for a detailed history should be looking elsewhere.) - This was sort of what I had in mind too. An article on the History of the Fourier transforms would be nice. For these articles however, I agree that it's best to focus on their mathematical properties.
2. I agree - this should go in the "completeness" section, right? - I was actually thinking of making a little "Convergence" section, and maybe briefly talk about Dirichlet's conditions. But since the article already exists on these conditions, a mentioning in passing might be sufficient.
3. Yay, we have consensus!
4. Yay, we have consensus!
5. Well, we need to figure out what is "deep" - Let's try to answer this question like this. Who would search on Wikipedia for "continuous Fourier transforms"? I think it's reasonable that this person would likely be a university student. This person would be familiar with calculus, but not necessarily all its subtleties. Let's try to gear the article towards this person. Yes, this is sort of vague, but maybe we'll try moving some stuff around, adding new sections, and see if that works out. If not, we can always revert.
6. Make a table of properties which contains
   • (We need to keep one normalization throughout...) - Well, maybe I'll try to make a table in my own sandbox and then once I'm done with it, I'll present it here for more discussion. I'm not surprised there is a good reason why there is resistance to a parcular normalization. Sometimes a non-normalized Fourier transform pair is better suited for particular applications. Can you direct me to the discussion page where the normalization scheme was chosen? I think its reasonable to feel that choosing a particular normalization scheme is a subtle form of "POV".
   • (..although I don't know the best place for it...Maybe in the Fourier transform page?) - I was actually thinking of making a separate page for this table, since I think it would be quite big. I don't want the table to take away from the content of the main article. On the Fourier transform page, we could write maybe 2 or 3 common properties for all the transforms, and then on a separate page, list all of them.
7. (...Strongly disagree - energy is a physics concept...) - I'd be satisfied with a mentioning in passing too.

Well, it looks like to me that we basically agree on all the changes. What is next you think? --HappyCamper 3 July 2005 20:26 (UTC)

Edit away. Just this article to start. Read the whole article carefully first. Start out with rearranging the sections, then fill in one section as much as you can. Wait until tuesday or wednesday when people get back to work and see what happens (so you don't waste a lot of time editing just to get reverted, and people don't have to dig through a million edits to figure out what happened). In other words, consider the people who have contributed to this article a lot over the last few months or years, and now there's been a change they need to understand. Make it easy on them. I'm not on vacation like many, so I'll agree or complain pretty soon. Then after the dust settles, it will probably be clear what to do next.

With regard to your responses - Concerning normalization, I think the DFT talk page is the one where the discussion got pretty heated. Not here, that was wrong. With regard to the university student statement, yes, a university student should be able to get good information, and that info should be up front, but the article should DEFINITELY include the real mathematics, and not just be a book of recipes. PAR 3 July 2005 21:07 (UTC)

Of course - none of the rigorous math should be removed. Well, I'll slowly add edits in, but I'll go at my own pace. I'll make sure to put in good edit summaries :-) --HappyCamper 3 July 2005 22:10 (UTC)

Normalization and notation

An appeal to change the normalization so to be the least inconsistent with physicists and engineers.

Greetings,

I have done a few edits to this article as it is, but I do not presume to have much "ownership" of it (I know, nobody owns WP articles, but I don't want to waste a bunch of time changing things just to get it reverted). Anyway, we've been discussing this a bit at Wikipedia talk:WikiProject Electronics and we feel pretty strongly that the normalization in the Continuous Fourier transform article should, for the principal definition, toss the entire ${\frac {1}{2\pi }}\$ scaling factor into the inverse Fourier Transform (but leave the notes of other scaling conventions, including the unitary one now shown as the principal definition, and also leave that that "generalization" section). The reasons are three-fold:

1. that unitary convention with ${\frac {1}{\sqrt {2\pi }}}\$ is not common in engineering and physics texts (none that I have seen, but I certainly recognize that I do not have an exhaustive library).

2. In the cases were the angular frequency version of the Fourier transform is perferable, we should have the Fourier Transform to be as much compatible with the common double-sided Laplace transform definition as possible. The F.T. would be a degenerate case of the bilateral L.T. with little change in notation and no messy scaling:

$F_{\mathcal {F}}(\omega )=F_{\mathcal {L}}(i\omega )\$ where $F_{\mathcal {L}}(s)\$ is the bilateral L.T. of $f(t)\$

3. Even though there are some nice symmetry properties (such as duality) with the unitary convention, there remain some inelegant scaling with convolution and such operations. (The common EE convention, shown below, is both unitary and has desirable scaling properties - some of us think it is superior in elegance and compactness of expression and manipulation, but we know it is rarely used outside of the engineering discipline. That's why it will be an ancillary article.)

So the principle definition we propose for the Continuous Fourier transform would be:

$F(\omega )\equiv {\mathcal {F}}(f,\omega )=\int _{-\infty }^{\infty }f(t)\ e^{-i\omega t}\,dt\$

$f(t)\equiv {\mathcal {F}}^{-1}(F,t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega )\ e^{+i\omega t}\,d\omega \$

We are also planning on introducing another article to link to this and to Fourier transform to present the unitary F.T. operator most often used in electrical engineering communications texts:

$X(f)=\int _{-\infty }^{\infty }x(t)\,e^{-j2\pi ft}\,dt\$

$x(t)\equiv \int _{-\infty }^{\infty }X(f)\,e^{+j2\pi ft}\,df\$

because with that symmetry of definition and no extraneous scaling factor, the use of the Duality theorem and Parseval's theorem (as well as the DC value vs. integral of transformed function theorem) are trivial. Also transforms of rectangular pulses, sinc functions, gaussian pulses, and chirp functions are trivial (and elegant). Note the different meaning of the symbol $f={\frac {\omega }{2\pi }}\$, which is non-angular frequency (likely Hz) so it is not used for the "time-domain" input to the Fourier Transform and, instead, we use $x(t)\$ and EEs use $j\equiv {\sqrt {-1}}\$ rather than $i\equiv {\sqrt {-1}}\$. But all this will be in the ancillary article we'll be making for the benefit of electrical engineers (in fields such as control theory, communications systems, and signal processing).

I know there has been some past discussion about this scaling, but the present principal definition is really a concern for us neanderthal engineers and we would like it to be a little more conventional. Can I get a little feedback on this before I put a lot of work into it? Thanks. r b-j 04:11, 12 December 2005 (UTC)

For what it's worth, I agree with r b-j. I am in favor of changing from the current convention of normalizing both the forward and inverse transforms to 1 over sqrt(2*pi) to the electrical engineering convention of normalizing only the inverse transform by a factor of 1 over 2*pi. I am also in favor of creating the ancillary article discussing the alternative form of the transform using frequency f in place of angular frequency omega such that omega = 2*pi*f. -- Metacomet 05:22, 12 December 2005 (UTC)

What you (r b-j) have said so far seems eminently reasonable to me and I can see no objections at the moment. Engineers are reasonable folk anyway (usually) and want to make things as simple as possible (but no simpler!). I dont know what reaction you may get from physicists and mathematicians tho'!--Light current 06:58, 12 December 2005 (UTC)

The $1/{\sqrt {2\pi }}$ normalization, contrary to the above statement, is probably the most common convention. (e.g. Kaplan, Weiner, etc.) Also in response to "Even though there are some nice symmetry properties (such as duality) with the unitary convention, there remain some inelegant scaling with convolution and such operations." - The duality and symmetry are the most fundamental properties of the transform, the convolution properties are not. Also, with regard to modifying the notation to conform with the Laplace transform, why do that? Nobody uses the Laplace transform, which is, by the way, a special case of the Fourier transform, not vice versa. The asymmetric normalization is for people too lazy to take a square root.

I beg to differ PAR with your statement that 'no one uses Laplace Transforms'. Electrical/electronics engineers use them all the time for solving steady state and transient network problems.--Light current 22:15, 12 December 2005 (UTC)

Ok - I'm done being aggravating. The point I am trying to make is that from my side (math-physics) you guys talk about engineers the way New Yorkers talk about New York city. There are the five boroughs, and then there is upstate (i.e. the rest of the known physical universe). There is a culture clash here, and I think that two articles may be the answer. I do think that the present article should be maintained as the math-physics angle, and not be changed to asymmetric normalization. PAR 10:30, 12 December 2005 (UTC)

Let's not argue pointlessly, let's find an effective solution

The truth is that both conventions exist, and although we could debate endlessly about which one is more common or which one is more standard, the fact is that both conventions are used frequently and widely. We will never achieve a consensus about which one is better or which one is more correct. I prefer the electrical engineering convention for many of the reasons that r-b-j has already mentioned. But it doesn't really matter which one I prefer, or which one anybody else prefers. We need to find a proper way to identify and describe both conventions, explain the differences between them, why someone might use one or the other, and a bit of the rationale for each. It is possible to do that in a single article, although it might be difficult or the article might become too long, or we could create two separate articles, one on each of the two conventions, and then a short overview article that would point readers to each of the two alternatives.

My recomendation is that we not spend a lot of time and energy debating which definition is right or which definition is better, but rather, let's discuss how best to present both definitions without confusing the reader and in as precise and concise a fashion possible.

BTW, I don't know about anybody else, but I use the Laplace transform all the time. So it is incorrect to say that nobody uses the Laplace transform! And I do think it is worthwhile to show the connection between Fourier and Laplace. -- Metacomet 12:28, 12 December 2005 (UTC)

I agree with every thing you have said above. I am leaning to the idea of two articles. The remark about the Laplace transform was meant to be a semi-humorous way to make a point (please read the next paragraph after that remark). PAR 13:56, 12 December 2005 (UTC)

since the discussion is about different scale factor, I think the best way is to write the article only about the most general version of the transform (the "generalized" one), adding a paragraph at the end saying which are the values of the parameters most commonly used and linking to other articles, containg more particular discussions of the Fourier transforms. This way other articles about different subjects could link directly to the page about the version of the Fourier transform they are using. Such a general approach might be quite difficult: somebody with a good knowledge of math should do it. The other more particular versions of the article will be quite easy to write after the general one has been finished. Alessio Damato 19:21, 12 December 2005 (UTC)

The resulting article will be somewhat obscure and difficult to read (even more so than now), I'm afraid. Let me point out that this same problem has been faced by every single textbook dealing with Fourier transforms, and none of them (to my knowledge) write all of the formulas in terms of a general scale factor as you propose (in fact, you need two general scale factors: one in front of the transform, and one to convert between f and ω as desired). Rather, they pick one convention that they happen to find convenient, and stick with it throughout — and, if necessary, they have one section which explains how to convert to other conventions. Similarly, I think Wikipedia should stick to one convention for the most part (I don't especially care which in this case since there is no clear consensus in the literature), and then have one section clearly explaining the different conventions that are common and how to convert between them (e.g. for the table). —Steven G. Johnson 20:03, 12 December 2005 (UTC)

This ain't going to be easy....there are many different ways to go, I am not sure which one makes the most sense. I do know that in any event, all of the FT-related articles need some general improvement, clarification, and clean-up (in my opinion). But it would be good if we could reach a consesus on the general direction we want to take before investing a lot of time and effort into revising these articles. -- Metacomet 23:53, 12 December 2005 (UTC)

Informal proposals

Proposal #1: three separate articles

how about 3 new articles:

Continuous Fourier transform (Mathematics)

$F(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)\ e^{-i\omega t}\,dt\$

$f(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }F(\omega )\ e^{i\omega t}\,d\omega \$

this would have most of the stuff from the present article.

