Talk:Dirac delta function/Archive 2

Archive 1

Archive 2

The frequency domain

I would like to see some specifics on the Dirac Delta in the frequency domain. The mere substitution of "x" by "f" in the current definition formulae would be misleading. This clarification is important since some other articles depend on it. For instance, the statement The Fourier transform of a Dirac comb is also a Dirac comb here.MaskedAce (talk) 03:31, 30 September 2012 (UTC)

The Fourier transform is already covered in detail in the relevant section. It's not clear what you specifically believe is inadequate in the present treatment. Sławomir Biały (talk) 21:24, 30 September 2012 (UTC)

In the equation that follows the expression "The inverse Fourier transform of the tempered distribution f(ξ) = 1 is the delta function. Formally, this is expressed": When I naively try to do some calculus I obtain $\int _{f=-\infty }^{\infty }e^{2\pi ift}df={\biggl [}{\frac {e^{2\pi ift}}{2\pi it}}{\biggr ]}_{f=-\infty }^{\infty }={\biggl [}{\frac {\cos {2\pi ft}+i\sin {2\pi ft}}{2\pi it}}{\biggr ]}_{f=-\infty }^{\infty }={\biggl [}{\frac {\sin {2\pi ft}}{2\pi t}}{\biggr ]}_{f=-\infty }^{\infty }=\lim _{f\to \infty }{\frac {2\sin {2\pi ft}}{2\pi t}}={\Bigg \{}{\begin{matrix}\lim _{f\to \infty }2f&{\textrm {as}}&t\to 0\\\in [-1,1]/(\pi t)&{\textrm {if}}&t\neq 0\end{matrix}}$

Is the Dirac pulse amplitude defined to be $2\infty$ so to speak? My question is if there is a reason that the definition doesn't contain a compensating factor of 1/2. I simply just don't understand and I believe that the article would benefit much if someone had any idea on how to explain the above in a pedagogical sense. Geo39geo (talk) 13:23, 12 February 2018 (UTC)

The value of the function $\textstyle {\frac {2\sin {2\pi ft}}{2\pi t}}$ at 0 is roughly the height of the hump; but did you think about the width of the hump? The integral of this function (over the whole real line) is equal to 1 (rather than 2), since $\textstyle \int _{0}^{\infty }{\frac {\sin at}{t}}\,dt={\frac {\pi }{2}}$ for all $a>0.$ Boris Tsirelson (talk) 18:53, 12 February 2018 (UTC)

Aha ok, I get it! This makes sense. Thank you very much for the succinct and fast reply. Geo39geo (talk) 22:10, 13 February 2018 (UTC)

Is Dirac delta function a function?

Hi all. Because Dirac delta is a distribution, it automatically means that it is a real-valued linear functional, right? But being a real-valued linear functional makes it a real-valued linear map. The latter makes Dirac delta a real-valued map. And a real-valued map is a function. So long story short, Dirac delta is a function. However, the article explicitly says it's not. It's like saying that a field is not a set. Could anyone fix it please? Thank you. Konstantin Pavlovskii (talk) 12:21, 9 August 2017 (UTC)

Hello D.Lazard,

I've decided to talk to you because apparently you reverted my recent contribution. I am extremely new to the Wikipedia and I haven't had enough time yet to get used to how things work here.

You wrote in the edit summary of your revert that "The article was not wrong" and that "the edit summary is mathematically wrong". I assume that the latter refers to the edit summary of my contribution. I think it would have been a good idea if you had left a note on the article's Talk page explaining the reason why you think my edit summary "is mathematically wrong" before actually reverting my contribution, exactly as Wikipedia recommends.

Because the contribution is already reverted by you, I'd still -- even more so -- appreciate a note from you on the article's Talk page explaining the reason why you think my edit summary is "mathematically wrong". This way you'd give me an opportunity to respond to your concern, so we could reach an editing consensus.

