Talk:Fourier transform/Archive 3

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Given that this method can only be feasibly useful in the era of computers, I did spend a long time wondering what the hell Fourier was up to when he came up with this stuff. He was hardly making mp3s.

So I eventually learned that it's some stuff to do with steam engines and working out why they keep exploding.

This article needs a history section for others curious why some ancient mathematician would come up with an algorithm which isn't feasibly useful without a computer! —Preceding unsigned comment added by 86.154.39.2 (talk) 20:47, 8 September 2010 (UTC)

For one thing, Fourier transformation is extremely useful and influential even as a purely analytical and conceptual tool, computation aside.

Even viewed only as a computational technique, however, it's not true that these methods (e.g. Fourier series) are infeasible without computers. Spectral methods (based on Fourier series and related expansions) converge exponentially fast for analytic functions, which means that even just computing a few terms (which is feasible by hand) can often tell you a lot. For this reason, they were arguably the dominant "computational" method for PDEs and integral equations before the computer era, and there are lots of classic pre-computer results, from Fourier's solution of the heat equation to Mie scattering, that use them. (In the case of the heat equation, all of the Fourier terms decay exponentially, with higher-order terms decaying faster, so that eventually it is dominated by just one term.) The limitation of this approach is that exponential convergence only comes for special high-symmetry geometries where all discontinuities and singularities can be accounted for analytically; in more general cases, the convergence is slower and the method is less practical as a computational method without computers.

— Steven G. Johnson (talk) 22:39, 8 September 2010 (UTC)

Characteristic functions in probability theory

I have added a brief reference to characteristic functions in probability theory. I think my brief reference should be expanded to include the convention for characteristic functions used in that wiki article in the table of other conventions. However, this is problematic due to differences in notation between the two articles. The notation can be made to align only in the case when there is a probability density function. JohnQPedia (talk) 21:29, 26 June 2009 (UTC)

There is already something on the discussion page of Characteristic_function_(probability_theory) that is related to this. There you will find comments to the effect that it is more intuitive to interpret the characteristic function as being acquired by taking the inverse Fourier transform of the probability density function (in contrast to the wiki article concerned, which says this is acquired by taking a Fourier transform). However, there are two points. First, even under that interpretation, the (forward) transform is still different from any of the three listed in the other conventions sections here, because under this interpretation it is the (forward) transform that is multiplied by ${\frac {1}{2\pi }}$, rather than the inverse. And second, some authors of textbooks currently in use may in fact refer to the characteristic function in probability theory as being produced by a Fourier transform (perhaps such as those listed in the references of that page, but not adequately cited inline), rather than by an inverse Fourier transform. In this case, I think the wiki should reflect actual usage. In either case, the other conventions section is incomplete with respect to characteristic functions in probability. JohnQPedia (talk) 01:06, 27 June 2009 (UTC)

Many textbooks on probability do simply state that the characteristic function is the Fourier transform (for example Billingsly's book), which is probably where this comment at the characteristic function page comes from. I have some added some information (with a reference) about the difference in conventions used for the characteristic function in the section of the Fourier-Stieltjes transform, which is where the characteristic function pages links to. And tried to rewrite the part you added to refer back to that. Unfortunately, I think it is hopeless to include the probability convention directly in the table with out redoing the table in some fundamental way. Thenub314 (talk) 14:35, 30 June 2009 (UTC)

In its article Characteristic Function, the MathWorld website confirms the interpretation of a characteristic function in probability as the Fourier transform of the probability density function, rather than the inverse Fourier transform. (Despite a compelling intuition to prefer an inverse Fourier transform, the mathematics are unaltered whether one considers a characteristic function to be acquired by a Fourier transform or an inverse Fourier transform.) Its article Fourier Transform also clears up any confusion, by explicitly saying that there are a number of conventions in widespread use. It lists five. It also gives explicit formulae for the Fourier transform and its inverse for arbitrary choices of constants. I have reproduced equations (15) and (16) from that page here

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

$f(t)={\sqrt {{|b|} \over {(2\pi )^{1+a}}}}\int \nolimits _{-\infty }^{\infty }F(\omega )\,e^{{-}i\,b\,\omega \,t}\,d\omega$

I put these here to indicate the kind of generality that could accompany the table in the section on other conventions. I believe such general formulae should be in the section for two reasons. First, it would make it clear how the specific examples in the table are derived. Second, it would provide a tool for the reader to interpret any sources that use conventions not listed in the table.

A further point is that the examples in the section on other conventions are generalized to apply to multidimensional integrals over ${\mathbb {R} }^{n}$. This means that a heavy edit is required to expand that section, while maintaining completeness of the table. (The MathWorld formulae given above are of course only one-dimensional.) I also think the multidimesional notation could be unclear to readers who are only up to speed on the most common circumstance, when the integral is one dimensional, and expressed in the form $\int \nolimits _{-\infty }^{\infty }f(x)\,dx$, rather than $\int \nolimits _{\mathbb {R} }f(x)\,dx$. This makes the table not generally accessible, and that undermines its utility. I suggest separate entries for the case when the integrals are one dimensional.

Thanks, all. JohnQPedia (talk) 03:17, 27 June 2009 (UTC)

Clarifying the contents

As an aid to the general public I propose to modify the first paragraph (I don't see how) in the following way:

"In Fourier transform, both domains are continuous and unbounded. Time intervals may be arbitrarily short and frequencies arbitrarily high. These infinite limits demand erudite mathematical tools not required in Fourier analysis. An example of an application more readily approached by Fourier analysis would be the inverse filter -- a band limited filter demanding an inverse of infinite bandwidth." —Preceding unsigned comment added by Jclaer (talk • contribs) 03:37, 3 July 2008 (UTC)

Hi all. I came here looking for a transform pairs table because I'm studying communication systems and ended up wondering why some Fourier transforms differed from the ones that I have tabulated in my textbook. I didn't understand why the 2pi factor was under a square root!! Anyway, I freaked out, googled a lot and finally read the article (it is kinda intimidating) finding out the answer my self (good job!). What I would like to notice is that it takes a while to locate and understand the 3 different conventions thing, and I think that it could be more simple if we give the reader a big picture wise look of that in the "contents". For instance, putting it like:
1 Definitions
1.1 Communications and signal processing (f) convention
1.2 Mathematics angular frequency (2pi) convention
1.3 Mathematics angular frequency (2pi unitary) convention

Finally, like some textbooks do, It would be really simplifying to say "in the rest of this article, we will use convention {whatever} unless stated otherwise". And stick to that in order to preserve consistency.

I would also like to say that I agree with Feraudyh (disgruntled editor) that this article is really hard to read for students like me, it seems really different from what we have on the communications textbooks. I understand that is a mathematical thing and it should be the most formal possible, but still I guess it could be more readable.--Dhcpy 01:41, 1 December 2007 (UTC)

The only way it could take "a while" to locate the 3 conventions is if you read the article haphazardly instead of linearly. I think you have to take some accountability for that yourself. I understand and sympathize with the suggestion to stick to one convention, like a textbook. That would probably serve beginners the best. But non-beginners would rather see their own pet convention presented along with the others, than not see it at all. So it's hard to please everybody.

--Bob K 22:50, 1 December 2007 (UTC)

I also came to this page (a while back) looking for transform pairs, and probably also skipped the first section (Definitions). However, since most of the results are (very helpfully) given for each convention, it seems pretty unambiguous. Is there a key result which is stated in only one version? --catslash 00:01, 2 December 2007 (UTC)

Ummm... just present the formulas in a table, with a column for each convention (duh).65.183.135.231 (talk) 20:26, 16 April 2008 (UTC)

Properties?

Why do we have a section on "# 4 Properties" and a section "# 6 Fourier transform properties". Maybe we should merge #6 into #4. Thoughts? Thenub314 00:28, 22 February 2007 (UTC)

I don't think there is a good reason for the separation. Somebody just one day decided to add "# 6 Fourier transform properties" as a stand-alone section. Maybe they were trying to stay out of the first person's space?

--Bob K 04:30, 22 February 2007 (UTC)

involution claim is wrong

Bo says: "By first conjugating the function, the forward transform equals the reverse transform, a so-called involution". But the forward transform produces ${\overline {X_{3}(f)}}\,$ and the reverse transform operates of $X_{3}(f)\,.$ In order for Bo's claim to be true, the reverse transform would have to operate on the same thing produced by the forward transform. I see no value in this involution idea. It's just smoke and mirrors.

--Bob K 01:25, 8 March 2007 (UTC)

I think what Bo intends to say is, ${\mathcal {F}}\{x\}(\omega )={\overline {{\mathcal {F}}^{-1}\{{\overline {x}}\}(\omega )}}$ but that is not an involution; so I removed the current statement. The correct statement would be something like: "by first conjugating the function and then conjugating the forward transform, the forward transform equals the inverse transform, a so-called involution", which makes my head spin :-) (it is always tough to "do" math in words").

I wonder though, if this property is worth mentioning on the page ... Abecedare 02:08, 8 March 2007 (UTC)

involution claim is correct

Bo, Ignore the above comments : both Bob and I realized too late what you meant. :-) Abecedare 02:20, 8 March 2007 (UTC)

Right. This transform:

$\int _{-\infty }^{\infty }[\cdot ]^{*}\ e^{i2\pi ft}\ df$

is the inverse of this one:

$\int _{-\infty }^{\infty }[\cdot ]^{*}\ e^{i2\pi ft}\ dt$

--Bob K 02:24, 8 March 2007 (UTC)

Exactly! Sorry for all my recent reversions on the article and talk page. I have tried to rewrite the word statement to make it clear that the Fourier transform can be deined in a form that is an involution. Please check if the point is now clear. Also I reverted to using the "overline" convention for conjugation, since it is used in the rest of the page and more importantly * is used for the adjoint operator (and AFAIK, there is no other "standard" notation for the latter). Regards. Abecedare 02:28, 8 March 2007 (UTC)

involution claim is original research

I've removed the "involution" definition, which is non-standard (not to mention non-linear). Bo, if you think this form should be included, please provide a reference to a published source that uses this definition. The question for Wikipedia is not whether the transform can be defined in such a fashion, it is whether it is defined in that fashion. Wikipedia is not the appropriate venue for notational reforms. Steven G. Johnson 05:00, 8 March 2007 (UTC)

No mathematical definition is standard, as no standardization organizations standardize mathematical definitions. The transformation is obviously a conjugate-linear map. That it is also an involution was just checked by editors Abecedare and Bob K. This is sufficient documentation for correctness. Steven G. Johnson's removal of the definition is a violation of Wikipedia:Resolving disputes # Avoidance. The value of the definition is that it avoids the risk of confusing forward and reverse transformation. That's why I want to share it. I'll leave it to Abecedare and Bob K to reinstall it if they like it too. Bo Jacoby 06:53, 8 March 2007 (UTC).