Continuous Fourier transform (Engineering and Physical Sciences)

$F(\omega )=\int _{-\infty }^{\infty }f(t)\ e^{-i\omega t}\ dt\$

$f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega )\ e^{i\omega t}\ d\omega \$

Continuous Fourier transform (Communications Engineering)

$X(f)=\int _{-\infty }^{\infty }x(t)\ e^{-j2\pi ft}\ dt\$

$x(t)=\int _{-\infty }^{\infty }X(f)\ e^{j2\pi ft}\,df\$

all three would have their definitions clearly stated, their properties (theorems) and, finally, a table of common transform pairs. the present article would have the generalized Fourier transform and then show what pairs of a and b are used for these three different common conventions, their definitions (on the same page for comparison), and then links to specific articles. how does that sound? r b-j 02:12, 13 December 2005 (UTC)

Some good news: I believe the the second and third versions that you have presented are mathematically equivalent forms, and that the difference in the argument, meaning $\omega \,$ versus $f\,$, and the difference in scaling, meaning the extra factor of $2\pi \,$, both have a physical interpretation in terms of units of measure and power spectral density. So I believe that you could cover both of these forms in a single article that could show the equivalence of the two forms and discuss the physical meaning of each in comparison with the other. You could also use a single table with three columns, one for the time domain, the second for the $f\,$ domain, and the third for the $\omega \,$ domain. -- Metacomet 11:02, 13 December 2005 (UTC)

they're all mathematically equivalent, differing only in some small detail in scaling on the "x-axis" and/or "y-axis". but those differences mean that there are scaling differences in the table of transform pairs and the table of theorems (like duality, Parsevals, and convolution). there are real scaling differences and if you are using one definition of the F.T. and applying the table applicable to the other, scaling errors will result. i think they should be in three different articles, all dispatched from the Fourier transform or Continuous Fourier transform article (after a definition and brief explanation of the differences) and then the specific articles (whether they be in math, science, or engineering) that refer to the F.T. or C.F.T. and use some particular theorem or transform pair, will link to the specific F.T. article with the definition and result being used. r b-j 19:00, 13 December 2005 (UTC)

What exactly is the physical interpretation of the ${1 \over {\sqrt {2\pi }}}$ scaling factor? -- Metacomet 23:05, 13 December 2005 (UTC)

no "physical" interpretation, but there's a mathematical interpretation: it's whatever it has to be so that you wind up with the same x(t) that you started with after inverse transforming from X(&omega) (which came from x(t)). it come from the fact (from Fourier series)that if you integrate x(t)=1 for a normalized period (that is with the fundamental angular frequency of 1), you get 2π your Fourier series will add up to something too big by a factor of2π, if you don't divide it out. then to get the Fourier integral, you let the period go to infinity, but that 2π factor stays with you.

There is a physical interpretation in quantum mechanics. Integrating the probability density of a particle's wave function throughout space should result in 1, representing a 100% probability that the particle exists somewhere. Without the ${1 \over {\sqrt {2\pi }}}$ factor, the total probability density represented by the function in the conjugate space is no longer 1.

Yes, this is why, to a quantum physicist, the unitary representation is the only representation that has physical significance.

I think that reinforces a point that I made about a week ago, which I will repeat here:

If we decide to have a single article, then I think the article should treat all three forms of the CFT on an equal foorting. The article should state that three different forms are in common usage, and it should explain that each form differs only in terms of normalization factors and/or domain scaling. The article should present the definition of each form, one after the other, and it should identify the purpose and major applications of each form. And then it can link to three separate sub-articles, one for each form, which would contain a table of common transform pairs, and a list of key properties and theorems, both using the notation of the particular form of the CFT. This approach would be consistent with the Wikipedia policy on maintaining a neutral point of view.

-- Metacomet 04:40, 2 January 2006 (UTC)

Let me revise my previous observation. I believe that not only are the second and third forms mathematically equivalent, I believe they are equivalent period, meaning they are completely the same, identical in every respect. -- Metacomet 23:09, 13 December 2005 (UTC)

well, then, my friend, you are most certainly mistaken. "every respect" means every respect, and if you interchange some operation, like convolution in frequency domain with its counterpart with the other form, you will get a scaling error. so there is a difference (quantitative, not so much qualitative) in some respect. r b-j 01:18, 14 December 2005 (UTC)

Proposal #2: one article, three separate tables

Is it really necessary to have three different articles on the Fourier transform? It seems to me that you just need three copies of the table — one for each convention. On a side note, I dislike r b-j's proposed names. I've seen mathematicians use the $d\omega /2\pi$ convention and physicists use the unitary convention. -- Fropuff 03:40, 13 December 2005 (UTC)

well if there are three tables with quantitatively different results for some transform pair or theorem, in the same article, i might think that would ostensibly lead to confusion. it should be absolutely clear which table goes with which definition. the problem for us EE guys is that we are agitating for some overhaul in the electronics/signal_processing articles and, before undertaking that, we would like to be able to point to a contiuous Fourier Transform article that is useful to us. but it really should not be totally disconnected from the F.T. as used by mathematicians (frankly, i haven't seen the unitary F.T. in physics texts of mine, but they are 25 years old). all physics and applied mathematics texts and some engineering texts that i have show the inverse F.T. with the ¹/_{2 π} factor (possibly some say F(i ω) instead of F(ω)). other engineering texts use the X(f) convention. i know i'm not a formal mathematician (but i do mathematics for a living) and i have never seen the unitary convention in the present article anywhere except in my CRC Handbook.

i would be happy for better names for the articles (these came off the top of my head). BTW both the first and last conventions are unitary, yet not quantitatively equivalent. r b-j 04:10, 13 December 2005 (UTC)

We could put the tables in three separate articles; as in

• Table of continuous Fourier transforms 1
• Table of continuous Fourier transforms 2
• Table of continuous Fourier transforms 3

The numerical labelling avoids stereotyping a group of people.

Almost all of the physics texts that I own use the convention presented in this article. The only exceptions that I can think of is Peskin and Schroeder's Quantum Field Theory. That being said, I really do prefer the $d\omega /2\pi$ convention. Mostly because I hate the square-roots. -- Fropuff 04:23, 13 December 2005 (UTC)

Having three completely separate articles would be a huge cut-and-paste job that would be difficult to maintain in the long term. Having three tables is okay, I guess. (I wouldn't label them as "mathematics", "communications" etcetera because any claim that a particular definition is restricted to a particular field is likely to be bogus. Nor would I call them "1", "2", and "3" (which is "1"?). Label them by the the normalization: "unitary normalization", "1/2π normalization", and "non-angular frequency" or some such.) —Steven G. Johnson 06:14, 13 December 2005 (UTC)

i don't want to stereotype either. i don't see anyone else using the "non-angular frequency" convention other than people (and students) doing communication systems engineering and/or signal processing. nonetheless it is a sorta venerated convention within those groups and it is also a unitary convention (the ${\frac {1}{\sqrt {2\pi }}}$ convention is not the only unitary operator so this "non-angular frequency" convention also has a trivial expression of the Duality Theorem and moreover has the advantage of no crappy scaling factors in convolution or multiplication of functions (also, $X(0)$ is the integral of $x(t)$ and vise versa - you can't get that with any "angular frequency" convention). i would like it if some good names for these three conventions were somehow composed and have the tables of properties (or theorems) and transform pairs to go with the definition in a separate article, so there is no legitimate reason for some Joe Schmoe to mix it up. r b-j 06:44, 13 December 2005 (UTC)

Proposal #3: one main article following single convention and 3 sub-articles of transform tables

I would suggest having one article that uses a single convention and derives (or at least describes) the various mathematical properties of the transform; since this article would be more mathematically oriented, I would suggest sticking with the 1/√2π convention here. (Duplicating this three times would lead to madness.) It would also have a short section summarizing the most common conventions. This would link to three articles for the different conventions, containing only the definitions of the transforms as well as tables of transform pairs. —Steven G. Johnson 02:46, 14 December 2005 (UTC)

i could go for that. in fact, i could go for about anything that will allow having the other two conventions (with their transform pairs) listed. if there is some rough consensus to do what Dr. FFTW has suggested, i might just go ahead and do it. any objections? r b-j 03:10, 14 December 2005 (UTC)

I like Steven's suggestion. You have my vote. What names do you propose for the alternate two articles? -- Fropuff 03:30, 14 December 2005 (UTC)

I would suggest something like Table of continuous Fourier transforms (1/2π convention) and Table of continuous Fourier transforms (non-angular frequency). (I wish there were a better term than "non-angular frequency", but I can't think of one. Just "frequency" is ambiguous because ω is often called "frequency" too.) —Steven G. Johnson 23:29, 14 December 2005 (UTC)

would the existing table of continuous Fourier transforms continue to be in the present article, or would it be spun into another article like the other two? perhaps: Table of continuous Fourier transforms (1/√2π, unitary convention), Table of continuous Fourier transforms (1/2π convention), and Table of continuous Fourier transforms (cyclic frequency, unitary convention)? how does that sound? r b-j 23:26, 15 December 2005 (UTC)

"Circular frequency" or "Cyclic frequency" maybe? PAR 23:46, 14 December 2005 (UTC)

Another idea would be to use the term "ordinary frequency" when referring to frequency in hertz, and the standard "angular frequency" for frequency in radians per second. See Proposal #4 below. -- Metacomet 15:26, 26 December 2005 (UTC)

Steven's suggestion is best. It seems clear that for efficiency and consistency, there should be only one theoretically oriented article, and that article should use notations and conventions geared to expressing these theoretical concepts. Separate articles can then reformulate these concepts and results in order to address the needs of a particular discipline. That has my vote. PAR 06:16, 14 December 2005 (UTC)

If we decide to have a single article, then I think the article should treat all three forms of the CFT on an equal foorting. The article should state that three different forms are in common usage, and it should explain that each form differs only in terms of normalization factors and/or domain scaling. The article should present the definition of each form, one after the other, and it should identify the purpose and major applications of each form. And then it can link to three separate sub-articles, one for each form, which would contain a table of common transform pairs, and a list of key properties and theorems, both using the notation of the particular form of the CFT. This approach would be consistent with the Wikipedia policy on maintaining a neutral point of view. -- Metacomet 15:38, 26 December 2005 (UTC)

Proposal #4: two articles, one math oriented, one signals oriented

I would like to suggest a different approach with two separate but linked articles. The first article would be called Continuous Fourier transform, and the second would be called Continuous-time Fourier transform:

• Continuous Fourier transform (CFT) would follow the definitions, conventions, and notation common in the mathematics literature, and would focus on the mathematical properties and origins of the Continuous Fourier transform.

In particular, this article would define the CFT using the unitary, angular frequency form, and would provide common transforms and properties using this form.

• Continuous-time Fourier transform (CTFT) would follow the definitions, conventions, and notation common in the signal processing and communications theory literature, and would focus on the engineering uses, applications, and origins of the Continuous-time Fourier transform.

This article would define two different forms of the CTFT:

> The non-unitary, angular frequency form (ω in radians per second)

>The unitary, ordinary frequency form (f in hertz)

-- Metacomet 19:38, 24 December 2005 (UTC), minor revision 15:06, 26 December 2005 (UTC)

I prefer my earlier proposal (have one article on the transform, and multiple transform-table articles for the multiple conventions.) First, your naming is not especially standard, and Wikipedia needs to refrain from introducing new terminologies. Second, as a pedagogical matter, I think it is a bad idea to suggest that these are two different transforms, as opposed to different notations. Third, the "engineering" applications of the transform have a lot of overlap with the "physics" and "mathematics" applications, and one should refrain from imposing false dichotomies. —Steven G. Johnson 21:59, 25 December 2005 (UTC)

Thank you for your comments. My goal was to come up with naming conventions that did not identify the names themselves with a particular community or application area, but to be able to essentially move in two separate directions, one more theory/math oriented, the other more application/engineering oriented. These of course partly reflect my own biases and ideas, although I was hoping to create a platform that would allow a diverse group of functional communities to coexist peacefully and to optimize each article for particular objectives. In other words, I was hoping to find a solution that would allow all of us to have our cake and eat it too. Maybe I did not succeed, and maybe your approach would be better. I don't know, but I thought I would propose it and see what you and others think. -- Metacomet 22:18, 25 December 2005 (UTC)

BTW, the terminology "Continuous-time Fourier transform" is, in my experience, the standard terminology, and again in my experience, far more common than "Continuous Fourier transform". On the other hand, that may simply be a result of my background in signal processing and communications theory, whereas the physics and math communities may see it the other way around. -- Metacomet 22:26, 25 December 2005 (UTC)

In addition, the terms "angular frequency" and "unitary" are also quite common and considered standard. The only really new terminology that I introduced was the term "ordinary frequency" as an alternative to the term "cyclical frequency" or "circular frequency" that others have suggested. IMO, I think "ordinary frequency" is a pretty good way of distinguishing frequency from angular frequency. Again, you are free to agree or disagree, but it's an idea. -- Metacomet 22:26, 25 December 2005 (UTC)

Decide by a vote

there are several good suggestions, we'll never agree completely. I think the only way to take a decision is to vote: let's point out which are the main choices, and anybody will say what he prefers. Alessio Damato 19:02, 13 December 2005 (UTC)

So what would be the question to vote on? Shall we create three articles of "equal standing" that depicts each of these definitions, and a table of theorems and of transform pairs that is correct for that specific definition, and have the main Fourier transform page "dispatch" the reader to each of these articles? maybe someone else can formulate a better way of wording a question for voting. Perhaps a second question (i'll put in here for political compromise) is shall the one article of the three that corresponds to the existing convention in Continuous Fourier transform remain in the same page of that name? (that would obviously boost that convention to a higher standing than the other two, but is perhaps what we have to do to get a solution acceptable to most.) r b-j 01:18, 14 December 2005 (UTC)

I believe that to this day this discussion has already ended with a good result. The article is really well presented the way it actually is now. When I studied Continious Fourier Transform, I was teached all three different forms on three different subjects, I think this is the best way to understand it and is exactly the way which the article uses: First, the mathematical unitary approach to the transform, then the non-unitary form from an engineering point of view, (including relation to the Laplace transform), and finally the frequency form. It is complete and formal. Nevertheless, I believe the article would need, as well as the initial formal mathematical approach, a less mathematical approach to the fourier transform using the "frequecy form", I would recommend using figures of simple signals in the time domain and in the frequency domain (the spectrum). I really think that an article about the continious fourier transform cannot be complete without having a single figure with a spectrum, as this is a much more intuitive way of understanding the fourier transform, much more accesible to interested people without requiring high mathematics understanding. I would recommend having a look at this page for ideas and inspiration: http://www.jhu.edu/~signals/

81.203.133.248 22:44, 9 May 2006 (UTC)

Should we create a new WikiProject?

This section is moot. See history if you want. -- Metacomet 17:28, 28 January 2006 (UTC)

Formal proposals

This section is moot. See history if you want. -- Metacomet 17:27, 28 January 2006 (UTC)

Error?

apologies for not knowing the protocol for adding a comment; this information is clearly not easy to track down.

But I write to mention that there must be an error in item 16 in the table of transform pairs, if the comment that e^(-x^2 / 2) is its own transform is true. (Probably the "a" in the denominator of the exponent of e should be replaced by a^2.) --daqu Daqu 19:40, 12 December 2005 (UTC)

I don't think there is an error - set a=1/2 and it works out. PAR 21:47, 12 December 2005 (UTC)

On a different note. Looking at fourier transform of $\sin(at)$. We should get something like (modulo factors involving $2\pi$) $\sin(at)=(e^{iat}-e^{-iat})/(2i)$. Hence the Fourier transform should be $(\delta (\omega +a)-\delta (\omega -a))/(2i)=-i{\frac {\delta (\omega +a)-\delta (\omega -a)}{2}}$. Which has a different sign from 27 of the table. But thought I would make sure I am not being silly before I changed it. Thenub314 01:40, 1 March 2007 (UTC)

Your proof is almost correct except that the Fourier transform of $e^{iat}$ is $\delta (\omega -a)$, not $\delta (\omega +a)$ (ignoring the scaling factors). That accounts for the sign difference between your result and the one in the table. Abecedare 02:14, 1 March 2007 (UTC)

Thanks, I knew I was being a little careless. Thenub314 14:54, 1 March 2007 (UTC)

Alternative forms

As originally submitted by Metacomet 23:41, 3 January 2006 (UTC):

angular frequency $\omega \,$ (rad/s)	unitary	$X(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }x(t)\ e^{-i\omega t}\,dt\$ $x(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }X(\omega )\ e^{i\omega t}\,d\omega \$
	non-unitary	$X(\omega )=\int _{-\infty }^{\infty }x(t)\ e^{-i\omega t}\ dt\$ $x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )\ e^{i\omega t}\ d\omega \$
ordinary frequency $f\,$ (hertz)	unitary	$X(f)=\int _{-\infty }^{\infty }x(t)\ e^{-j2\pi ft}\ dt\$ $x(t)=\int _{-\infty }^{\infty }X(f)\ e^{j2\pi ft}\,df\$

As modified by Bob K

angular frequency $\omega \,$ (rad/s)	unitary	$X_{1}(\omega )\equiv {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }x(t)\ e^{-i\omega t}\,dt\ ={\frac {1}{\sqrt {2\pi }}}X_{2}(\omega )={\frac {1}{\sqrt {2\pi }}}X_{3}({\frac {\omega }{2\pi }})\,$ $x(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }X_{1}(\omega )\ e^{i\omega t}\,d\omega \$
	non-unitary	$X_{2}(\omega )\equiv \int _{-\infty }^{\infty }x(t)\ e^{-i\omega t}\ dt\ ={\sqrt {2\pi }}\ X_{1}(\omega )=X_{3}({\frac {\omega }{2\pi }})\,$ $x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X_{2}(\omega )\ e^{i\omega t}\ d\omega \$
ordinary frequency $f\,$ (hertz)	unitary	$X_{3}(f)\equiv \int _{-\infty }^{\infty }x(t)\ e^{-j2\pi ft}\ dt\ ={\sqrt {2\pi }}\ X_{1}(2\pi f)=X_{2}(2\pi f)\,$ $x(t)=\int _{-\infty }^{\infty }X_{3}(f)\ e^{j2\pi ft}\,df\$

I am not sure that the scaling factors you have used in converting from one form to another are correct. For example, I am pretty sure that the scaling factor in converting between Form 2 and Form 3 should be 2π, not 1. -- Metacomet 14:55, 10 January 2006 (UTC)

No problem. Using the definitions, look what happens if we evaluate function $X_{2}(\omega )\,$ at $\omega =2\pi f\,$:

$X_{2}(2\pi f)=\left.X_{2}(\omega )\right|_{\omega =2\pi f}\equiv \int _{-\infty }^{\infty }x(t)\ e^{-j(2\pi f)t}\ dt\equiv X_{3}(f)$   Q.E.D.

Note that the variable of integation $(dt)\,$ does not change. (Were you thinking otherwise?)

--Bob K 19:03, 10 January 2006 (UTC)

The end result is not correct, and I believe the error in the math has to do with the chain rule. I am pretty sure (95 percent) that the correct result is:

$X_{2}(\omega )=2\pi X_{3}(f)\,$

where

$\omega =2\pi f\,$

I am working on a simple proof of this result, but it is not quite finished yet.

The scaling factor is also related to conservation of energy and Parseval's theorem.

-- Metacomet 19:58, 10 January 2006 (UTC)

<sigh> What do you think that is I just did?   Proofs don't get much simpler, and I didn't use the chain rule.  And what it proves is that your 95% certain result is impossible!

--Bob K 02:28, 11 January 2006 (UTC)

I realize that you didn't use the chain rule. And that, I believe, is the problem. I think that the proof that you have offered, although simple, is incorrect, because it does not account for the chain rule.

All I can say is that I am 100 percent certain that the conversion between Forms 2 and 3 requires a scaling factor of 2 π in amplitude, as the example below illustrates. The 5 percent uncertainty is that I am not entirely sure why it is required or how to prove it, but I suspect it is related to the chain rule and Parseval's theorem.

Why would I need the chain rule, when I'm not doing any derivatives? --Bob K 03:53, 11 January 2006 (UTC)

I don't know. I am not in the mood to argue about it, and it really doesn't matter to me. We still don't even have a consensus on how or even whether to present multiple forms of the CFT, or what to call them, or how many articles we should have, or how to link the articles, or what tables we should present, or yah-da-yah-da-yah.... So I am not sure that we should spend a lot of time right now worrying about how to convert from one form to another until we settle these other questions. As I said, I don't have the energy to fight about it, so whatever... -- Metacomet 04:08, 11 January 2006 (UTC)

Example

This issue is closed. See history if you are interested. -- Metacomet 17:30, 28 January 2006 (UTC)

The proof is incorrect

To Bob K and PAR:

You are right, and I am wrong. After reviewing the arguments that you both have made, and after doing some reading on my own, I have concluded that the proof that I presented is flawed, and that the result is incorrect. I am sorry that I ever brought it up, not because it is wrong (which it is), but because of the unbelievable level of hostility that some editors (not you) have shown toward me personally. Unfortunately, the mode of operation on WP often includes insulting people's intelligence, attacking their character, and questioning their motives. Obviously, I am exagerrating to make a point, but this particular situation is not unique. It is all too common, and I have no interest in taking part anymore. -- Metacomet 17:52, 27 January 2006 (UTC)

Thanks for the closure, a very gracious jesture. That is not sarcasm, and I never use sarcasm in writing, because once you start down that path, people often can't tell your literal remarks from your sarcastic ones. I am sorry you are upset with Steve. I enjoy his contributions immensely, so I am certainly willing to cut him a lot of slack for terseness. He earns it. He certainly means well, he is well-informed, and his advice (to read a book) was valid. The truth sometimes hurts. I viewed it as bluntness, not an insult. --Bob K 18:25, 27 January 2006 (UTC)

Proof of equivalence

This issue is closed. It's in the history if anyone wants to see it. Meanwhile, I have a few crow feathers to pull out of my mouth... -- Metacomet 20:40, 27 January 2006 (UTC)

Two Questions about Fourier Transforms

I've done two exerxise of Riley's Physical Math.