I partially agree with both of you. (1) the "definition" is antilinear but correct in that if the Fourier transform was so defined it would be involution, (2) it is non-standard, i.e. it is not covered in mainstream Fourier transform texts that I have seen, although obviously the property ${\mathcal {F}}\{x\}(\omega )={\overline {{\mathcal {F}}^{-1}\{{\overline {x}}\}(\omega )}}$ on which the "definition" is based is both well-known and trivially derived.

So Bo Jacoby, can you please provide some reference(s) in which the $\int _{-\infty }^{\infty }[\cdot ]^{*}\ e^{i2\pi ft}\ dt$ is offered as a definition of the Fourier transform? ... else including it would be a violation of wikipedia's guidelines against original research. Regards. Abecedare 07:23, 8 March 2007 (UTC)

Thank you, Abecedare. It is not the definition, but it is a useful variant form. I agree that the nice properties are well-known and trivially derived, so there is no question about reliability, which is the reason behind the general ban against original research in wikipedia. It may be considered original research on my part, but not on yours, so I cannot reinstall it, but you can. Just refer to me. Bo Jacoby 08:05, 8 March 2007 (UTC).

Sorry Bo Jacoby, can't do that, since as per WP:RS we need well-regarded published sources not only to vouchsafe the correctness of the "definition" (which is not really the issue here), but also its mainstreamness (to invent a word, since this is only talk-space). And given that 100's of books and 1000s of papers (very conservative estimate) have been written on Fourier transforms, we will need not just isolated references, but sufficiently "gold-plated" ones (example, text-books used in top-ranked university courses, or review articles on Fourier transforms written in prominent journals) before we can add this as a definition. Unless those are forthcoming, there is nothing I or any other editor can do about this. Regards. Abecedare 08:24, 8 March 2007 (UTC)

I am grateful that my humble observation is honored by the name of original research. I approve Bob K's formulation: "a non-standard type of Fourier transform that is an involution, which means that the inverse transform is the same operation as the forward transform". This does not make any promises of gold-plated mainstream references but gave a straightforward and clear explanation. The corresponding FFT is foolproof because only one procedure is needed for transforming forwards and backwards. For real x(t) it coincides with one of the other definitions. Too bad that only very few people got the chance to read before removal. I can do no more about it. Any of you are welcome to undo the edit "04:56, 8 March 2007 Stevenj" when you realize that our readers deserve the best. Bo Jacoby 13:53, 8 March 2007 (UTC).

There is a standard involution on the Banach algebra L¹(G, Haar) for any locally compact group G. For G abelian the involution is f*(x) = complex congugate(f(-x)), quite different from your definition. The operation you define isn't even an involution operation on the Banach space L¹ (G) for non-finite G. It will be very confusing to overload the term "involution" by calling your transformation an involution also.--CSTAR 15:06, 8 March 2007 (UTC)

Bo, I don't know if your transform is an involution or not. I agree that it is kind of cool and interesting, but I don't see how it furthers the cause of this article, which is to help people understand the standard Fourier transform(s). In fact, I think it is counterproductive. I wish there were fewer forms, not more. If indeed yours is an involution, then why not add your transform to that article?

--Bob K 16:00, 8 March 2007 (UTC)

See the definition in involution is simply: "a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f". Bob K checked that the variant 4 of the fourier transform satisfies this condition, just above. The forward and backward transformation is the same. There are many involutions: the identity, the sign change, the reciprocal, the complex conjugation, &c. Bo Jacoby 16:33, 8 March 2007 (UTC).

Well no, I did not say that $f\,$ and $t\,$ are in the same domain.

And $\int _{-\infty }^{\infty }[\cdot ]^{*}\ e^{i2\pi ft}\ dt$ is not a function of a variable. Rather it is a transform of a function.

And if that's not enough, the transform is not its own inverse, because the inverse substitutes $df\,$ for $dt\,$.

--Bob K 18:12, 8 March 2007 (UTC)

The variables f and t are not in the same domain, but the functions X and x are in the same domain. The fourier transformation transforms x into X. The transform maps the function x of time to the function that maps the frequency f to the integral $\int _{-\infty }^{\infty }{\overline {x(t)}}\ e^{i2\pi ft}dt$

$x\mapsto \left(f\mapsto \int _{-\infty }^{\infty }{\overline {x(t)}}\cdot \ e^{i\cdot 2\cdot \pi \cdot f\cdot t}\cdot dt\right)$

In this expression both t and f are dummy variables. Other variable names can be substituted for t and f. They may even be swapped:

$x\mapsto \left(t\mapsto \int _{-\infty }^{\infty }{\overline {x(f)}}\cdot \ e^{i\cdot 2\cdot \pi \cdot t\cdot f}\cdot df\right)$

Using $f\cdot t=t\cdot f$ you get

$x\mapsto \left(t\mapsto \int _{-\infty }^{\infty }{\overline {x(f)}}\cdot \ e^{i\cdot 2\cdot \pi \cdot f\cdot t}\cdot df\right)$

So the transformation that transforms a function of t into a function of f also transforms a function of f into a function of t. Bo Jacoby 22:38, 8 March 2007 (UTC).

But transforms are not functions. So your transform cannot be "a function that is its own inverse". --Bob K 23:50, 8 March 2007 (UTC)

According to the article Function, a transform is also a function. It is a very general concept. The domain need not be a set of numbers, but can be a set of other functions too. Bo Jacoby 00:12, 9 March 2007 (UTC).

A transform is a function on the (appropriate) space of functions. For instance ${\mathcal {F}}$: L²(R) → L²(R). However all the discussion about the correctness of the involution "definition" is moot because of the WP:OR concerns (the policy specifically talks about original research that "defines or introduces new terms (neologisms), or provides new definitions of existing terms"). This can be settled only by references (see WP:ATT) and no amount of debating on the talk page can overcome that. Regards. Abecedare 02:37, 9 March 2007 (UTC)

And besides, a brand new definition does not help people understand "Fourier transform". It's bad enough that we already have three to worry about. A fourth would be counterproductive. Why not use the involution article to showcase your involution? Why here? This article is not about involutions.

--Bob K 04:01, 9 March 2007 (UTC)

Bob, please note that no Wikipedia article should be used by any editors to "showcase" their X, for any X that is not found in reputable sources. Let's not foist our problems onto other unsuspecting articles. —Steven G. Johnson 15:11, 9 March 2007 (UTC)

Right. Point taken. --Bob K 15:26, 9 March 2007 (UTC)

smoother introduction to the 3 different conventions

I propose the following section to replace sections Fourier_transform#Definition and Fourier_transform#Normalization_factors_and_alternative_forms. One thing that bothers me about the current version is the abrupt appearance of the ${\sqrt {2\pi }}\,$ factors. I think it can be done in a more natural looking progression.

Another thing that bothers me is the paragraph:

Then the inverse transform can be written:

$x(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }A(\omega )e^{i(\omega t+\phi (\omega ))}\,d\omega$

which is a recombination of all the frequency components of x(t).  Each component is a complex sinusoid of the form e^iωt whose amplitude is proportional to A(ω) and whose initial phase angle (at t = 0) is φ(ω).

Wouldn't it be more logical if the components being recombined had amplitudes equal to A(·)?   Well this way they do:

--Bob K 00:46, 11 April 2007 (UTC)

Definitions

There are several common conventions for defining the Fourier transform of a complex-valued Lebesgue integrable function, $x.\,$  In communications and signal processing, for instance, it is often the function:

$X(f)=\int _{-\infty }^{\infty }x(t)\ e^{-i2\pi ft}\,dt$,   for every real number $f.\,$

When the independent variable $t\,$ represents time (with SI unit of seconds), the transform variable $f\,$ represents ordinary frequency (in hertz).  The complex-valued function, $X,\,$ is said to represent $x\,$ in the frequency domain.   I.e., if $x\,$ is a sufficiently smooth function, then it can be reconstructed from $X\,$ by the inverse transform:

$x(t)=\int _{-\infty }^{\infty }X(f)\ e^{i2\pi ft}\,df$,   for every real number $t.\,$

Other notations for $X(f)\,$ are:  ${\hat {x}}(f)\,$  and  ${\mathcal {F}}\{x\}(f)\,$.

The interpretation of $X\,$ is aided by expressing it in polar coordinate form:  $X(f)=A(f)\ e^{i(2\pi ft+\phi (f))}\,$, where:

$A(f)=|X(f)|,\,$   the amplitude

$\phi (f)=\angle X(f),\,$   the phase

Then the inverse transform can be written:

$x(t)=\int _{-\infty }^{\infty }A(f)\ e^{i(2\pi ft+\phi (f))}\,df$

which is a recombination of all the frequency components of $x(t).\,$   Each component is a complex sinusoid of the form e^{i 2 π f t} whose amplitude is A(f) and whose initial phase angle (at t = 0) is φ(f).

In mathematics, the Fourier transform is commonly written in terms of angular frequency:  $\omega =2\pi f,\,$  whose units are radians per second.

The substitution $f={\frac {\omega }{2\pi }}\,$ into the formulas above produces this convention:

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

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

which also happens to be a bilateral Laplace transform evaluated at s = i ω.

The 2π factor can be split evenly between the Fourier transform and the inverse, which leads to another popular convention:

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

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

A transform whose inverse has the same multiplicative factor is called a unitary transform.

Variations of all three conventions can be created by conjugating the complex-exponential kernel of both the forward and the reverse transform. The signs must be opposites. Other than that, the choice is (again) a matter of convention.

Comments on Bob's version

I like your version and support replacing the "Definitions" and "Normalization Factors and alternative forms" by the one above. Some comments:

• We can replace "sufficiently smooth" by continuous, since that is the condition for the stated point-wise convergence ("for every real number $t.\,$").
• The description "A transform whose inverse has the same multiplicative factor is called a unitary transform." is not clear to me, since I don't know what "multiplicative factor" of a transform means. IMO perhaps we can just state that the third form of the transform is unitary, and leave the explanation (F*F = FF* = I where F* is adjoint operator) to the "Completeness section".

Aside: I just noticed that properties section was missing the $1/{\sqrt {2\pi }}$ factor without which the all the completeness, parseval's convolution etc results in the section were incorrect! I have corrected that now. Abecedare 01:36, 11 April 2007 (UTC)

Proof of inverse fourier transform

Why is the inverse fourier transform true?

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

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

now what can I do? How can I reduce the right side of the equation so that it equals the left? —Preceding unsigned comment added by Iownatv (talk • contribs)

The simple derivation, often found in engineering texts, proceeds as follows:

$\int _{-\infty }^{\infty }x(\tau )\left(\int _{-\infty }^{\infty }\ e^{-j2\pi f(\tau -t)}\,df\right)\,d\tau =\int _{-\infty }^{\infty }x(\tau )\delta (t-\tau )\,d\tau =x(t)$

A more rigorous derivation (needed to justify the above steps) requires careful consideration of the functional space to which $x(t)$ belongs and the convergence properties of the various integrals. For this reason the article does not delve into deriving the various Fourier transform results and properties. Such analysis, at varying level of rigor, can be found for example in Mallat's "A wavelet tour of signal processing"; Bracewell's "The Fourier Transform and Its Applications" or Stein and Weiss's "Introduction to Fourier Analysis on Euclidean Spaces". Abecedare 05:28, 3 May 2007 (UTC)

Thanks Iownatv 20:20, 5 May 2007 (UTC)

Heuristic "Derivation"?

Would it be appropriate to add a heuristic "derivation" of the fourier integral and inverse as a limiting process of the exponential Fourier series of an arbitrary periodic function to aperiodic by letting the period tend to infinity? This at least makes it somewhat intuitive and plausible, before one goes on to the convergence and existence conditions and all of that. DivisionByZer0 20:35, 14 June 2007 (UTC)

Range and domain

What about the range and domain of the Fourier transform? The article does not say anything about which functions (or generalized functions) can be transformed or what is the range of the transform. --KYN 20:52, 3 August 2007 (UTC)

That is a tough question. Certainly some things can be said like the fourier transform maps $L^{2}\mapsto L^{2}$, Schwartz functions to schwartz functions, tempered distributions to tempered distributions. But after that it gets a little more messy, a complete description image of $L^{1}$ under the fourier transform is still an open question. For $2<p<\infty$ the Fourier transform of a function in $f\in L^{p}$ is not a function, but for $1\leq p\leq 2$ the Fourier transform is a function in $L^{q}$, with ${\frac {1}{q}}+{\frac {1}{p}}=1$. Thenub314 (talk) 12:15, 23 April 2008 (UTC)

Ok, I don't expect a complete presentation on this issue, but at least the very basic could be presented in the article. Here is a start:

The domain and range of the Fourier transform cannot be described as two well-defined sets of functions. Instead, they can be chosen in several different ways depending on exactly what we mean by a function and an integral. Furthermore, for some pair of domain and range which can be described for the Fourier transform, it may sometimes be of interest to consider the restriction of the transform to some proper subset of the domain. In general, however, we want to describe as "large" sets as possible for the domain. Such extensions can be done in different ways and may lead to domains where either one is not a subset of the other. Some examples of domains and ranges which are described in the literature are

• The set of Schwartz functions is closed under the Fourier transform. Schwartz functions are "proper" functions but does not include all proper functions which we could be interested in having a Fourier transform for.

• $L^{2}$ is closed under the Fourier transform. Note that $L^{2}$ does not consists of "proper functions" but rather Cauchy sequences of proper functions.

• The set of tempered distributions is closed under the Fourier transform. Tempered distributions are also a form a generalization of functions, but is not compatible with $L^{2}$.

Something along these lines anyway. --KYN (talk) 07:59, 24 April 2008 (UTC)

Your absolutely correct we should say something. A list like this might be nice. I might make the following suggestions. For Schwartz functions, it should be enough to say functions instead of "proper" functions. Which may confuse some people. By the same token, I think it would be ok to provide a link to the page on $L^{p}$ spaces when we mention $L^{2}$. If we do make an issue of the definition of $L^{2}$ functions, I am not sure if it is better to use equivalence classes of Cauchy sequences versus delareing two functions to be equal if they agree almost everywhere. I am pretty open minded on the subject though. Also I might like to mention the integral makes sense for $L^{1}$ functions. Thenub314 (talk) 20:28, 24 April 2008 (UTC)

I am confused by the statement at the end of your insert where you say tempered distributions "not compatible with $L^{2}$.". I didn't notice it the first time I responded. You may be correct, I am just not sure what you mean. Thenub314 (talk) 20:27, 29 April 2008 (UTC)

With incompatible I simply mean that "tempered distributions" and "L^2" are sets of generalized functions which are defined in two completely different ways and even if some elements in one set can be identified with elements in the other, this is (as far as I understand) not the case for all elements of both sets. --KYN (talk) 20:58, 29 April 2008 (UTC)

It turns out every element of L² can be identified as with a tempered distribution. And, the natural question for this article is what about their Fourier transforms... But if you think of an L² f as a tempered distribution and take the Fourier transform of that distribution, that is the same as first taking the Fourier transform of f and regarding that L² function as a distribution. So everything is self consistent. Yosida's Functional Analysis, pages 150-153 has a nice proof of this. Thenub314 (talk) 01:10, 30 April 2008 (UTC)

I see, I was not aware of this correspondence between L2 and tempered distributions. Is it mentioned anywhere in Wikipedia? To me, it appears to be a sufficiently interesting fact to mention in the Lp or the tempered distr articles. --KYN (talk) 10:00, 1 May 2008 (UTC)

I think it would be prefect to mention in the tempered distribution article, but I have not read it so I am not sure if it is there. It turns out any function in L^p for p≥ 1 corresponds to a tempered distribution. —Preceding unsigned comment added by Thenub314 (talk • contribs) 13:10, 1 May 2008 (UTC)

I am following the discussion here with interest. I haven't looked at the Yosida reference, but allow me to take a shot at why this is true. If S is the Schwartz space, then L^p is the dual of the normed space (S, ||.||_q), where q is the Holder conjugate of p. On the other hand, the usual Frechet-Schwartz topology on S is stronger than the L^q topology for any q. The dual of S with this stronger topology, the space of tempered distributions, contains the dual with respect to any weaker topology. Thus in natural way L^p is contained in the space of tempered distributions. silly rabbit (talk) 13:47, 1 May 2008 (UTC)

I thinik this works fine. Another way to look at it is to take $f\in L^{p}$ and $\varphi \in {\mathcal {S}}$ and consider

$\int _{\mathbb {R} ^{n}}f(x)\varphi (x)\,dx=\int _{\{x\mid f(x)<1\}}f(x)\varphi (x)\,dx+\int _{\{x\mid f(x)\geq 1\}}f(x)\varphi (x)\,dx$.

The second term is bounded by $\|\varphi \|_{\infty }\|f\|_{p}$. The first term is bounded by

$\|\varphi (x)\cdot (1+|x|)^{n+1}\|_{\infty }\int _{\{x\mid f(x)<1\}}(1+|x|)^{-n-1}f\,dx\leq \|\varphi (x)\cdot (1+|x|)^{n+1}\|_{\infty }\|(1+|x|)^{-n-1}\|_{q}\|f\|_{p}$.

Because $\|\varphi (x)\cdot (1+|x|)^{n+1}\|_{\infty }$ and $\|\varphi \|_{\infty }$ are bounded by semi-norms defining the topology on S we get that f defines a tempered distribution in this way. Thenub314 (talk) 15:41, 1 May 2008 (UTC)

I request that the sentence "The domain and range of the Fourier transform cannot be described as two well-defined sets of functions." be made more precise. I can't disambiguate the following possible meanings for it: 1) the description of the domain and range depends on the measure, 2) description of the domain and range is a topic of ongoing research, 3) a known uncomputability result prevents in some sense from describing the domain and range, 4) something else? Thanks. 129.110.242.7 (talk) 23:05, 30 September 2008 (UTC)

The real case

Many, particularly engineers and physicists, may be interested in only the real case and may be confused by the use of complex exponentials instead of trigonometric functions. The MoS guideline Wikipedia:Make technical articles accessible says that articles should begin with easier sections the reader can relate to. I suggest that a section using sines be added to the beginning before going on to complex exponentials. Loom91 20:15, 13 August 2007 (UTC)

Engineers and physicists, really? I don't recall ever having seen the Fourier transform taught without using complex exponentials in some form. The Fourier series, yes, and we do have the trig expressions in that article, but by the time anyone gets to Fourier transform on the real line they are usually talking about complex exponentials.

The problem is, even if you write it in terms of sine and cosine, you still almost have to deal with relatively advanced concepts such as delta functions just to show that the inverse works. Using trig just makes these concepts more complicated to express, but doesn't really allow someone without the math background to "understand" the transform. (The case of Fourier series is quite different, in that it is elementary to show that the inverse formulas work. Proving convergence is more difficult, although L2 convergence is pretty easy for the series case.)

On the other hand, the introductory paragraph is way too vague: "the Fourier transform is ... a certain linear operator that maps functions to other functions."? Someplace in the introduction it should say that the Fourier transform expresses a function an terms of a linear combination of sines and cosines or sinusoids or something of that sort.

And someplace it should explicitly say that sines and cosines, for mathematical convenience, are written in terms of complex exponentials via Euler's identity. And way too much of the introductory sections is devoted to the trivial topic of normalization conventions, which sheds almost no light on the fundamental concepts of Fourier transforms.

In short, I agree that some rewriting would be very helpful to make this article more accessible. I'm not inclined to think that the major problem lies with trig. vs. complex exponentials, though. —Steven G. Johnson 21:17, 13 August 2007 (UTC)

regarding "linear combination":

Do you disagree with [1] ??

--Bob K 00:01, 14 August 2007 (UTC)

Yes, I disagree with that. While there may be a formal context in which you might shy from using the term "linear combination" to refer to an infinite series or continuous integral, colloquial usage has no such restriction. And while each "individual" frequency has measure zero in the integral unless it is weighted by a delta function, conceptually this is the simplest way for a nontechnical person to think of the Fourier transform (essentially, as a limit of a Fourier series). And after all, the current text talks about "frequency components", which has much the same problem except that it is more vague about what the "components" are. (And it is not as if the concept of a "sinusoidal component" cannot be made rigorous, by integrating against a test function with any prescribed narrow bandwidth around a given frequency.) The topic paragraph should be handwavy and conceptual, should be informal for the benefit of the nontechnical reader, and should try to convey a picture of what is going on without worrying about distributions and measurable sets. All of those precise distinctions can be made later in the article. Calling it "a certain linear operator that maps functions to other functions" conveys next to nothing (this description applies equally well to multiplying functions by 2). —Steven G. Johnson 18:09, 14 August 2007 (UTC)

regarding normalizations:

Still, I think it is an important topic, because it is the source of much confusion and disagreement. There aren't many places a person can go and find all three conventions harmoniously and impartially explained, including a reasonably logical progression from one to the next and the next. And formulas for converting any convention to any other.