The questions are on [1]

Can anyone help me?

They're ok by my teacher's help.--HydrogenSu 19:30, 29 January 2006 (UTC)

In Physics

Even in Physics,${\frac {1}{\sqrt {2\pi }}}$ still exists in front of The Fourier Transforms and those of Inversion. Those formulas were written in Riley's Mathematical Methods for Physics and Engineering.

Wonderful!

I love how this page explains the unitary/non-unitary transform pairs. It's wonderful - thank you! --HappyCamper 17:16, 23 May 2006 (UTC)

table of transforms

I really like how the article is, now, but I noticed that the table of Fourier transforms is growing, and it might become the main part of the article. Since a complete table of Fourier transforms can be a useful reference, I suggest to keep on improving it but moving to a different page such as Table of Contiuous Fourier transforms, keeping on the main article just a few as an example. What do you think about it?? Alessio Damato 10:40, 15 June 2006 (UTC)

• Sounds good to me. Also, we have created a maintenance problem regarding the row numbering. I think it is internally consistent now, except for the missing numbers. But there are probably some references in other articles that are out-of-date. Is there a better way to do it? --Bob K 12:45, 15 June 2006 (UTC)

Orthogonality Section incorrect.

After the article points out that we make extend the to $\mathbb {R} ^{n}$ it uses some incorrect normalizations, also by this point since $t$ no longer refers to time I am changing it to $x$.

Definition

I added the notation to denote the fourier transform by using a "hat". It has been suggested that this is not standard and conflicts with the hilbert transform notation. I disagree, it is extremely popular thought mathematics, and within mathematics it the hilbert transfrom is never denoted with a hat that I have seen. Thenub314 19:16, 16 July 2006 (UTC)

Here are examples of ${\widehat {f}}(t)={\mathcal {H}}\{f\}(t)\,$:

• http://www.phy.auckland.ac.nz/assets/pdf/chap5_00.pdf
• http://w3.msi.vxu.se/exarb/mj_ex.pdf

I also found an example of ${\widehat {f}}(t)={\mathcal {F}}\{f\}(t)\,$:

• http://iria.pku.edu.cn/~jiangm/courses/IRIA/node24.html

I assume there are more examples to be found. So what I suggest is that we do mention it as an alternative notation (as you have done), but let's not just stick it into the middle of an equation. Let's give it its own sentence somewhere.

--Bob K 21:04, 16 July 2006 (UTC)

Sounds good. 128.135.197.2 15:42, 17 July 2006 (UTC)

Plancherel's theorem and Parseval's Theorem

We seem to have them mixed up, even the the articles they point give them the opposite attribute. Thenub314 16:04, 17 July 2006 (UTC)

Plancherel is the more general theorem, Parsevals is the special case of Plancherel, right? PAR 21:59, 17 July 2006 (UTC)

Ok, I have corrected this. Parseval is a special case of Plancherel. (Google: Plancherel Parseval "special case") PAR 03:50, 19 July 2006 (UTC)

I have done a little research into the matter, it seems we are far from the only one's who disagree. For example Rudin uses the usage I suggested as does Grafakos. But there are examples of other fine texts (For example Zygmund) Attributes the equality of the L² norms for Fourier Series to Parseval, and attribute the equality of L² norms for Fourier transfroms to Plancherel. As far as which is "more general" as it is trival to use either theorem may prove the other. But in some sense the statment about equality of the inner product uses that the fourier transfrom of an L² function is in L²

The most common usage in research articles that I have read is to attribute the result about equality of L² to Plancherel and to refer to the equality of inner products as Parseval's Identity. I will make a note of this in the article.

Thenub314 19:22, 19 July 2006 (UTC)

That sounds good, I didn't realize there was a disagreement. By the way, how do you prove the equality of the inner products using just the equality of the norms? PAR 20:06, 19 July 2006 (UTC)

Sorry par, I didn't notice the question until today form some reason. Let's let $f,g\in L^{2}(\mathbb {R} )$, and to set notation let $\|\cdot \|_{2}$ be the $L^{2}$ norm and ${\hat {h}}$ be the Fourier transform of the function $h$. We have

$\|f+g\|_{2}^{2}=\|{\hat {f}}+{\hat {g}}\|_{2}^{2}$, also $\|f\|_{2}^{2}=\|{\hat {f}}\|_{2}^{2}$ and $\|g\|_{2}^{2}=\|{\hat {g}}\|_{2}^{2}$. Now use that $\|f+g\|_{2}^{2}=\langle f+g,f+g\rangle =\|f\|_{2}^{2}+\|g\|_{2}^{2}+2\langle f,g\rangle$ and (using the above inequalities) $\|{\hat {f}}+{\hat {g}}\|_{2}^{2}=\langle {\hat {f}}+{\hat {g}},{\hat {f}}+{\hat {g}}\rangle =\|f\|_{2}^{2}+\|g\|_{2}^{2}+2\langle {\hat {f}},{\hat {g}}\rangle$.

Now we have $\|f\|_{2}^{2}+\|g\|_{2}^{2}+2\langle f,g\rangle =\|f\|_{2}^{2}+\|g\|_{2}^{2}+2\langle {\hat {f}},{\hat {g}}\rangle$ so we may cancel and divide by 2.

Possible problem with above proof:

$\|f+g\|_{2}^{2}=\langle f+g,f+g\rangle =\|f\|_{2}^{2}+\|g\|_{2}^{2}+2{Re}\langle f,g\rangle$

$\|{\hat {f}}+{\hat {g}}\|_{2}^{2}=\langle {\hat {f}}+{\hat {g}},{\hat {f}}+{\hat {g}}\rangle =\|f\|_{2}^{2}+\|g\|_{2}^{2}+2{Re}\langle {\hat {f}},{\hat {g}}\rangle$

so matching and canceling terms we have only proven that ${Re}\langle f,g\rangle ={Re}\langle {\hat {f}},{\hat {g}}\rangle$, while the inner-rpodect equality result is stronger. There may be derivation of the inner product result, from the norm result, but I don't believe the above derivation is the correct one. Am I missing something ? Abecedare 05:14, 5 July 2007 (UTC)

Hi Thenub, Regarding (→The Plancherel theorem and Parseval's theorem - Changed special case to equivalent. See discussion.); I have the same problem as Abecedare. Does 'See discussion' mean that you are about to give us a proof for the imaginary parts of the inner product? I'd like to see this before accepting equivalent instead of special case. --catslash 11:28, 5 July 2007 (UTC)

Given the proof that the real part of the inner product is the same, replace g with ig and repeat the proof, in which case you get the result that the imaginary part of the inner product is the same. Q.E.D. —Steven G. Johnson 17:48, 5 July 2007 (UTC)

Thanks, that works and is (as usual) obvious with hindsight. Abecedare 18:22, 5 July 2007 (UTC)

Yes indeed. Thanks. --catslash 23:15, 5 July 2007 (UTC)

Notation Question

Wouldn't it be better to represent complex conjugation with a bar instead of an asterix? Here in the US it is the usual way it is taught in high school and calculus classes. I understand that using an asterix is more common in physics and engineering, and and has nice connections to the dual when we are working in hilbert spaces. But I personally feel representing it as a bar makes it more readable for a wider audience. But what do people think? Thenub314 16:25, 17 July 2006 (UTC)

It's a bigger mess than that. According to http://physics.nist.gov/Pubs/SP811/sec10.html, section 10.1.2, the "correct" symbol for conjugation is neither asterisk nor bar. It is ^. And checkbox #7 at http://physics.nist.gov/cuu/Units/checklist.html shows bar used to designate the average value of a set of numbers. I am familiar with that usage. I have also used it extensively (in the U.S. school system) to designate a vector. (http://physics.nist.gov/Pubs/SP811/sec10.html, section 10.2.1 requires boldface, italic for vectors, but that is a little impractical for homework problem-solving.) The asterisk, as you point out, is commonly used for conjugation in engineering. But we also used it that way in the (U.S.) schools that I went to. Maybe I am forgetting something, but I don't recall ever using the bar for conjugation. --Bob K 03:19, 18 July 2006 (UTC)

You're misinterpreting the use of ^ in the article you link. To quote "Examples: ^ (conjunction sign, p ^ q means p and q)" Here they are not speaking of conjugation they are speaking of a concept from logic.

Thank you sir. Exactly right. I should have listened to the voice in my head that was saying "Why in the world do they mention conjugation in the same breath with "p^q means p and q". --Bob K 07:09, 20 July 2006 (UTC)

Now, I will concede the point that bar is frequently used for averages. Let me leave the notation for vectors aside. Now it may be my experience is not typical. Since becoming a math graduate student I have very rarely seen anyone use a asterisk to denote the complex conjugate of a number. During my life as a signal processor, (If that makes sense to say) I would frequently see both notations. Now you may present excellent arguments as to which is a better notation, but I am interested in the poorly defined question "which is more common?" But I welcome any thoughts on the matter. Thenub314 20:45, 19 July 2006 (UTC)

Me too. But I fear that we are well below the radar here. Your question may as well be invisible. Good luck! --Bob K 07:09, 20 July 2006 (UTC)

I have decided to go ahead and make the change. I noticed the article on complex numbers uses a bar to denote conjugation, so it seemed like a good idea to change it. Thenub314 19:17, 21 July 2006 (UTC)

Fourier Transform Properties

Not sure about these, differ from given definitions in article Duality "Proof": $X(\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}\,dt\,$

Set w=t and t=-w

$X(t)=\int _{-\infty }^{\infty }x(-\omega )e^{j\omega t}\,d\omega \,$

$X(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }2\pi x(-\omega )e^{j\omega t}d\omega \,$

Therefore $X(t)\iff 2\pi x(-\omega )\,$

Convolution in the time domain: $(x*v)(t)\iff X(\omega )V(\omega )\,$

Multiplication in the time domain: $x(t)v(t)\iff {\frac {1}{2\pi }}(X*V)(\omega )\,$

Merge?

Although I used the merge tag, I don't mean that all of the content of Continuous Fourier transform should be merged with Fourier transform. It may make sense to keep both articles, but the current distinction between these two and other related articles is unsatisfactory. In particular, we have the following inconsistencies:

• Continuous Fourier transform contains many properties and theorems that apply to all or most Fourier-related transforms - I therefore think they should be kept in Fourier transform. (An example is the statement on eigenfunctions which just got deleted from the latter. Granted, a discrete Gaussian is not the same as the continuous function, but if we wanted to maintain that distinction then we would have to duplicate all articles on functions with a discrete version.)
• Fourier transform contains a list of Fourier-related transforms, which is largely redundant with List of Fourier-related transforms.
• Fourier transform contains half page descriptions of various Fourier-related transforms, which are redundant with the articles themselves and might better be kept in an article dedicated to comparing them (List of Fourier-related transforms might be appropriate, if we don't take "list" to literally.)