--Bob K 00:13, 14 August 2007 (UTC)

I'm not saying that the article should omit this information. Just that it is not central to understanding Fourier transforms, and shouldn't occupy the dominant position that it does now in the introductory section. The introductory section should pick whatever convention is used in most of the article (unitary in ω, it looks like), and mention that other conventions are discussed in the section (e.g.) "Conventions and Normalizations" below. —Steven G. Johnson 18:09, 14 August 2007 (UTC)

Loom91, were you referring to Sine and cosine transforms, which can avoid complex functions altogether (albeit at the cost of requiring even/odd symmetry of the transformed function) ? If so, those are linked from the pages "See Also" section and in the List of Fourier-related transforms. Abecedare 21:26, 13 August 2007 (UTC)

The lead section already says Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its frequency components, and the inverse transform synthesizes a function from its spectrum of frequency components, and makes no mention of normalization. This could be elaborated in an Introduction inserted before the Definition section - but a precise definition (including normalization) is pretty central, and ought to follow immediately after. I agree with Bob K, that the handling of the (possibly contentious) normalization issue is now very good. And by the way, even us engineers were taught complex algebra at school --catslash 21:26, 14 August 2007 (UTC)

fourier transform of an integral

hi! does somebody know where i can find a proof of $\int _{-\infty }^{t}f(u)\,du\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{i\omega }}F(\omega )+\pi F(0)\delta (\omega )\,$

thanks! --217.117.224.177 09:57, 21 September 2007 (UTC)

${\begin{aligned}\int _{-\infty }^{t}f(\tau )\,d\tau &=\int _{-\infty }^{\infty }u(t-\tau )\cdot f(\tau )\,d\tau \\&=u(t)*f(t)\quad \longleftarrow \quad {\mbox{convolution}}\end{aligned}}$

Therefore:

${\begin{aligned}{\mathcal {F}}\left\{\int _{-\infty }^{t}f(\tau )\,d\tau \right\}&={\sqrt {2\pi }}\cdot {\mathcal {F}}\{u(t)\}\cdot {\mathcal {F}}\{f(t)\}\\&={\sqrt {2\pi }}\cdot {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)\cdot F(\omega )\\&=\left({\frac {1}{i\omega }}+\pi \delta (\omega )\right)\cdot F(\omega )\\&={\frac {1}{i\omega }}F(\omega )+\pi \delta (\omega )F(\omega )\\&={\frac {1}{i\omega }}F(\omega )+\pi \delta (\omega )F(0)\end{aligned}}$

I assumed the unitary convention, but if $F(\omega )\,$ is the non-unitary transform, this expression is also a non-unitary transform, because the non-unitary transform of $u(t)\,$ is ${\sqrt {2\pi }}\cdot {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right).\,$  I.e., the ${\sqrt {2\pi }}\,$ factor comes from $U(\omega )\,$ instead of from the transform property.

--Bob K 21:03, 25 September 2007 (UTC)

Archiving parts of this talk page

This talk page is now so long that I believe its practical use is reduced. Therefore I propose to archive the earlier parts of this talk page, starting from 2004 until 2005, leaving Talk:Fourier transform#Alternative forms as the first posting that remains in the talk page. I propose to use the "Subpage archive method/cut and paste procedure" described here. In principle I would like to archive a larger chunk, let's say including all 2006 postings, to make the page more manageable but there appears to be some discussions which were initiated in these early postings but still have attracted answers or comments in 2007. Any optinions? --KYN 15:21, 23 September 2007 (UTC)

You can archive as much as you want. If it is needed you may post here a summary of the most interesting/active part of the archive. At any point anyone can restart any part of the archived discussion. (Igny 03:05, 24 September 2007 (UTC))

All discussions started 2006 or earlier have now been moved to "Archive 2" which, together with "Archive 1", now is reachable from the Archive icon at the top of this page. If there are active threads which have been removed, please reinsert them on this page. --KYN 09:56, 1 October 2007 (UTC)

Distribution 312

According to my signals and systems book (Signals and Systems by Oppenheim, Willsky and Nawab) the fourier transform for the signal $x(t)=\sum _{n=-\infty }^{\infty }\delta (t-nT)\,$ is $X(j\omega )={\frac {2\pi }{T}}\sum _{n=-\infty }^{+\infty }\delta \left(\omega -{\frac {2\pi n}{T}}\right)$. However the wikipedia article says that the fraction in front of the sum is ${\frac {\sqrt {2\pi }}{T}}$. I changed this a few days ago, but it was changed back, so I thought I'd start a discussion about it. There is a pretty good example in my book that shows the derivation of this (p. 299) so I'm pretty sure that square root shouldn't be there, but obviously someone thinks it should be there. —Preceding unsigned comment added by Fkp1 (talk • contribs) 21:31, 31 October 2007 (UTC)

Does your book use the same normalization convention (Fourier transform#Definitions) as is used in the article? The article table uses the unitary angular frequency convention... check that they match. That's usually the source of factor-of-${\sqrt {2\pi }}$ mismatches... ǝɹʎℲxoɯ (contrib) 21:53, 31 October 2007 (UTC)

PS- That sounds like a book oriented towards engineers, which makes it quite likely that they use the non-unitary convention (which is asymmetrical, but saves slightly on computational effort). ǝɹʎℲxoɯ (contrib) 21:55, 31 October 2007 (UTC)

I was the one who changed it. I checked it directly by numerically integrating on a computer, so I'm quite confident in having reverted it. You should check the conventions defining Fourier transforms in your book, there's likely a factor of (2*pi)^(1/2) differing between the definition used here and the one in your book.

By the way, if it was incorrect, it would be a good service to change it in both the ω and the f columns; after changing just one, they were inconsistent, which was why I was checking in the first place. --Steve 02:04, 1 November 2007 (UTC)

Ahh, that explains it. Sorry for the trouble! Fkp1 16:23, 1 November 2007 (UTC)

Why is distribution 311 a distribution?

Unless I am mistaken, $e^{-at}u(t)$ is a square-integrable function when $a>0$. Why then is this Fourier transform pair in the distribution section?

Butala 13:46, 1 November 2007 (UTC)

Removed a paragraph of disgruntled editor

I removed this from the "Definitions" section and reproduce it here because that is where it belongs

Wikipedia is supposed to be for everyone. This is only useful to mathematicians. Mathematicians have many books containing this material. This is an opportunity to make Fourier transforms understandable to lay people. This is what Wikipedia used to be like. Now the mathematicians have subsumed Wikipedia into their own domain, and use it as a place to lay down the same formalities that are available in their many books---books that are the reason that people need Wikipedia to remain a place for the non-specialist to go for understanding. After all, when deciding what should be here, what non-mathematician could know better than mathematicians? And how many non-mathematicians will show up here to participate in the decision? Below, you can see the result. Wikipedia used to be useful. It used to be by the people for the people. This page is an example of how Wikipedia pages about math are only useful to mathematicians. If you edit this out, then replace it with something that will provide an understanding of Fourier transforms to somebody who doesn't already understand them. Doing that elegantly is a greater challenge than creating the formalisms below. Wikipedia is not a repository for enshrining formalities. Use your books for that. It is for people to learn from. Nobody who comes here to find out what a Fourier transform is will understand this. Shame on the mathematicians for their lack of consideration and focus on their own interests.

Feraudyh 10:40, 6 November 2007 (UTC)

Hello. I am an engineer, not a mathematician. And I have done a lot of work recently to try to mitigate this very problem. For one thing, the very first sentence contains a link to Fourier analysis, a more remedial article, with the recently-added section: Fourier Analysis#How it works (a basic explanation). I considered adding a statement to the effect that it was "prerequisite" information, but that does not seem to be a Wikipedia tradition. So I am wondering if "disgruntled editor" has taken Fourier analysis into account.

I have also tried to improve the consistency in terms of using SI units (hertz) and unitary normalizations. And I have tried to emphasize the similarities and relationships between the continuous FT and its "special cases".

In theory, the Wikipedia strategy is to start the article at the lowest level and end at the highest level, with smooth transitions. But that is challenging at best and impractical at worst, especially at these pay levels :-). I appreciate a lot of the things the mathematicians have contributed, and I appreciate that I can find them online, because if I had to go to a college library, I wouldn't bother. My inclination, if I made the "rules", would be to encourage multiple levels of the same article: Fourier transform 101, Fourier transform 201, etc. No plan is without drawbacks, and a drawback of that plan is the same inconsistencies one sees between textbooks (conventions, definitions, terminology, symbols, notation). But hey, that is reality, and Wikipedia's charter (for better or worse) is to document reality, not "fix" it.

--Bob K 12:00, 6 November 2007 (UTC)

Did you mean Fourier tranform for dummies? Containing myths, misconceptions, trivia, etc? It is hard for me to picture more than two levels of difficulty for most math articles. Someone should start dummipedia or something where each article (mathematical and others) should start with In a kingdom far far away... (Igny 03:57, 7 November 2007 (UTC))

• Dummipedia started! — PM Poon (talk) 16:02, 22 February 2008 (UTC)

No matter, because that is not how Wikipedia works, and I don't expect it will change anytime soon. But FYI, I was assuming that what we have now, minus a few expendable pieces, would be the 101 level. I don't understand why "disgruntled" says: "This page is an example of how Wikipedia pages about math are only useful to mathematicians." That is why I asked for more specifics.

--Bob K 05:44, 7 November 2007 (UTC)

Dear Feraudyh. I completely agree with you. The article should be readabel to you. I regret that it is not. I have tried to mend it, but I am opposed by another WP-editor who believes that he can do better and that WP should contain 'standard' mathematics even if that is incomprehensible mathematics. See: User:Bob_K#two_articles_for_discrete_fourier_transform_.3F. I am sorry. Keep complaining. It helps. Bo Jacoby 13:49, 6 November 2007 (UTC).

Can we get more specific? I don't actually see a lot of high falootin' stuff in the article. It looks pretty basic to me. What would you remove? "Generalization" perhaps? It's kind of interesting, but admittedly not very consequential. What about "Completeness"? I always skip over that one, but maybe that's just me. I don't seem to have much use for "Localization property" either, but it looks well-written. I think the article would look less intimitating without all the square root operators, which result from using radian frequency instead of ordinary frequency. But lots of books do prefer radian frequency, so we're probably stuck with it.