— Sebastian (talk) 02:21, 22 August 2006 (UTC)

Wanting to check something re. Fourier transforms, I blundered into the Fourier transform page. I was surprised that I needed to go this page instead. Perhaps this page should be called 'Fourier transform', and the current Fourier transform page should be called 'Variants of the Fourier transform', since a section of that name currently occupies the greater part of it. The current arrangement seems to sacrifice clarity to achieve generality. --catslash 12:17, 4 September 2006 (UTC)

No - Its real simple the way it is. Main article is Fourier transform which should be general, then separate articles for each variant. None of the variants should lay claim to being "The" Fourier transform. Historically, the Fourier sine and cosine series are what Fourier discovered, but to call these "The" Fourier transform would just be confusing and not according to present usage. PAR 02:32, 6 September 2006 (UTC)

Hmm! - these were called Fourier series when I was at school (admittedly a long time ago). I take you point about generality, but see Wikipedia:Naming conventions. --catslash 01:48, 9 September 2006 (UTC)

Touché!
Regarding PAR's point about generality: I agree that the curent structure appears simple on first glance. However, in reality, scopes and boundaries of the articles are so ill defined that they have significant structural inconsistencies, as shown above. Until we solve these issues, we can not claim that it is "real simple". I hope that accomplished editors like PAR would acknowledge this problem and help solving it. — Sebastian (talk) 17:45, 13 September 2006 (UTC)

Well, since you put it that way, how can I refuse? The point I wanted to make was not that the articles are good as they stand, but that the heirachy should be maintained - Fourier transform as a main article and Fourier series, continuuous Fourier transform, etc. as sub-articles. How do the "significant structural inconsistencies" change this? Also, Catslash, regarding the naming conventions, can you say what you were referring to more exactly? If I got touchéd there, I didn't feel it. PAR 19:57, 13 September 2006 (UTC)

The inconsistencies per se may indeed not change the overall hierarchy of the articles. It seems to me that they are rather a symptom of the problems we have with the hierarchy. I'll try and explain what I mean in section #Topology of the article series below. — Sebastian (talk) 00:27, 24 September 2006 (UTC)

I'm not fencing! It's up to you to decide whether a point has hit home. I was referring to...

Generally, article naming should give priority to what the majority of English speakers would most easily recognize, with a reasonable minimum of ambiguity, while at the same time making linking to those articles easy and second nature.

Another way to summarize the overall principle of Wikipedia's naming conventions:

Names of Wikipedia articles should be optimized for readers over editors; and for a general audience over specialists.

--catslash 20:34, 13 September 2006 (UTC)

I just finished a chess game (notice I didn't say "won"). And now I feel like I am spectating one. Alas, I have nothing useful to add to the discussion at this time. But I think it's going in a good direction, on average. --Bob K 23:06, 13 September 2006 (UTC)

Hmm..., the way I learned the history of the Fourier Transform, it's "discovery" was also due to Fourier, not just Fourier Series. He seemd to have some strange notion of approximating a line by a really large circle. Which he didn't justify very well. On the other hand he didn't justify Fourier Series very well. I personally don't like the current division of the general theory of Fourier Transoforms. I have to agree with Sebastian that some work needs to be done on the divison to clean it up but the overall idea of having a General Fourier transform article that links to more specific instances seems fine. But really I just wanted to make a comment on the history. Thenub314 23:51, 13 September 2006 (UTC)

OK, I'm in full agreement with the naming conventions. Can someone propose a new set of article names and reasons the present ones are not good? Presently it is (or should be):

• Fourier transform (general)
  • Continuous Fourier transform (Continuous -> Continuous)
  • Fourier series (Continuous -> Discrete)
  • Discrete Fourier transform (Discrete -> Discrete)
  • Discrete-time Fourier transform (Discrete -> Continuous)
  • (Related transforms - Not Fourier)

PAR 21:02, 13 September 2006

Well of course we dare not deviate from tradition. But for consistency's sake, it would be nice if tradition had given us Discrete-frequency Fourier transform instead of Fourier series.   So how about Discrete-frequency Fourier transform (aka Fourier series)?   No, I didn't think so, but thanks for reading.

--Bob K 03:42, 14 September 2006 (UTC)

I should say that I think the Fourier transform pages are bloody brilliant (in common with many of the maths pages). The Continuous Fourier transform page gave me the result I was looking for in the normalization I preferred. ...But then Sebastian's merge?-tag moved me to point out that "continuous fourier transform" isn't a recognized term (

Search term	Google hits
"fourier transform"	7,450,000
"fourier series"	1,690,000
"discrete fourier transform"	699,000
"discrete time fourier transform"	45,900
"continuous fourier transform"	23,100

) and so if I understand policy correctly, it should be called "Fourier transform", and open with something like

In mathematics the term Fourier transform most commonly refers to...

with links to other meanings (variants). I don't think this is an important issue though, so I won't be commenting further. --catslash 11:41, 14 September 2006 (UTC)

Another excellent point. When are you going to give me something I can disagree with?    ;-)

I think we have a concensus on the traditional names now. And the policy is clear. But that leaves the traditional problem. There is an elegant underlying connectivity, symmetry, duality, and cohesion that is largely obscured by the inconsistent names that arose sequentially, instead of all-together in a grand master plan. (Is my software background showing?) Some people (alas, I speak from personal experience) fail to see that beauty right away (possibly forever), because we tend to assume that if it was there, the names would be chosen accordingly. Mindless tradition sometimes needs a little help.

The kind of help I'm talking about is Fourier_transform#Family_of_Fourier_transforms, which is excellent. But it could be even better if Fourier series was replaced or accompanied by Discrete-frequency Fourier transform.

I guess it's obvious, but that table clearly illustrates the raison d’être for the non-traditional term continuous Fourier transform.

--Bob K 23:34, 14 September 2006 (UTC)

"Fourier Transform on R" ?

One naming option might be to say something like "The Fourier Transform most commonly refers to..." Then point out you can talk about the Fourier transform on different groups, then have an article for each group (R, T, Z, etc) that has traditionally been important and keep the Fourier Transform article general. But, "continuous Fourier transform" I personally dislike. (What does the continuity refer to?) Mathematicians simply use the terminology (when it is needed) "The Fourier Transform on R" (or R^n as the case may be.) Thenub314 14:46, 15 September 2006 (UTC)

Hint: The word discrete in DFT refers to the fact that both domains are discrete-valued. But yeah, I had that very same question when I first got here. And the pattern that it fits nicely into is obscured by one misfit: Fourier Series. --Bob K 04:52, 16 September 2006 (UTC)

Thanks, this still seems a misnomer. Other interesting groups may well have a continuous fourier transform in this sense. I think it is fine to stick with standard terminology of Fourier Series, Discrete Fourier Transform, etc. when available. But as it was pointed out this name is rarely used. But I have heard the phrase "The Fourier transform on R". And I think it is the right thing to say. The function is a function of a real variable, this is what distinguishes the Fouier transfrom on this page. This language is also used on Harmonic analysis. If I were to make a case for this language it would be that on mathscinet I get the following results

Search term	mathscinet hits
"continuous fourier transform"	10
"the fourier transform on"	229

Thenub314 13:23, 16 September 2006 (UTC)

The phrase Fourier transform on R does not distinguish between the cases:

• periodic   time domain $\rightarrow$ discrete frequency domain (Fourier series)
• aperiodic time domain $\rightarrow$ continuous frequency domain

--Bob K 02:04, 17 September 2006 (UTC)

Sure it does. If the time domain is periodic then it is a circle not a line (or possibly thought of as an interval). In either case the integrals defining the Fourier transform for the "periodic time" case are not taken over all of R. As a general statement I thought it was the _compactness_ of T, Z_n etc that leads to the discrete frequency domain. Thenub314 13:14, 17 September 2006 (UTC)

Any periodic function, with period $\tau \,$ can be modeled like this:

$x_{\tau }(t)=x(t)*\sum _{n=-\infty }^{\infty }\delta (t-n\tau )=\sum _{n=-\infty }^{\infty }x(t-n\tau )\,$

Its Fourier transform is:

$X(f)\cdot {\frac {1}{\tau }}\sum _{k=-\infty }^{\infty }\delta (f-k/\tau )={\frac {1}{\tau }}\sum _{k=-\infty }^{\infty }X(k/\tau )\cdot \delta (f-k/\tau )\,$

which is zero except at the discrete set of frequencies: $f=k/\tau \,$

(Note that periodicity is the only required time-domain property.)

The inverse Fourier transform is:

$x_{\tau }(t)\,$	$=\int _{-\infty }^{\infty }\left[{\frac {1}{\tau }}\sum _{k=-\infty }^{\infty }X(k/\tau )\cdot \delta (f-k/\tau )\right]e^{i2\pi ft}\,df$
	$={\frac {1}{\tau }}\sum _{k=-\infty }^{\infty }X(k/\tau )\cdot \int _{-\infty }^{\infty }\delta (f-k/\tau )\cdot e^{i2\pi ft}\,df$
	$={\frac {1}{\tau }}\sum _{k=-\infty }^{\infty }X(k/\tau )\cdot e^{i2\pi kt/\tau }\,$

which is also called the expansion of $x_{\tau }(t)\,$ into a Fourier series. The expansion coefficients are:   ${\frac {1}{\tau }}\cdot X(k/\tau )\,$.

So the Fourier series "transform" is just the non-zero valued coefficients of the Fourier transform of a periodic function. By discarding the zero-valued coefficients, it becomes a discrete-frequency Fourier transform, the dual of the DTFT. I think it is a shame that so many treatments, including the naming convention, deliberately obscure that elegant relationship.

--Bob K 16:43, 17 September 2006 (UTC)

I think that there are some subtle reasons why what you said above does not always work. For example, there exists $f\in L^{1}(\mathbb {R} )$, continuous, so that ${\hat {f}}\in L^{1}(\mathbb {R} )$ with the properties that ${\hat {f}}(0)=1$; ${\hat {f}}(n)=0$ for all integers $n\neq 0$; $f(2\pi n)=0$ for all $n$. If we let $x(t)=\sum _{n=-\infty }^{\infty }f(t+n2\pi )\,$, then $x(t)$ is in $L^{1}(\mathbb {T} )$. On the one hand x(0)=0, by properties of $f$, but if I tried to use the inverse fourier transform, I would get $1/(2\pi )$. Forgive me if I am not careful, an exact treatment is in Katznelson's book, chapter 6 section 1 (In particular exercise 15). Also it may very difficult to make the argument you gave completely rigorous. In order to take the Fourier Transform of a periodic function on R, you need the machinery of Fourier transform of measures or tempered distributions, and not everything that seems to work out symbolically always works out.) I stand by my statement that identifying the group identifies the transform. But I do acknowledge that when looking quotient group of a larger group some identifications may be made (if done with care).Thenub314 22:20, 17 September 2006 (UTC)

We have what we need in Row 25. Wikipedia contains many instances of Fourier transforms of periodic functions (such as $\cos(\omega t)\,$). To suddenly deny it here is inconsistent and hypocritical. (I'm not talking about anyone personally. I'm just saying that it's hypocritical of the editorial "staff" to deny here what is acceptable elsewhere, e.g. Nyquist–Shannon sampling theorem.) --Bob K 23:56, 17 September 2006 (UTC)

I am straying a bit from my point by continuing, but bare me just a bit longer. I have not denied anything. Many places in Wikipedia use the Fourier transform of periodic functions, when they do they are really using the theory of tempered distributions and their Fourier transforms (or something similar.) And many (including the article on the Nyquist–Shannon sampling theorem) point it out. My point is that the study of fourier transforms of periodic functions on R is not the same as the study of Fourier series. One assumes much more background. So I feel the phrase "Fourier Transform on R" is the most descriptive of the possible titles for the section I have heard.
Thenub314 01:17, 19 September 2006 (UTC)

OK, but "Fourier Transform on R" does include periodic functions (as you just said), and therefore it does not distinguish between the cases:

• periodic   time domain $\rightarrow$ discrete frequency domain (Fourier series)
• aperiodic time domain $\rightarrow$ continuous frequency domain

which you disagreed with, apparently on the shakey grounds that it requires more background to make the case.

--Bob K 17:09, 19 September 2006 (UTC)

I am sorry I was unclear, I disagree because I don't particularly believe it. A periodic function on R has a Fourier transform which is a distribution on R. You seem to be claiming that every periodic function has a Fourier transform on R which is supported in some discrete set and we know exactly what it is. This can't quite be true as stated, for trivial reasons. But I am cautious of more bizzare counter examples. I tried to show that what you had written doesn't exactly work as stated by giving an example where the two sides on some equality were different. If it is true you certainly haven't proven it. So let me say again, the study of fourier transforms of periodic functions on R is not the same as the study of Fourier series. At least, I won't believe otherwise unless I read a proof. Thenub314 20:41, 21 September 2006 (UTC)

You don't have to believe it, because I conceded that one requires more background than the other. My derivation of the Fourier series (above) is overkill. The additional background it requires includes the amazing Row 25, which I didn't prove, but I don't have to. The rest of the proof is straightforward, so unless you have a problem with the Dirac comb, the suspicion falls on your "bizarre counter-example". And frankly, if "bizarre counter-examples" are the only obstacle here, then I ask you to weigh the merits of those vs the merits of a naming convention that emphasizes the unity and duality of the Fourier transform family. There is nothing wrong with naming something "periodic-time Fourier transform" with a footnote. And the footnote would explain the bizarre circumstances under which "Fourier series" is the better alternative.

--Bob K 01:49, 22 September 2006 (UTC)

My point is that you cannot relgate the study of Fourier Series to a subset of Fourier transforms of periodic functions. For example, an important part of the subject is the Fourier series for measures (or distributions). In which case your argument breaks down because the convolution of distrbutions is not defined, and your in trouble at the first line. Also you very freely exchange infinite sums and integrals, in what sense is any of this supposed to converge? So I do think it makes sense to unambiguously speak of the "Fourier transform on R". For better or worse it is the most common language used in Harmonic Analysis. And as stated above the phrase continuous Fourier transform even appears rarely on google. Also, care should be used when pointing to wikipedia to back up your argument. In this case I think it is fine, but for an example I will point out Row 21 is technically false as stated. I do not suggest that it be changed, but just point out why caution might be necessary. Thenub314 03:03, 23 September 2006 (UTC)

Wikipedia has mistakes. I know. But the particular concepts I am using have survived the review of many editors, including yourself. I am not the first to apply Row 25 in a convolution. And if it isn't good for that, then please tell me what is it good for? And why doesn't the article make the distinction clear? I.e. please consider adding what you think is missing or fixing what you think is broken. And why do you say on the one hand that "the convolution of distributions is not defined" and on the other hand "In this case I think it is fine"? Aren't you contradicting yourself? And if it really is fine in this case, then what are we arguing about? I am willing to settle for that. My primary point is that if the widely used term discrete-time Fourier transform is valid, then so should be discrete-frequency Fourier transform (or equivalently periodic-time Fourier transform). What mathematical technicality am I missing that validates one and invalidates the other? The answer in none, because they are mathematical duals. --Bob K 03:56, 23 September 2006 (UTC)

I say the convolution of a distribution with a distribution is not defined because in general it isn't. By the phrase "In this case I think it is fine" I meant I don't think there is an error in Row 25. I do think the sense in which row 25 is true is non-trivial. If you're missing no technicalities, what about the counter example pointed out above? Thenub314 00:40, 28 September 2006 (UTC)

I thank you for the counter-example. But unfortunately I am not as interested in it as you are. I think that enough books feature convolution with Dirac combs that despite whatever quibbles you might have, it's good enough for Wikipedia. I did not invent the concept. --Bob K 02:29, 28 September 2006 (UTC)

A term that would distinguish the periodic case is discrete-frequency Fourier transform, as would periodic-time Fourier transform. As I have shown, these also can be reduced to a special form that we call Fourier series, which is an alternative description with its own advantages and disadvantages. I don't think we should suppress either description. We should treat them as alternatives and point out the merits of each. (One requires less background, for instance.)

--Bob K 17:09, 19 September 2006 (UTC)

On a different point, regarding periodic time domains and discrete fourier transforms. Many things one may think of as a "time domain" are not usually thought of as being periodic, but they are compact. It is this compactness, at least I thought, that implied the discrete nature of the Fourier transform. But I think that is a better discussion for the Fourier Transform page, and not so relevent here.
Thenub314 01:17, 19 September 2006 (UTC)

But the time domain is not compact. In my example above, $x(t)\,$ may be 0 outside of $[0,\tau ]\,$. But the Fourier series representation of $x(t)\,$ is not 0 outside of $[0,\tau ]\,$. In fact it is identical to the periodic function $x_{\tau }(t)\,$. In other words, you cannot represent just $x(t)\,$ in a discrete series. It's not periodic, so it takes a continuous Fourier transform.

--Bob K 04:44, 19 September 2006 (UTC)

This was not refering to your specific example but was a general comment. The study of Fourier Series generally looks at functions defined on the circle a compact interval. Or starts with a periodic function but only uses the restriction of that function to an interval. But, you may also think of the Fourier transform of functions defined on the torus, say and the same principle applies, that is the space on which the fourier transform is defined is discrete. If we are to discuss this more let's do it on my talk page, it doesn't seem relavent to the merge/title questions. Thenub314 20:41, 21 September 2006 (UTC)

Topology of the article series

* Disclaimer: I am not a mathematician. I am using the term "topology" to clarify that I have little to say about where exactly the border between different articles of this series should be, but I'm rather looking at the number of articles and their overall hierarchical structure. — Sebastian (talk) 00:27, 24 September 2006 (UTC)

PAR wrote above "The heirachy [of the article series] should be maintained - Fourier transform as a main article and Fourier series, continuuous Fourier transform, etc. as sub-articles. ... Presently it is (or should be):

• Fourier transform (general)
  • Continuous Fourier transform (Continuous -> Continuous)
  • Fourier series (Continuous -> Discrete)
  • Discrete Fourier transform (Discrete -> Discrete)
  • Discrete-time Fourier transform (Discrete -> Continuous)
  • ..."

At first glance, this looks nice and orderly. But it seems to me that it contains a fatal flaw. The way I see it, any discrete-time FT can be seen as a special case of a continuous FT (by using some magic like the Dirac comb). This reminds me of the way one can see a rectangle as a parallelogram, and the above hierarchy feels to me as if we had:

• Parallelogram (general)
  • Parallelogram (unequal sides, unequal angles)
  • Rectangle (unequal sides, equal angles)
  • Rhombus (equal sides, unequal angles)
  • Square (equal sides, equal angles)

The point of my merge request was to eliminate the duplicate "parallelogram" entries. — Sebastian (talk) 00:27, 24 September 2006 (UTC)

I don't think we would want to eliminate the duplicate "parallelogram" entries if the link between the four cases was rather obscure to anyone but a serious mathematician, and if the common description of the four cases was quite different. This is the case with the various Fourier transforms. I think the elegant link between the four transforms should be discussed in the main article and the four kept separate. PAR 13:20, 24 September 2006 (UTC)

It seems to me you are making the following assertions:

• The common description of the four cases is quite different. True. I assume you mean the fact that they use sums vs integrals and finite vs infinite endpoints. These different descriptions are indeed in use and we can not gloss over them.
• This difference is an argument for keeping two general articles. That goes against Occam's razor, so I would need to see a good example where Wikipedia keeps two articles for the same fact, just because of some differences in its description. (This is my favorite bullet since it best fits the general level I intended for this "topology" section.)
• The link is obscure. It depends on what you mean by "link": (Case 1: by "link", you mean differences in description as mentioned above:) You may be right that the mathematics behind that are obscure to non-mathematicians. But I fail to see why that would justify keeping two articles for the general topic. (Case 2: by "link", you mean the factual correspondence between discreteness and periodicity; in other words, that the FT of a periodic function looks like a comb:) This is not obscure, but a fact well known among people who use the FT.
• The link is separate from the transforms. It depends as above: (Case 1:) The connection between the different descriptions may indeed merit coverage in a separate article. But it seems to me that this is a general calculus issue, rather than one that's limited to Fourier transforms. (Case 2:) This is a property of the the continuous FT and can be covered there.
• We would want: We all want the same: Wikipedia to be as accurate and understandable as possible. Finding out how to achieve this is the purpose of this discussion. — Sebastian (talk) 21:36, 25 September 2006 (UTC)

What do you propose as the proper set of articles? PAR 01:40, 26 September 2006 (UTC)

One idea would be to add an article defining Fourier transform on Locally compact groups. That would be a natural place to discuss properties general to all fourier transforms, then in the generalizations sectoin of the other fourier transform articles, link back to this one. Of having the name of the article match (for better or worse) the most standard name.

• Fourier Transform on Locally compact abelian Groups
  • Fourier transform (Continuous -> Continuous)
  • Fourier series (Continuous -> Discrete)
  • Discrete Fourier transform (Discrete -> Discrete)
  • Discrete-time Fourier transform (Discrete -> Continuous)
  • ..."

Thenub314 14:33, 27 September 2006 (UTC)

So "Fourier Transform on Locally compact abelian Groups" becomes the "general" article. That sounds good to me. PAR 15:41, 27 September 2006 (UTC)

I would prefer a treatment where the periodic and/or discrete cases are initially viewed as simplifications of the general case. I.e., integrals are replaced by summations:

Fourier transform

• <These subsections would cover the general case (aka continuous Fourier transform).>
• ...
• ...
• Simplified forms for special cases
  • Transform of functions that are periodic:  the Fourier series
    • ...
    • ...
  • Transform of functions that are sampled:  the discrete-time Fourier transform
    • ...
    • ...
  • Transform of functions that are both sampled and periodic:  the discrete Fourier transform
    • ...
    • ...