--Bob K 17:52, 6 November 2007 (UTC)

I'm afraid I have to side this time with the non-mathematicians complaining about the (lack of) accessibility of this article. It is true that there is still a huge scope to improve the presentation – and not only for the benefit of non-mathematicians but mathematicians as well. Without any attempt at completeness, I would like to make the following observations and make a few suggestions:

• Terminology: My modest proposal would be to use systematically Fourier transformation for the operation (function) of forming the fourier transform of a function (or distribution...). And then mention that many authors use transform for both.
• Title of the article: Fourier transform is likely to be the first article someone interested in the topic looks at. Fourier transform(ation)s are however much more general than the one for functions on the real line. Therefore this article should be renamed something like "Fourier transform on R" or "Fourier transform on the real line". This is particularly the case if "Fourier analysis" currently contains both the general discussion and the "down to earth" discussion. Thus that article appears a better candidate to bear the title of this one.
• The lead does not provide practically any motivation for having the Fourier transformation. General statements such as "transforms certain hard problems about the original function to more tractable ones about its transform", "provides a way represent a complicated function by means of simpler ones" and so forth would be a good start, to be followed by slightly more technical comments. Moreover, the "time domain / frequency domain" terminology in the lead is hardly helpful to people without physics/engineering background. This includes mathematicians: an aspiring number theorist looking forward to a brief introduction to Fourier transformations would have hard time with time and frequency, I'm afraid.
• Definition is immedetiately in terms of an explicit formula with no discussion, background etc. Reference is immediately made to Lebesgue-integrable functions, and then for some unexplicable reason the discussion moves into the realm of engineering with the sudden introduction of SI units (I'm again thinking about my poor aspiring number theorist...). More generally, my humble opinion is that the context provided by harmonic analysis (dual groups,...) in fact helps both mathematicians (larger dose of detail and formalities) and typical non-mathematicians (less detail) to understand the Fourier transformation. The case may be different for engineers and some physicists, who typically have a practical reason to use the transformation, and may be happy to get quickly to the mechanics of applying it. Renaming this article and addressing the more general set-up (first less, then more technically) in a new Fourier transformation article should help here.
• Unexplained formulas fill most of the article. Too little explanation, too much reliance on collection of formulas. Currently the article seems to aim at being a collection of facts to refresh memory than an aid to understanding.
• Too many tables fill the article. While it's great that effort has been put into providing an online reference to the formulas, their place is hardly in the article itself. They are more "list-like" material, that it would be good to move to a separate article and reference here.

The above may look harsh, but it is definitely not meant as critique of the editors that have worked on the article. However, in order to move the article (rather the topic) to a new level I'm afraid that something like the above should be addressed. In other words, work remains to be done. Stca74 14:24, 7 November 2007 (UTC)

Your insights are appreciated. Thanks for taking the time. I frankly doubt that a single article can perfectly accommodate both the number-theorists and the EEs, or even do it "well". Because of the inherent instability of Wikipedia, a compromise strategy will lead to endless thrashing. I suspect it already has. The whole structure is just wrong. Math books don't look much like EE books, and there are insurmountable reasons for that. So we're just wasting our time here trying to fit a square peg in a round hole. Can we split off into separate articles? Or will someone find a policy/rule violation and force everyone back into the same foxhole? If we split, and the two articles really do belong together, that will become apparent over time, and someone can merge them. But in the meantime, separate groups can work independently, unconstrained by the need to compromise with each other.

--Bob K 15:18, 7 November 2007 (UTC)

OK, that went well. So nobody really wants to deal with the whole interdisciplinary problem. The only hope for an in-depth treatment is to divide and conquer, which is also what universities do, by department. The time/frequency view is sufficient for EEs, so that is what they teach, along with real-world units such as hertz. Wikipedia's charter is to reflect that reality (and others), not to merge it all together into an abstract generality.

So instead of working on the actual article, how about we step back and work on the bigger picture first? Apparently there needs to be an overview article about "Fourier transformations" that would itself be linked to from even higher-level articles (on transformations, representations, analysis, etc). The overview article would try to explain that specialized treatments of the same basic subject have evolved to best suit the needs of various practitioners and theorists. Then it would provide links to the appropriate, but separate, articles. We might take a stab at predicting what those articles will turn out to be, but it will probably evolve on its own over time as articles tend to split and merge. Perhaps just two will be sufficient (one for mathematical formalities, and another focussed on the time/frequency domain implications). But perhaps not.

Of course, the same points, such as the duality of convolution and multiplication, will end up being made in multiple places, but each time it will be in terms of the conventions familiar to the interested reader. The page-count will be non-minimal, but each reader only needs to read the thread that applies to himself.

--Bob K 16:21, 9 November 2007 (UTC)

I am new to Wikipedia and am not sure that this is the proper way to contribute to the discussion on this talk page. I hope so!. To start, I was drawn in by Googling the Wiki page on FTs. I was, frankly, very disappointed in the material presented. I am a math/physical scientist professional and was looking for something a little more entertaining and, for the layman or number theorist, more inspiring than a Xerox of a textbook. Been there. I agree wholeheartedly with the "disgrundtled editor" in his comments, and the others who followed, that Wiki should, in my phrasing, be a conduit to the community to the understanding of higher, interesting concepts. This "window" into a fascinating world is guided by those of us, in any field (math, history, music, etc.), who have commited their energies to gaining a deep understanding of their field, can now gladly offer their insights over the remarkable new medium of the Internet. Kind of a win-win situation.

My own thoughts on the FT presentation will be brief since this whole business seems to have such tremendous inertia. Other than reiterating my intial comment that it doesn't really do any one any good to regurgitate textbook material on Wiki ( send them to a couple of decent web sites

http://www.youtube.com/watch?v=QQbEOjGOY5I

or sic them on Bracewell's book. Try to come up with some original material. My own personal approach to teaching the idea of the FT and transforms in general is the idea of projections and orthogonality. Vectors projecting on the x-y plane leading to random functions projecting onto orthogonal periodic functions (sin and cos). Etc.

Randy Patton —Preceding unsigned comment added by Randypatton (talk • contribs) 06:19, 31 March 2009 (UTC)

Categories of Fourier transforms

This new section is important, but somewhat confusing. The dash in 'aperiodic-discrete' is confusing. It perhaps means that the function is aperiodic and discrete. Why not write that? As to "The Fourier Transform for this (Aperiodic-Continuous) type of signal is simply called the Fourier Transform". No, it is called the fourier integral. The reader wants to know a reason for this trouble; why not a single fourier transform to cover these cases? The problem is that a 'discrete function' as the limit of continuous functions does not exist, and the solution in terms of distributions is too advanced. Bo Jacoby (talk) 12:08, 20 January 2008 (UTC).

Useless

I've cited this article, chosen pretty much at random as many others are similar, as an example of one that I consider useless in that it is written at too high a level to be generally accessible, in an a discussion I have sought to provoke about how Wikipedia's mathematical articles are structure in the Village pump/Policy area in the Community Portal.

David Colver (talk) 15:15, 2 February 2008 (UTC)

Invertibility condition

The text says "If x is Hölder continuous, then it can be reconstructed from by the inverse transform". The linked page on Hölder continuity tells me that anything bounded is Hölder continuous with exponent 0. But this includes bounded discontinuous functions, which cannot be reconstructed from the inverse transform. It appears to me that one of the two pages (and I don't know which) must be in error. —Preceding unsigned comment added by 80.177.154.109 (talk) 12:33, 7 February 2008 (UTC)

I am not completely sure about the Hölder continuity, but functions that are bounded and are jump discontinuous (other forms of discontinuities as well) can be fourier transformed, and also inversely transformed. However, the inverse transform and the original function are not going to be the same everywhere, only almost everywhere. This means that the two functions can differ in a null set. So one usually speaks of the two functions as the same if they are the same almost everywhere. --Sillerpojken —Preceding undated comment was added at 12:46, 3 August 2008 (UTC)

Your correct there is a lot of confusion around this sentence. Not every bounded function has a Fourier transform defined by integration. More general methods for defining the Fourier transform could be applied, but they also have to be applied to define the inverse transform. I believe the sentence here is means of f is Hölder with positive exponent, then the resulting function is integrable and the integral defining the inverse transform converges almost everywhere to f. But this seems to be there for the sake of adding some conditions to the inversion formula. I feel this this it is a bit early in the article for this level of detail. Perhaps it would be better to say "Under suitable conditions on F, f may be reconstructed from F by the inverse transfrom:" Thenub314 (talk) 14:12, 28 August 2008 (UTC)

Article is target for vandalism

Folks from the popular 4chan boards and other sites are targeting the LaTeX source in higher math articles for minor vandalism, I could see this hurting homework grades and exam prep for quite a few students, something to keep an eye out for.

search google for "Vandalize_Every_Equation" to see a link to a wiki on another popular site with details (it's blacklisted on wikipedia's spam filters so I can't link to it directly)

140.247.10.17 (talk) 06:04, 16 February 2008 (UTC)anonymous

Bessel function

Sorry, but I'm not sure that Bessel function are square integrable (the norm L2 of its transform diverge). If is not so, please tell me why? (Sorry if I don't sign this message) --penaz

Bessel has been added to the table of Square-integrable functions 71.37.40.153 (talk) 23:38, 24 March 2008 (UTC)

Moved from Talk:Fourier transform/Comments

'THERE ARE ERRORS IN THE EQUATIONS FOR MEAN AND VARIANCE OF FOURIER TRANSFORM INTEGRALS!!. YOU MUST DIVIDE THE FIRST AND SECOND MOMENTS(The Equations currently given) BY THE NORM TO OBTAIN THE CORRECT RELATIONSHIPS.'71.254.7.2 15:06, 17 June 2007 (UTC)

The mean and variance are defined in the article under the assumption that $\|f\|=1$. So it is unnecessary to divide by the norm (although it would be otherwise). Silly rabbit (talk) 23:30, 11 March 2008 (UTC)

Some Fourier transform properties: Integration

I think defining the heavy side function at this point breaks up the flow a little. It makes this entry different form the others in this section. I think for the sake of simplicity it is best make the point there is a convolution here. Thenub314 (talk) 03:37, 25 April 2008 (UTC)

Notation

I would like to raise some questions about notation that I was thinking about in reading this article. Particularly in defining the Fourier transform we use x(t) as the signal and f. By the time we are done with the section we are ω. And starting in the second section we are using f as a function. And when we discuss multi-dimensional Fourier transforms x becomes the variable. I would like to move that we change the definition. At the very least let's call the function and variable in the first definition something else. I of course would like to say more about the use of "hat's" e.g. ${\hat {f}}$ and ${\check {g}}$ (we mention the first, but don't say anything about the second.) I feel this notation is a standard in mathematics, and not uncommon in Engineering or Physics. Back to the main point, for the sake of first year calculus students that read this page, I would like see the first definition changed. Thenub314 (talk) 02:52, 14 May 2008 (UTC)

I wish the real world weren't so messy, with three common standards, but it is. It is not our job to pick one and ignore the others. I like the intro as it is. $f,\,$ of course, is also commonly used for functions, but so are many other letters, especially with using $f\,$ for frequency in hertz. Else we might have something ugly like:

$F(f)={\mathcal {F}}\{f\}(f)=\int _{-\infty }^{\infty }f(t)\cdot e^{j2\pi ft}dt$

As a matter of common sense, I use $f\,$ for functions only when I am not using it for frequency.

FWIW, I have not seen ${\check {g}}$ in my engineering books. And the only ${\hat {f}}$ I can recall was a Hilbert transform.