I realize there are other ways to describe the special cases, such as "Transform of compact functions" and "Transform of functions of a discrete variable (or domain)", and these would have to be explained in the fine print. But I think most readers will appreciate the use of the common terms periodic and sampled (or maybe digitized).

--Bob K 18:49, 27 September 2006 (UTC)

Yeah, and add to that:

• Generalizations
  • Fractional Fourier transform
  • Chirplet transform
  • ...

— Sebastian (talk) 22:04, 27 September 2006 (UTC)

It's great, but as you said there is enough material for several articles. Assume for the sake of discussion that Fourier series, DTFT, DFT, Fractional Fourier transform, and Chirplet transform are just links to separate articles. Then the only section with any meat is Fourier transform, which is large enough and with enough subsections to obscure the topology you are hoping to reveal... except (I must admit) for the automatically-generated Contents box, and maybe that is sufficient. It seems the only other choice is to make the meaty section a separate article also, leaving just an outline of links. But that brings us right back to two articles that could both be called Fourier transform. One is the outline, and the other is the top level subsection of the outline. Do you have a different vision than that? --Bob K 06:50, 28 September 2006 (UTC)

I can follow your argumentation up to "the only other choice". But first, let me clarify a possible misunderstanding: I don't think the relation between generalizations and special cases deserves especially prominent treatment. My point was merely that the current structure is confusing. — Sebastian (talk) 02:52, 1 October 2006 (UTC)

FWIW, I will try to explain what I meant. My understanding of the outline/structure above is that what we are currently calling continuous Fourier transform gets elevated to a higher organizational level than the more specialized and generalized forms... they just become sections 7 and 8, in terms of the current Content box. In the process we drop the nonstandard name, and continuous Fourier transform just becomes the default meaning of Fourier transform, which is pretty much the convention everyone else follows. Combining the current Contents box with the structure implied above, the new Contents would look like this:

Contents

Fourier transform

1 Definition

2 Normalization factors and alternative forms

3 Generalization

4 Properties

4.1 Completeness

4.2 Extensions

4.3 The Plancherel theorem and Parseval's theorem

4.4 Localization property

4.5 Analysis of differential equations

4.6 Convolution theorem

4.7 Cross-correlation theorem

4.8 Tempered distributions

5 Table of important Fourier transforms

5.1 Functional relationships

5.2 Square-integrable functions

5.3 Distributions

6 Fourier transform properties

7 Simplified forms for special cases

7.1 Transform of functions that are periodic: the Fourier series

7.1.1

7.2 Transform of functions that are sampled: the discrete-time Fourier transform

7.2.1

7.3 Transform of functions that are both sampled and periodic: the discrete Fourier transform

7.3.1

8 Generalizations

8.1 Fractional Fourier transform

8.1.1

8.2 Chirplet transform

8.2.1

9 See also

10 References

11 External links

Then I made the assumption that sections 7.1, 7.2, 7.3, 8.1, and 8.2 all become just links to separate articles, leaving just the standard Fourier transform content, sections 1-6. Since we don't actually number the sections and subsections here, I think it would be hard for people to see the hierarchical structure (except by looking at the Contents box). To make the structure more apparent, if that is one of our goals, I was thinking that we would also have to relocate the content of sections 1-6 and link to it, which actually doesn't make much sense now that I really think about it. So, nevermind. I guess I will just wait and see what actually happens. Carry on!

--Bob K 08:16, 1 October 2006 (UTC)

Now, back to your main argument: In the likely case that we agree that the article is getting too full, we have more than just one way to shorten it. The worst possible choice is to split an article into two articles that both could be named the same. This is bound to cause the inconsistencies which started this discussion. We should be looking for better ways. I think your concerns can be addressed - maybe the problems that come up in the merge discussion are really blessings in disguise, motivating overdue improvements? Anyway, they're not in the way of the proposed merge. — Sebastian (talk) 02:52, 1 October 2006 (UTC)

I am a bit confused, when you say "<These subsections would cover the general case.>" do you mean to discuss first how things work on a general group? Or do you mean specifically the case of the fourier transform on R^n? or just R? Thenub314 00:32, 28 September 2006 (UTC)

I meant just R... what we have been calling "continuous Fourier transform". By general case, I meant functions that are not periodic or sampled, so neither of the integrals (forward or reverse transform) can be reduced to a summation. --Bob K 01:52, 28 September 2006 (UTC)

Thanks for clarifying your entry above. I think the article on Fourier Series is fine, we should at best just link to it from the Fourier Transform article. I don't think it would be wise to re-write these articles to stress that they can in special cases of anything else. And definitely not rewrite them to the point that more usual way of looking at them is fine print. But including a section that describes the relationship would be fine by me.

Thenub314 03:10, 28 September 2006 (UTC)

I don't think it would be wise to split of sections 1-6 their own article. The structure of the Generalizations is a bit confusing. Why are we including info here about the chirplet transform other then a link? Why not information about wavelets, curvelets, H^p atoms, Fourier transfroms on LCA, Fourier-Stieltjes transforms, etc. The more we talk about it the better I think a merge would be. Thenub314 13:31, 1 October 2006 (UTC)

Just to make sure we're on the same page, sections 1-6 currently are their own article (this one, in fact), and the outline above represents a logical (not necessarily physical) merge with related topics. The problem with a physical merge is that the result is enormous, so we end up linking/outsourcing. chirplet transform is definitely a good candidate. But where do we stop? If Fourier series, etc. all become links, then the only meat left is sections 1-6, and we end up with what we have now plus the new links. So all we have accomplished is to add some hierarchical information to Continuous Fourier transform and rename it: Fourier transform. But the hierarchical information is hard to see amidst the detail of sections 1-6, exacerbated by the lack of section numbering (except in the Contents box).

--Bob K 16:47, 1 October 2006 (UTC)

Well shoud the article on the fourier transform outline the whole subject? Isn't that a bit closer to the goal of the Harmonic analysis article? I think the article about the fourier transform should just be about the fourier transforms, and include links to related topics. Yes, fourier series, chirplete transforms, wavelet transforms, etc. should be their own articles. Thenub314 21:39, 1 October 2006 (UTC)

For starters, Harmonic analysis makes the claim: "The basic waves are called harmonics", which is true of Fourier series, but not Fourier transform. Implying otherwise would just add to the confusion.

I agree that Fourier series, DTFT, and DFT should have their own articles, as they do now. So the burning question is this: How are they related to Fourier transform?

Wikipedia's current stance is that Fourier transform is a superset, which also includes continuous Fourier transform as a subset. But that is a non-standard name, so there is a movement afoot to replace it with something like Fourier transform on R. And at the same time there is a movement to merge continuous Fourier transform with Fourier transform.

• If we maintain the current stance that Fourier transform is a superset, then I think the merged outline would have to look something like that proposed above. In particular, the specialized transforms become special-cases of the integral transform.
• The alternative appears to be that Fourier transform is only what we are now calling continuous Fourier transform, a mere peer of the other three forms. If so, then what is to become of the current Fourier transform article?
  • Does it and its unifying role just vanish? In that case, the merge is really just a purge, and I'm skeptical of its value vs. what we have now.
  • Or does it get a different name? In that case, in what sense is it a "merge"?

--Bob K 18:58, 2 October 2006 (UTC)

Let me make a suggestion. Rename continuous Fourier transform to Fourier transform (which seems to be the most common name for that integral transform) and rename the current Fourier transform article to Fourier analysis. I would then consider merging harmonic analysis into Fourier analysis or vice versa, but this need not be done immediately (for now, harmonic analysis can stick with the more abstract while Fourier analysis can be the more concrete and familiar four cases).

Bob is, of course, correct that only the Fourier series and the DFT employ "harmonics" in the classic sense of integer frequency multiples. However, the field of "harmonic analysis" has since broadened far beyond that, and the name is a mere historical accident at this point. e.g. Katsnelson's book on harmonic analysis defines it as: the study of objects (functions, measures, etc.) defined on topological groups. The group structure enters into the study by allowing the consideration of the translates of the object under study, that is, by placing the object in a translation-invariant space. (The familiar Fourier series corresponds to discrete translational symmetry, the Fourier transform on the real line corresponds to continuous translational symmetry, and the DFT to the finite cyclic group; the corresponding group structures lead naturally to the transform definitions via projection operators onto irreducible representations of the groups. But this is too abstract for most readers.)

—Steven G. Johnson 01:35, 3 October 2006 (UTC)

I agree completely with Steven. Thenub314 13:00, 3 October 2006 (UTC)

Sounds good. Now here is Sebastian's original complaint, with the proposed new names substituted for the current ones:   --Bob K 13:36, 3 October 2006 (UTC)

"In particular, we have the following inconsistencies:

• Fourier transform contains many properties and theorems that apply to all or most Fourier-related transforms - I therefore think they should be kept in Fourier analysis. (An example is the statement on eigenfunctions which just got deleted from the latter. Granted, a discrete Gaussian is not the same as the continuous function, but if we wanted to maintain that distinction then we would have to duplicate all articles on functions with a discrete version.)
• Fourier analysis contains a list of Fourier-related transforms, which is largely redundant with List of Fourier-related transforms.
• Fourier analysis contains half page descriptions of various Fourier-related transforms, which are redundant with the articles themselves and might better be kept in an article dedicated to comparing them (List of Fourier-related transforms might be appropriate, if we don't take "list" to literally.)"

BTW, http://en.wikipedia.org/w/index.php?title=Special:Whatlinkshere/Fourier_transform&limit=500&from=0 reveals about 325 internal links to Fourier transform. We can't redirect Fourier transform to Fourier analysis, if we plan to rename continuous Fourier transform $\rightarrow$ Fourier transform, so I hope there is an automated way to fix those links! --Bob K 15:22, 3 October 2006 (UTC)

• On second thought, what percentage of those articles actually thought they were linking to the equivalent of continuous Fourier transform? Probably a high percentage. Then we would actually be fixing more links than we break by renaming continuous Fourier transform $\rightarrow$ Fourier transform (and of course redirecting continuous Fourier transform $\rightarrow$ Fourier transform). --Bob K 16:03, 3 October 2006 (UTC)

To confirm Bob's speculation, I did a random sampling of the pages linking to Fourier transform, and every single one of them clearly meant continuous Fourier transform.