--Bob K (talk) 04:32, 14 May 2008 (UTC)

Well, your expression is part of my point. In this article (starting with the second section) f is used repeatedly as a function. It is also used repeatedly for frequency. To take an example, if we condense what this article says about Perceval's theorem at the end of section 2, just to the case for ordinary frequency, it reads

$\int _{-\infty }^{\infty }f(t)\cdot {\overline {g(t)}}\,dt=\int _{-\infty }^{\infty }F(f)\cdot {\overline {G(f)}}\,df\,$     (ordinary frequency).

Which I think is confusing if your new to the subject. We should certainly explain the different notations, I think we all struggled with different notations at some point. But I think it makes sense to change the letter f at least for the first definition we give. These previous few sentences are my main point, but I would like to switch gears momentarily. As you say it is not our job to mention some notation and ignore others, so we should mention that some people use ${\check {g}}$ to denote the inverse Fourier transform of g.

I understand that using a ${\hat {f}}$ to denote the Fourier transform of f is not common in some branches of Engineering, but there are still plenty of text books that use this particular notation. On the flip side of the coin in mathematics it has become standard across many disciplines to use ${\hat {f}}$. Which is why I advocate ${\hat {f}}$ to denote the Fourier transform.Thenub314 (talk) 06:53, 14 May 2008 (UTC)

I am not defending the use of f as a function in this article. Someone else put it there, and I just let it be, because it was not confusing (at least to me). However, you have written some things using f as a function and hat for FT that I did find unnecessarily confusing.

I have no problem with "we should mention that some people use ${\check {g}}$ to denote the inverse Fourier transform of g", other than the regret that the world is so messy. In fact I am the one most-responsible for the equal treatment given to all three conventions in the definitions section, the summary table, and the properties section. I think it is part of the job of the encyclopedia to do that and perhaps point out the advantages and disadvantages of different conventions. If we were space-limited, that would be different.

But it is impractical and less useful to maintain that level of parallelism throughout the whole article. So different editors end up choosing their own favorite conventions to make points unrelated to convention, and mild chaos ensues. A solution, if the will exists, would be to write and maintain the article in the different conventions, just like people now do in about 25 different languages. That would be fine with me.

--Bob K (talk) 14:13, 14 May 2008 (UTC)

Starting in elementary algebra it is an extremely common to use f to denote a function. I expect most readers would be more familiar with that usasge. I think we would still be maintaining an equal treatment if we choose a different letter to denote ordinary frequency. Thenub314 (talk) 15:38, 14 May 2008 (UTC)

It wouldn't be a major crisis. I've had my say, and I am getting no help from the "community". So I'm going to sit down and shut up now. FWIW, I'd greatly prefer  $\nu ,\,$  instead of  $\xi .\,$

--Bob K (talk) 08:21, 15 May 2008 (UTC)

I am happy with ν, it is also very common. I probably won't have time to make any changes to the article today. But I will do so tomorrow, if changes aren't already made by then. Thenub314 (talk) 11:39, 15 May 2008 (UTC)

Sorry for the lack of support Bob; I was on holiday. For readers unfamiliar with Fourier transforms, it would make it more readable if we used t and f to flag up what's time-domain and what's frequency domain ($\nu$ isn't unknown, but will be less widely recognized). On the other hand, you can use any letter to represent a function g(t), h(t), u(t), v(t) are all clearly functions of time.

If we are going to use $\nu$, then could we use <math> for the inline-expressions? or somehow specify the font so they look similar? Otherwise it comes looks like a v on my browser.--catslash (talk) 13:31, 17 May 2008 (UTC)

I am not too particular if we use ν or $\nu \,$ (though we shouldn't force the output like I just did). I find html leads to paragraphs that are slightly easier to read, even if it the formulas look a bit different. Over all I would not object either way. I disagree that using f as the frequency variable is more readable for those unfamiliar with Fourier transforms. (Most of my reasons are stated above.) Thenub314 (talk) 21:56, 17 May 2008 (UTC)

Nicer form for formula 106+107

Perhaps the following way of putting parentheses is nicer?

$\left({\frac {d}{dt}}\right)^{n}f(t)\rightarrow (i\omega )^{n}F(\omega )$

$t^{n}f(t)\rightarrow \left(i{\frac {d}{d\omega }}\right)^{n}F(\omega )$

I suspect that anyone who makes it to that point in the article will recognize the nth-derivative equally well with or without parentheses. If you have a preference (aesthetic, intuitive, or whatever else) for one way or the other, I say go for it. :-) --Steve (talk) 17:08, 23 July 2008 (UTC)

Distribution 309

I calculated the Fourier transform of 1/x in two ways, and I didn't get the minus sign out front. Can anyone confirm this? —Preceding unsigned comment added by 59.167.144.142 (talk) 11:50, 28 August 2008 (UTC)

Okay I just realized the table uses a different definition of the Fourier Transform, problem solved. —Preceding unsigned comment added by 59.167.144.142 (talk) 12:22, 28 August 2008 (UTC)

Table entry #601

$\omega$ and $T$ are undefined. And according to Poisson_summation_formula, $F$ is not the unitary transform.

--Bob K (talk) 20:15, 31 August 2008 (UTC)

What is meant by the unitary tranform is $F(\omega )\!\ {\stackrel {\mathrm {def} }{=}}\ \!{\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!f(t)e^{-i\omega t}\,dt$

This is consistent with the other tables. Mr. PIM (talk) 01:45, 3 September 2008 (UTC)

And as for the redundancy of the Parseval's Theorem, most of the properties in Fourier transform#Some Fourier transform properties are repeated in the summary tables of Fourier transform#Tables of important Fourier transforms. Mr. PIM (talk) 01:51, 3 September 2008 (UTC)

In the intro sentence,

In the table below, $F\left(\omega \right)$ is used to denote the Fourier transform $\ldots$

$\omega$ is very clearly defined as the transform domain variable. $T$ is an arbitrary value, the same way that $a$ and $b$ are used in 101 thru 104. If it helps you to better understand the Poisson summation formula using the notation of $a$ rather than the notation $T$, go ahead and change the $T$'s to $a$'s. Mr. PIM (talk) 12:19, 3 September 2008 (UTC)

Almost aribtrary. T=0 would cause a problem. Perhaps the valid possible choices for T should be mentioned in the in the remark. Thenub314 (talk) 10:06, 4 September 2008 (UTC)

I have added to extra factor of ${\sqrt {2\pi }}$ to correct for the discrepancy between the def of the unitary transform and the transform as provided in the article Poisson summation formula. Mr. PIM (talk) 12:31, 3 September 2008 (UTC)

Comment on non-formulaic description of the Fourier transform

Dear Frequent Editors,

I am both a user of Fourier transforms in medical imaging and signal processing, and an occasional instructor to students with various backgrounds in mathematics (medical students are usually on the less mathematical end of the spectrum). I can empathize with some of the comments from previous posters about the need for a conceptual (i.e. non formulaic) section introducing the Fourier transform.

Here are a few pages with examples:

http://www.e-mri.org/image-formation/fourier-transform.html

http://www.e-mri.org/image-formation/2d-fourier-transform.html

Another useful analogy is the comparison to musical notation (although strictly speaking this is more like a joint time-frequency transform).

And a very good reference for Fourier transform related identities:

Linear Systems, Fourier Transforms, and Optics

By Jack D. Gaskill

http://books.google.com/books?id=ZcUPAAAACAAJ

Curtcorum (talk) 04:22, 9 September 2008 (UTC)

Easy enough to add those links to the article. Also, I don't think this is what you have in mind but you might take a look at http://wiki.riteme.site/wiki/Fourier_analysis#How_it_works_.28a_basic_explanation.29

--Bob K (talk) 12:33, 9 September 2008 (UTC)

I think certainly we could begin a bit softer, instead of directing the reader to somewhere the might find a more simple explanation. I do not like the first image because it gives the false impression that the amplitude of the continuous Fourier transform of a sine wave would be finite, and even equal to the amplitude of the sine wave. Though it might make a nice picture for Fourier series.

Here is my attempt at a rewrite of the first paragraph. Let me know what you think...

In mathematics, the continuous Fourier transform is an operation that transforms one function of a real variable into another. The new function, often called the frequency domain representation of the original function, describes which frequencies are present in the original function. This is in a similar spirit to the way that a chord of music that we hear can be described by notes that are being played. The continuous Fourier transform is similar to many other operations in mathematics which make up the subject of Fourier analysis. In this specific case, both the domains of the original function and its frequency domain representation are continuous and unbounded. The term Fourier transform can refer to both the frequency domain representation of a function or to the process/formula that "transforms" one function into the other.

Thenub314 (talk) 13:31, 9 September 2008 (UTC)

Sheet music is analogous to a spectrograph/spectrogram (frequency vs. time). And the frequencies have finite durations. A Fourier transform is a static distribution of the frequencies that all have infinite duration.

--Bob K (talk) 19:07, 9 September 2008 (UTC)

Sure, exactly as Curtcorum pointed out. Though I think it is common to still use the analogy. I have changed it again to reflect your criticism. Let me know if it is an improvement. Thenub314 (talk) 20:12, 9 September 2008 (UTC)

Yes, I think it is better now.

--Bob K (talk) 05:25, 10 September 2008 (UTC)

The "magnitude" of the delta function in the link

http://www.e-mri.org/image-formation/fourier-transform.html

is problematic. It does hold for an FFT/DFT, but not in the continuous limit. Most MRI data processing is dealing with FFTs and DFTs.

--Curtcorum (talk) 01:58, 13 September 2008 (UTC)

Possible pictorial example

Here is a possible transform pair example. Heavy line is real component. Dashed is imaginary. I am not sure if it is better to show axes to scale or not.

• 
  Time domain signal
• 
  Frequency domain peak

--Curtcorum (talk) 03:15, 13 September 2008 (UTC)

The time signal having a nonzero imaginary component is not oscillating but rather rotating. One needs to know the signal for negative values of time too. Bo Jacoby (talk) 07:57, 13 September 2008 (UTC).

It is a fine point, but I'll fix the caption (delete "oscillating") for clarity.

The imaginary (dispersion) component in the frequency domain results from the lack of negative time information. Curtcorum (talk) 02:37, 15 September 2008 (UTC)

Well, its more then missing negative time information. If you wanted the Fourier transform to be real then you would need a very specific structure of the signal for negative time, which is completely determined by it's values for positive time. I interpreted (or possible mis-interpreted) Bo Jacoby comment about negative time to mean that the Fourier transform is defined for functions whose domain is the entire time axis. If you want to take a Fourier transform of something else you have to treat the missing parts as 0 (or the function periodic with very large period, etc). So for illustrative purposes it might be nice if the picture included some of the axis for negative time. But maybe he can correct me if I misunderstood. Thenub314 (talk) 09:38, 15 September 2008 (UTC)

(I think the figure shows that the Fourier transform is not real. There are both real and imaginary parts. Bo Jacoby (talk) 10:11, 17 October 2008 (UTC))

Here is a clarification:

$y(t)$ and $Y(f)$ is essentially table entry #205 with an additional frequency shift.

I think showing the negative time axis is a good idea. Also showing the formulas? An abbreviated table of transform pairs with graphical examples might be good? Is there a link to one online that anybody knows of? Curtcorum (talk) 22:07, 15 September 2008 (UTC)

Wikipedia 0.7 articles

This article is apparently on a list of Wikipedia articles to be released on DVD. But we have a few maintenance tags to take care of. The deadline is October 20th. The two tags ask us to clarify and use inline citations. The too technical tag came after an ip user added the statement that every thing after the lead was for "advanced mathematicians". The nofootnotes tag was added back in November of 2007. I would like to make the article more accessible. Do people have suggestions as what the best changes might be? Thenub314 (talk) 11:07, 19 September 2008 (UTC)

Here are some comments that I am collecting that might help, and I invite comment. The first sentence in the definitions section is:

There are several common conventions for defining the Fourier transform of a complex-valued Lebesgue integrable function f :R→C.

Could be shortened to:

There are several common conventions for defining the Fourier transform of a integrable function f :R→C.

And we cite "A Friendly Guide to Wavelets" Gerald Kaiser, which has a nice discussion of the different conventions on pages 31-32.

Also the suggestion is made in our rating that:

Probably too much space is taken by the tables of explicit transforms — as such tables can be useful reference, they should perhaps be preserved in a linked page.

And I think I agree with this comment.

The section on "Generalization" doesn't really generalize, but just shows how to choose different constants. It does not cite a references, and since it follows by change of variables I don't know of any references to cite for it. At best it is not very enlightening and at worst it is "original research". I am in favor of killing the section, but if we don't we should at least reference it and move deeper into the article. Thenub314 (talk) 16:44, 21 September 2008 (UTC)

Another thought to simplify the article for the reader would be to stick with one convention throughout the article and simply have a section on the different notational conventions that one finds in the literature. Presenting them all back to back in the definitions sections seems like it may be a lot to absorb at once. Thenub314 (talk) 18:19, 21 September 2008 (UTC)

I think the back-to-back section is very well done. If you want to pick one for the subsequent sections (which is a good idea), pick the simplest one, which is ordinary frequency. It's unitary and it avoids those ugly ${\sqrt {2\pi }}$ factors.

--Bob K (talk) 21:39, 21 September 2008 (UTC)

This may be missing the point. I suspect that the disgruntled editors User:Feraudyh, David_Colver and others did not study maths at school, or perhaps have not finished school yet and consequently are unfamiliar with integral calculus. There was a comment somewhere (I can't find it; perhaps it is in the archive), that the article ought to be comprehensible to a 16 year old. I reckon even an 18 old would have to work hard to read this, as there is no motivation, no mental picture, only algebra. What's wanted is not a pruning of useful information, but the addition (near the start) of some explanation in words and perhaps pictures. It needs to say that the point is to express an arbitrary signal as the sum of an infinite number of sinusoids of all possible frequencies, and to reverse the process ...etc. Notice that the section on the Plancherel theorem, mentions fancy stuff like Hilbert spaces, isometric autimorphisms and Pontygrain duality. What it doesn't say is that the power in a signal had better come out the same whether it's calculated in the time-domain or the frequency domain - that's the sort of point that might help the reader get an intuitive grasp. --catslash (talk) 23:38, 21 September 2008 (UTC)

I actually wasn't aware of these two editors. I am bad about reading old threads. The comment from the tables was made by Stca74 a bit more then a year ago. The comments about lacking inline citation were by a Wikipedia admin. Your right that the comments about lacking clarity can best be handled by down some down to earth explanation and good pictures certainly don't hurt, and I'll give it a shot as I can. I also think clarity can be improved on a smaller scale by better organizing the information we already have. I didn't really want to trim useful information. The only section I am interested in possibly trimming is the section titled "Generalizations", but I am not very adamant about that. Thenub314 (talk) 08:11, 22 September 2008 (UTC)

I hope the following revision improves the article. I have tried to add pictures, explanation, citations, and improve structure. I am sure there are a few mistakes, but I think I could use a few more pairs of eyes.

After having read the article I feel that the template is incorrect. It leads to an essay about avoiding vague statements. That does no seem to be a problem. Some editors seem to have complained about the math knowledge required or difficulty of reading the formulas. That is nearly the opposite problem to vague statements. Common language expressions avoiding math formulas may be easy to read but would then be very vague.Ht686rg90 (talk) 20:41, 3 October 2008 (UTC)

signal processing notation

Now that we have eradicated all semblance of typical signal processing notation, i.e. $x(t),\,$  $X(f),\,$  and $\int _{-\infty }^{\infty },\,$  (instead of $\int _{\mathbb {R} ^{n}}\,$), do we just stop here? It seems to me that an encyclopedia should try to represent the signal-processing perspective, because that is why a lot of readers will come here. Do we now create article Fourier transform (signal processing)?

--Bob K (talk) 19:50, 8 October 2008 (UTC)

If it is just notational conventions, then why not create a subsection for signal-processing notation in the "Other conventions" section?Ht686rg90 (talk) 22:27, 8 October 2008 (UTC)

That might not be what other signal processing editors want. So I'm asking. It's not what I want, but if I'm the only one who cares, then heck with it.

--Bob K (talk) 01:16, 9 October 2008 (UTC)

Speaking as an electrical engineer who deals occasionally with signal processing, this page reads completely as though it were written by a mathematician and not an engineer, as it only barely mentions the conventional signal processing notation, and spends most of its time on what seems (to me) like highly technical math that few engineers are interested in. It could probably be improved for the engineering audience, do any other engineers share this opinion? How much of the audience for this article is math vs. engineering? Mattski (talk) 23:41, 20 January 2010 (UTC)

Could you clarify the conventional signal processing notation? - if you mean using the symbols t and f rather than x and ξ then I would support that on the grounds that time/frequency-domain correspondence is the area of application where most people would first encounter Fourier transforms. I can't see very much fancy mathematics on the page though (also speaking as an electrical engineer who deals occasionally with signal processing). --catslash (talk) 10:22, 21 January 2010 (UTC)

For me the issue is not so much that I prefer one notation over another, but that we should use something that is consistent throughout the article. The convention that the article uses is quite well-established in the literature. Claims that the article should be rewritten so that electrical engineers and signal processors can read it are unconvincing. There are classic engineering-oriented books that use similar conventions to the article: Bracewell's classic textbook is perhaps the most prominent among these. Sławomir Biały (talk) 13:43, 21 January 2010 (UTC)

Being well established in the literature is not unique to this convention. Why was the original well established convention rewritten into this one? Just because somebody preferred it this way. But others prefer it the way it was. The best compromise I can think of is what I suggested 15 months ago... new article Fourier transform (signal processing).

--Bob K (talk) 21:23, 21 January 2010 (UTC)

Well, I haven't yet seen anyone give a clear alternative to the current convention. What, for instance, is the "signal processing" or "electrical engineering" notation for the integral

$\int _{\mathbb {R} ^{n}}f(x)e^{-ix\cdot \xi }\,dx?$

What seems likely is that someone changed conventions because there may be no reasonably well-established way to denote this in signal processing, although it is well-known in other areas like quantum mechanics and probability theory. A separate article does seem like one possibility, but some effort should be made to include content there that would justify a new article (not just a copy-paste of the one-dimensional sections in this article with a different notation). Sławomir Biały (talk) 23:40, 21 January 2010 (UTC)

The way that I would write the transform is as follows

$\int _{-\infty }^{\infty }f(x)e^{-i2\pi fx}\,dx$. Mattski (talk) 09:19, 22 January 2010 (UTC)

But that's just the one-dimensional transform. Sławomir Biały (talk) 11:54, 22 January 2010 (UTC)

The multi-dimensional form is no longer an issue (for me), because the article no longer leads with it (as it did 15 months ago). So I don't think the current objections are about that. But in a "signal processing" article, with this one as backup, I would:

• avoid the multi-dimensional form
• use f for ordinary frequency (Hz), not the function name
• use s (as in signal) for the function name, to avoid confusion with the x in this article
• use t for the variable in the domain being transformed to the f domain
• use S (as in Spectrum) for the function name in the f domain.
• When another function name is needed, I would use r and R.

--Bob K (talk) 21:38, 22 January 2010 (UTC)

What the L

This bit is unclear to me: If ƒ is a function for [−L/2, L/2], then for any T ≥ L we may.... There doesn't seem to be any other reference to the quantity L; is it needed? --catslash (talk) 12:48, 9 October 2008 (UTC)

Your correct, that sentence makes no sense. I have tried to fix it, it simply should have said that ƒ is a function supported in [−L/2, L/2]. It is only implicitly used in the equality ${\hat {f}}(n/T)=c_{n}=\ldots$.Thenub314 (talk) 13:08, 9 October 2008 (UTC)

Simple Analogy

Many people are familiar with the rising and falling "waterfall" bar displays on a stereo, showing a real-time view of the music spectrum, from bass (low-frequency) on the left, to treble (high-frequency) on the right.

Comment. The FT of a piece of music is fixed for all time, and does not change, the waterfall bars are more related to a windowed Fourier transform. Thenub314 (talk) 19:01, 10 October 2008 (UTC)

However, the audio signal being amplified and fed to the speakers is simply the intensity ("loudness") of the music at any given instant - older audio equipment had just a needle display ("VU meter") that represented the sound volume -- and the actual signal -- at any given time.

So if the tuba hits a note at the same instant as the piccolo, there's only one value of the signal at that moment, which is what the needle shows -- and the speaker plays! How does the display "know" to make both a left bar and a right bar - but not much in between - rise sharply?

The Fourier transform can be thought of as a "black box" which, given the input of that stream of single values representing sound volume over time, produces a stream of many values, simultaneously: the level of each frequency present at that moment in time. So - even though the original signal had the input from the tuba and piccolo added together into a single value, the Fourier transform can tease apart the low notes from the high notes, much as humans can easily zero in on a single conversation at a noisy cocktail party.

The reason it's called a transform is that it transforms the information contained in a single signal value of intensity, which varies over time (called a "time-domain representation") into a set of values representing the distribution of frequencies at each moment (hence: "frequency-domain representation").