If everyone agrees, I'll go ahead and make the moves (continuous Fourier transform $\rightarrow$ Fourier transform, and Fourier transform $\rightarrow$ Fourier analysis) in a few days. —Steven G. Johnson 21:55, 3 October 2006 (UTC)

I agree. Thanks everyone for contributing to this solution. — Sebastian (talk) 05:49, 4 October 2006 (UTC)

I agree. Go for it. Thenub314 12:42, 4 October 2006 (UTC)

After that, I think Sebastian's original 3 concerns are still valid. One noteworthy thing about the "redundant half-page descriptions", is that they try not to start with a definition of the transform, $X(f)\,$, $X(k)\,$, etc. Rather they start with the conjecture that $x(t)\,$, $x(n)\,$, etc can be represented in terms of sinusoids, which is the inverse transform. For a total newbie, that might be more appealing than a sudden definition. For others, who are expecting the definitions, it's probably less appealing. Anyhow, it's different... not as redundant as it might first appear. --Bob K 14:06, 4 October 2006 (UTC)

Good point. This problem of two possible approaches to a topic seems like a general problem with mathematical articles. Has there ever been a discussion on how to solve this? (Maybe if a reader could change the skin between "newbie" and "math geek" .... just kidding.) Anyway, I think in the Fourier analysis article, starting with the mathematical formula is justified. In fact, I think we don't need much more than just the formula in this article. I'll go ahead and edit it to show what I mean. This is an example for the "outsourced" articles discussed below. — Sebastian (talk) 00:08, 8 October 2006 (UTC)

Yes I agree, please make that change. The term 'Fourier transform' generally refers to what wikipedia currently calls 'continuous fourier transform'. Wikipedia is out of step with general usage and this is potentially confusing. So the page currently labelled 'continuous fourier transform' should be labelled 'fourier transform'. There is a case for a general article pointing out the links and similarities between all these things (FT, FS, DFT...) and calling it 'Fourier analysis' seems reasonable, I cant think of anything better. I dont think links will break if a redirect is used. Paul Matthews 15:23, 4 October 2006 (UTC)

1. If the redirect you have in mind is Fourier transform $\rightarrow$ Fourier analysis, please reread. I.e., we aren't planning on doing that.
2. One of the issues on the table is that there are not one, but two, articles of a general nature: (1) the current Fourier transform, and (2) List of Fourier-related transforms.

--Bob K 16:51, 4 October 2006 (UTC)

Ways to shorten this article

This came up during the merge discussion, but it applies regardless: This article and the Fourier Transform article are quite full. I propose therefore the following to shorten them (or it, if we merge):

Sections that already exist as independent ("outsourced") articles

One way to shorten the article is simply to trim existing, already "outsourced" sections, such as this, this and this. I don't see any reason why these should contain more than a "seemain" reference, a short paragraph of text and a definition formula. (BTW - Why did you include the elipses under your subsections? This makes me regret having written "Yeah" under your post!) — Sebastian (talk) 02:52, 1 October 2006 (UTC)

For example, the section called Transform of functions that are periodic: the Fourier series (and numbered 7.1 above) represents the current Fourier series article, which currently comprises 8 subsections. Those subsections would logically become 7.1.1, 7.1.2, 7.1.3, .... I am not implying that we would actually move the Fourier series article. I am just showing where its subsections would fall logically, not physically, in a new hierarchy based on the promotion of continuous Fourier transform to the level of Fourier transform.

--Bob K 16:16, 1 October 2006 (UTC)

I see. Thanks! — Sebastian (talk) 00:08, 8 October 2006 (UTC)

I believe that that alone would get the merged article already to an agreeable size. — Sebastian (talk) 02:52, 1 October 2006 (UTC)

To sum up, here are some such moves that we might consider:

• Fourier analysis#Fourier transform --> article: Fourier transform
• Fourier analysis#Multi-dimensional version --> article: Fourier transform
• Fourier analysis#Fourier series --> article: Fourier series
• Fourier analysis#Discrete-time Fourier transform --> article: Discrete-time Fourier transform
• Fourier analysis#Discrete Fourier transform --> article: Discrete Fourier transform
• Fourier analysis#Fast Fourier transform --> article: Fast Fourier transform (This shouldn't even be a section Fourier analysis).
• Fourier analysis#Family of Fourier transforms --> as a section into article: List of Fourier-related transforms
• Fourier transform#Convolution theorem --> article: Convolution theorem
• Fourier transform#Cross-correlation theorem --> article: Cross-correlation
• Fourier transform#The Plancherel theorem and Parseval's theorem --> article: Plancherel theorem or Parseval's theorem

Sections we may want to move to separate articles

A good article needs balance between its subsections. So we probably will find it necessary for good balance to outsource other subsections. — Sebastian (talk) 02:52, 1 October 2006 (UTC), modified 00:08, 8 October 2006 (UTC)

This includes the following sections:

• Table of important Fourier transforms

Variance

Shouldn't the equation for the variance be $\Delta ^{2}A\equiv \langle A^{2}\rangle -\langle A\rangle ^{2}$, rather than $\Delta ^{2}A\equiv \langle A^{2}-\langle A\rangle ^{2}\rangle$

-- Robert L 16:31, 24 September 2006 (UTC)

They are the same thing, since $\langle x+c\rangle =\langle x\rangle +c$ when c is a constant. PAR 18:13, 24 September 2006 (UTC)

Another equivalent expression, which is arguably better because it suggests the reason for this definition of variance, is $\Delta ^{2}A\equiv \langle (A-\langle A\rangle )^{2}\rangle$. —Steven G. Johnson 18:19, 24 September 2006 (UTC)

I agree - thats the best definition, the others seem to be derivative. PAR 01:40, 26 September 2006 (UTC)

Eigenfunctions

Hermite polynomials are eigenfunctions of the FT. It might be nice to see them represented in the table which lists the transforms of particular functions.

Actually, it is the Hermite functions that are eigenfunctions of the FT, not the Hermite Polynomials -- they need to be multiplied by the normal distribution to be eigenfunctions. And yes, that would be nice. Aaron Denney

Localization property

As currently written this sections does not make sense (to me, at least). Would anyone like to take a stab at it, or should I go ahead ? Abecedare 22:03, 4 November 2006 (UTC)

In what way where you thinking of changing it?Thenub314 00:22, 5 November 2006 (UTC)

I have three primary concerns:

(1) The notation $\Delta x$ and $\Delta \omega$ is not clear since it does not make explicit the dependence upon the functions f and F.

(2) Gaussian functions have two interesting properties related to the Fourier transform (a) with appropriate choice of mean and variance a Gaussian function is an eigenfunction of the Fourier transform; (b) all Gaussian functions satisfy the Fourier Uncertainty Principal with equality. It is the 2nd property of Gaussian functions that is of importance in this section. The first should perhaps be moved to a more relevant section (e.g. one pointing out other eigenfunctions of the Fourier transform such as, the comb function (distribution to be exact) ), or simply left as a side-note here.

(3) The relationship between the 'smoothness' of a function (degree of differentiability) and (asymptotic) rate of decay of its Fourier transform should be expanded, perhaps into a sub-section of its own.

I also have a couple of questions for the more experienced wikipedians here:

(1) Is there a way to number the equations on the page ? This will remove the necessity of using vague descriptors like "this", "above" etc.

(2) How mathematically detailed should the results on Wikipedia be ? For instance, do we need to specify the space of functions for which the uncertainty principal is well defined. Abecedare 02:43, 5 November 2006 (UTC)

1 - It seems to me the relation is clear - there are some missing steps, but no missing definitions. Just a series of substitutions to come up with the result. Are the missing steps the problem here?

2 - I agree, this material in the beginning does not seem relevant to the section, although it should be said somewhere.

3 - I agree for the same reason.

4 - I think thats a good idea. If this section becomes too detailed, perhaps it will deserve its own article.

PAR 04:35, 28 November 2006 (UTC)

The defintion of the $\Delta$ operator isn't entirely clear to me - I assume it's just the square root of the $\Delta ^{2}$ operator, but this could be made explicit. Also, defining the expected value of a function seems an unnecessary generalization if only the identity function $A(t)=t$ is relevant. --catslash 18:26, 21 January 2007 (UTC)

I don't understand this excerpt:

Define the expected value of a function A(t) as:

$\langle A\rangle \ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }A(t)f(t){\bar {f}}(t)\,dt$

and the expectation value of a function B(ω) as:

$\langle B\rangle \ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }B(\omega )F(\omega ){\bar {F}}(\omega )\,d\omega$

Also define the variance of A(t) as:

$\Delta ^{2}A\ {\stackrel {\mathrm {def} }{=}}\ \langle (A-\langle A\rangle )^{2}\rangle$

and similarly define the variance of B(ω). Then it can be shown (citation needed) that

$\Delta t\,\Delta \omega \geq {\frac {1}{2}}.$

The equality is achieved for the Gaussian function listed above, which shows that the gaussian function is maximally concentrated in "time-frequency".

Based on what's defined, one concludes that:

$\Delta t={\sqrt {\Delta ^{2}t}}\quad {\stackrel {\mathrm {def} }{=}}\quad {\sqrt {\langle (t-\langle t\rangle )^{2}\rangle }}=\quad ???$

--Bob K 12:24, 5 April 2007 (UTC)

Bob K, can you take a look at the new version and let me know if it is clearer now ? The older version, incidentally, was not "incorrect", but perhaps confusing since it did not make explicit (1) the dependence of $\Delta$'s on f or F, (2) it did not differentiate between the variable t, and the identity operator on $\mathbb {R}$ and used t for both, (3) as catslash pointed out above, it started by defining average of general functions A(t), even though the final result only uses average of t and t-u_f. Hope the new version is an improvement! Abecedare 21:58, 9 April 2007 (UTC)

That looks fine to me now. Thanks!

--Bob K 23:16, 9 April 2007 (UTC)

Consistent notation

I noticed that PAR edited the Properties section to "replace x with t to conform to rest of article" which I thought was a good idea. However the edits were reverted by Thenub314 who felt that "Changing x to t shouldn't be done uniformly". Can we perhaps develop a consensus on the issue and aim for a consistent notation on the page ? Here is my proposal:

• Use t as the independent 'time' variable and f as the corresponding frequency variable.
• Use g(t) (and if needed h(t), p(t), q(t) ) for the time-domain signal, and G(f) (or $G(\omega )$) for its Fourier transform.

This way we avoid using, (1) f for both the frequency variable and function name, and (2) x as both an independent variable and function name. What do other editors think ? Abecedare 03:46, 28 November 2006 (UTC)

I reverted back to replacement of x with t. It seems staggeringly obvious to me that this is the right thing to do. Could anyone explain why it is preferable to have randomly mixed notations in this article?

Also, t  is the standard symbol for time, and ω is the standard symbol for angular frequency, so I think t  and ω should definitely be used. f  is the symbol for circular frequency ω=2πf  and things like exp(ift) would be not good. Alternatively, x  is the standard symbol for distance, and k  is the standard symbol for spatial frequency, so things like exp(ikx) could be used as well. Things like exp(itx) are way bad. PAR 04:35, 28 November 2006 (UTC)

I agree with both of you. Let's replace f(t) with something else. And I am still trying to get used to Thenub314's ${\bar {g}}(t)\,$ (instead of $g^{*}(t)\,$ ), but it's a struggle. --Bob K 18:27, 1 December 2006 (UTC)

My real complaint about changing x to t was that it is done in cases where t represents an n dimensional quantity. The best arguemtn for t over x is that often people are thinking of a time varying signal when taking FT's. So I don't mind the change through out most of the article. But in sections when we are discussing the fourier transform on an n dimensional space then it doesn't seem to make a lot of sense to me. (PS I have been away awhile, sorry about the slow reply) Thenub314 13:27, 18 December 2006 (UTC)

I agree with that too. n-dimensional time doesn't make sense. --Bob K 16:36, 18 December 2006 (UTC)

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