I put this here for the moment because I didn't want to simply RV A_Doon's contribution, but I also didn't want to leave something that confused the Fourier transform with a time-frequency transform. Thenub314 (talk) 18:54, 10 October 2008 (UTC)

TeX v HTML

At the risk of encouraging conflict, may I point out that the WP:MOSMATH guideline on the question of TeX versus HTML appears to be the unilateral decision of User:Oleg_Alexandrov. See Help:Displaying a formula for a more balanced assessment. --catslash (talk) 00:13, 17 October 2008 (UTC)

Well, I don't see any particular conflict at the moment, it sounds like a good time to try to reconsider WP:MOSMATH. Maybe this should be discussed on the project page or on the discussion page of WP:MOSMATH. Thenub314 (talk) 06:49, 17 October 2008 (UTC)

Convolution Integral

109 Fourier Transform of $f(x)g(x)\ \neq$ $({\hat {f}}*{\hat {g}})(\omega ) \over {\sqrt {2\pi }}\,$

but

109 $f(x)g(x)\ =$ $({\hat {f}}*{\hat {g}})(\omega ){\sqrt {2\pi }}\,$

If we put in all the values the ${\sqrt {2\pi }}$ comes in the numberator not denominator. Please check and let me know

Wilkn (talk) 21:54, 21 October 2008 (UTC)Wilkn@yahoo.com

I think the entry is correct, unless I made a careless error. We can calculate the Fourier transform of $f(x)g(x)$ by:

${\mathcal {F}}(f\cdot g)(\omega )={\frac {1}{\sqrt {2\pi }}}\int f(x)g(x)e^{-i\omega x}\,dx=$
${\frac {1}{\sqrt {2\pi }}}\int f(x){\frac {1}{\sqrt {2\pi }}}\int {\hat {g}}(\eta )e^{i\eta x}\,d\eta e^{-i\omega x}\,dx=$
${\frac {1}{2\pi }}\iint f(x){\hat {g}}(\eta )e^{-ix(\omega -\eta )}\,dx\,d\eta =$
${\frac {1}{\sqrt {2\pi }}}\int {\hat {g}}(\eta ){\frac {1}{\sqrt {2\pi }}}\int f(x)e^{-ix(\omega -\eta )}\,dx\,d\eta =$
${\frac {1}{\sqrt {2\pi }}}\int {\hat {g}}(\eta ){\hat {f}}(\omega -\eta )\,d\eta ={\frac {1}{\sqrt {2\pi }}}({\hat {f}}*{\hat {g}})(\omega )$

Let me know if this answers your question. Thenub314 (talk) 17:55, 22 October 2008 (UTC)

Thank you for your response. I was wrong sorry about that. I am new to wikipedia so please use your best judgement to delete the page or not. My research is centered on FT, would it be possible for you to give me your contact information ? Wilkn (talk) 00:58, 23 October 2008 (UTC)Wilkn—Preceding unsigned comment added by Wilkn (talk • contribs) 19:21, 22 October 2008 (UTC)

No problem, we all make mistakes. Wikipedia already gives you my contact info, if you click on my name from any of my posts, it will take you to my user page. That is like any other page, there is a discussion page where you can leave me messages. There is also a "toolbox" on the left hand side of the page, you can click on the link "Email this user" to send me an email. Thenub314 (talk) 08:24, 23 October 2008 (UTC)

Ball multiplier

(in the section Multi-dimensional version)

I have tried to clarify the statement about the fact that the indicator function of the ball is not a multiplier for L^p, by introducing f_R defined by

$f_{R}(x)=\int _{|\xi |<R}{\hat {f}}(\xi )\mathrm {e} ^{2\pi ix\cdot \xi }\,\mathrm {d} \xi ,\quad x\in \mathbf {R} ^{n},$

but it is not satisfactory: the real thing is that for some f ∈ L^p, this function f_R is simply not in L^p, so the question of f_R converging to f is pointless. I don't exactly see what the original intention was. Does anybody know? Bdmy (talk) 14:31, 29 November 2008 (UTC)

Well I should, as I introduced it in this edit. Without using the multiplier theorem I thought it was not clear if f_R was in L^p? (at least I can't think of how to see it immediately) Somehow I was trying to give some historical context for why one would be interested in the boundedness of the multiplier for the ball. I choose the Duoandikoetxea reference because on pg 17 he introduces the problem by asking if f_R converges to f in L^p, and explains that this is equivalent to having that the multiplier for the characteristic function of a ball of radius R needs to be bounded in L^p with norm independent of R. So one way to interpret Fefferman's result is that you can't hope to recover a function from its Fourier transform by taking a limit of the truncated integrals. Now this being said, I am being a little silly, because as Fefferman points out in the intro to his paper, it is not hard to limit the problem to an interval around 2 by some simple calculations. Thenub314 (talk) 20:01, 19 December 2008 (UTC)

compaction (in uncertainty princple section)

I didn't even realize it was an english word! (And I looked it up, and it is) In some ways the use of this word is brilliant, but I feel the word is fairly uncommon. Is there any objection to me trying to rework this sentence? Thenub314 (talk) 20:19, 23 December 2008 (UTC)

Of course I don't have an opinion on this. What about the small changes in maths? Bdmy (talk) 20:27, 23 December 2008 (UTC)

The only mathematical change I noticed is that we now require ƒ to be integrable. I don't think this is necessary for the theorem, but technically speaking this section is before the section where we describe how to extend the Fourier transform to L², so it seems like a fine addition to require it. Is that the reason you inserted the condition? Thenub314 (talk) 20:59, 23 December 2008 (UTC)

Of course. I also slightly rephrased the equality case: if f is given, you don't need to give the Fourier transform with a second constant C_2, you just have to compute... Also I put the equality case right after the first relation with D_0, because if you allow translations like in the lines after, you have many other equalities. I put a square to f in definition of D_0. OK? Bdmy (talk) 21:20, 23 December 2008 (UTC)

Yes. It all looks great.

Notation angular frequency omega versus nu

In the table of Fourier pairs, the third and fourth column refer to conventions of the Fourier transform that both employ angular frrequency but differ by their normalization constant. I do not really understand, why different symbold (omega) and (nu) are used to denote the angular frequency. Wouldnt it be clearer to choose consistent notation here? —Preceding unsigned comment added by 91.66.240.116 (talk) 11:36, 29 December 2008 (UTC)

Yes, nu usually refers to the ordinary frequency 2πω, using it here is confusing. -- Army1987 – Deeds, not words. 12:41, 29 December 2008 (UTC)

Notation: Isn't epsilon a strange choice for a natural number?

I am used to epsilon denoting infinitesimally small real values in "epsilon-delta proofs" and the like, and was therefore a little confused by the formulas in section "Definition". If this isn't standard notation (and to me it seems like it isn't: neither of the standard textbooks I checked use it) I would strongly suggest we change it Thomas Tvileren (talk) 10:35, 8 January 2009 (UTC)

It's a xi rather than an epsilon. I would prefer f (and t rather than x), but see Notation discussion above. Picking up a couple of books within reach, I see that both use f and t; one uses f(t) and F(f) (using f twice), the other (Numerical Recipes) uses h(t) and H(f) (but that looks a bit like the Heaviside step function). --catslash (talk) 11:12, 8 January 2009 (UTC)

Do you mean the letter that denotes the frequency variable? In which case it is not an not the greek epsilon ($\epsilon$) but the greek letter xi ($\xi$). It is one fairly standard notation, for example Stein & Shakarchi, Pinsky and a few other books in the references section follow this notation. Other common notations include using $\omega$ and $f$ (in which case you need to use a different letter for your function.) Denoting the frequency variable by ƒ is especially common in engineering contexts.

Also, the frequency variable doesn't need to be a natural number, it can be any real number. In the subject of Fourier series, the variables are often natural numbers (or something times a natural number) and there you often see n, m, k, etc. which are more typical choices for natural numbers. Hope that helps. Thenub314 (talk) 11:21, 8 January 2009 (UTC)

Oops, it said real number, right… I thought it said natural for some reason. And of course it is a xi and not an epsilon… Well, that was embarassing. Thanks anyway. Thomas Tvileren (talk) 12:34, 8 January 2009 (UTC)

Caption of sync function is wrong!

The caption of sync function is wrong! It says that it is bounded and continuos but not Lebesgue integrable. Actually this is not true! If it is bounded and continuous is even Riemann integrable, and so it is Lebesgue integrable! Probably author meant that there is no analytic expression (and if I remember well analytic function is sin(x)/x whose integral should have such an expression), but this doesn't mean that the function is not integrable! —Preceding unsigned comment added by 85.134.152.225 (talk) 23:38, 26 October 2009 (UTC)

No. The sinc function has an improper Riemann integral on (−∞,∞) but is definitely not Lebesgue integrable. A measurable function is Lebesgue integrable if and only if its absolute value has a finite Lebesgue integral. But the Lebesgue integral of |sin x/x| is infinite, by comparison with the harmonic series. 71.182.240.148 (talk) 10:41, 27 October 2009 (UTC)

Removed entry from table

I removed the following entry:

	Function	Fourier transform unitary, ordinary frequency	Fourier transform unitary, angular frequency	Fourier transform non-unitary, angular frequency	Remarks
310	${\frac {1}{x^{n}}}$	$-i\pi {\frac {(-2\pi i\xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi )$	$-i{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )$	$-i\pi {\frac {(-i\nu )^{n-1}}{(n-1)!}}\operatorname {sgn}(\nu )$	Generalization of rule 309.

The function 1/xⁿ fails to be a distribution. The correct statement probably holds for the tempered distribution

$S[f]=\int _{-\infty }^{\infty }\left[x^{-n}\left(f(x)-f(0)-f'(0)x-\cdots -f^{n-1}(0)x^{n-1}/(n-1)!\right)\right]\,dx.$

I'm not sure if this belongs in the article, though. At any rate, it needs to be verified. Sławomir Biały (talk) 22:25, 18 November 2009 (UTC)

Another way to go about it is to define the distribution by

${\frac {1}{x^{n}}}:={\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n-1}}{dx^{n-1}}}p.v.{\frac {1}{x}}.$

Then the above follows easily from earlier properties. However, 1/xⁿ doesn't usually refer to the tempered distribution obtained by distributionally differentiating the principal value of 1/x. Sławomir Biały (talk) 12:27, 19 November 2009 (UTC)

For #310, I see you added the range from 0 to 1. What is the situation with 1/x^a, where a is fractional (say, 5/3 or 7/3) and greater than 1? Any clues?

Also, rather than remove it entirely, I recommend that you restore the original 310 for integer 1/x^n, somewhere else in the tables, even if that is not suitable for the distributions section. —Preceding unsigned comment added by 74.70.96.16 (talk) 00:25, 21 November 2009 (UTC)

Well, the function x⁻ⁿ for n ≥ 1 does not define a distribution, so there isn't even a question of existence of the Fourier transform in any usual sense of the term. This includes |x|^−a for a > 1. One game that is usually played is to regularize this function by subtracting off a high enough jet, but then the object denoted by |x|^−a isn't what one usually means in the normal sense. If the content is restored in the article, the notation should at least be explained, and a table seems to be a very unsuitable way to do that. Sławomir Biały (talk) 02:31, 21 November 2009 (UTC)

I have put it back in, with this caveat. Sławomir Biały (talk) 16:20, 21 November 2009 (UTC)

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