Thank you, I hope to talk to you soon. Konstantin Pavlovskii (talk) 10:16, 10 August 2017 (UTC)

To editor Konstantin.pavlovskii: The Dirac delta function is not a function, as clearly stated in the article about it. This fact is sufficiently known and accepted by all mathematicians, for not needing any discussion here. Thus, this is your assertion in the edit summary that is wrong, not the result of your edit. I apologise for not having done the distinction in my own edit summary. Nevertheless the revert was justified, because it is always useful to recall that the Dirac function is not a function, despite its name. As the point is not the distinction between a function and a distribution, I have edited the sentence for less emphasizing on this point. D.Lazard (talk) 12:42, 10 August 2017 (UTC)

To editor D.Lazard: Hello D.Lazard, what makes you think that a Dirac delta function that is a distribution, is not at the same time a function? Any distribution is a linear functional, so any distribution is a real-valued map. By definition any real-valued map is a function. I've already pointed it out in the Dirac delta function talk page the corresponding article that you refer to has been recently edited by the community to remove the incrorrect statement that the Dirac delta function is not a function. You could have a look at the new Dirac delta article yourself. I'd appreciate if you undid the harmful reverts of my contributions, please. And could please discuss my contributions on the corresponding talk pages first before reverting them. And thanks for that prompt reponse. Konstantin Pavlovskii (talk) 13:34, 10 August 2017 (UTC)

I think Konstantin's perspective is reasonable. I don't claim to fully agree with it, but I'm inclined to tone down the "not a function" aspect at least. Russian mathematicians even call it a "generalized function" ;-) I think we should clarify both the sense in which it is and is not a function, rather than relying on possibly hidden meanings of the word "function". Sławomir Biały (talk) 13:43, 10 August 2017 (UTC)

A distribution is a linear functional, that is, a map from a space of functions to a space of functions to the reals. Thus Dirac delta function is not a real-valued function, although it may be considered as a "function-valued" function. As, for everybody, a function, without further specification, means a real- or complex-valued function, you are wrong by saying that Dirac delta function is a function. D.Lazard (talk) 13:10, 10 August 2017 (UTC)

D.Lazarad, a linear functional is always a map from a vector space to its underlying field. In this particular case, it is a map from a vector space of real-valued functions of a real variable. The latter is a vector space whose underlying field is reals. Therefore the distribution (linear functional) used to define the Dirac delta function is a real-valued linear functional. And once again, a distribution has to map a vector space to its own underlying field not any other field. Therefore the Dirac delta function is a distribution, which is a real-valued functional, and -- you already know -- it automatically makes Dirac delta function a real-valued map (e.g., a real-valued function). Konstantin Pavlovskii (talk) 18:13, 10 August 2017 (UTC)

OK, Dirac delta function is a map from a unspecified vector space of infinite dimension to the reals. Moreover, the domain of this map depends on the context (either the functions, in the usual sense, that are defined for x = 0, or the functions that are defined everywhere, or the continuous functions, etc). Thus a correct formulation would be: "although they are generally considered as a generalization of real-valued functions of one variable, distributions and Dirac delta function are not such functions". In all the articles you have edited, Dirac delta function appear in contexts (for example Fourier transform), which are generally devoted to real-valued functions of a real or complex variable. In such a context, "function" means function of a real or complex variable. It is thus important to recall to non-experts that Dirac delta function is not a function in this usual sense. Thus I would agree to replace "is not a function" by something like "is not a function in the usual sense". But I disagree to remove the caveats. D.Lazard (talk) 19:00, 10 August 2017 (UTC)

First of all I did not understand why you brought up the question of dimensionality of a vector space the DDF maps to real and what follows it -- all these other things you have there up to ". Thus a". I certainly miss the reason you wrote them. What's helpful is to make sure that the users, should they have any doubt about what the Dirac delta function really is, could in one click reach a page where the fact that the DDF does not take any real value anywhere on the real line -- is expicitly written and I would strongly recommend against using word patterns that support the weird idea that the DDF is a misnomer, for example: "DDF is not a function, in <certain sense>". An engineer perceives this as you if you were saying that DDF is only a function to a certain extent (and not to the full extent). Sort of like a peanut being not exactly a nut. This pattern will continue to generate and suppport that "misnomer" that I even saw on someone's personal page here. And also could I ask you D.Lazard again to revert the reverts of my contributions that you reverted and bring any concerns you have about the reader confusing DDF with a function of a real variable to the corresponding talk pages before reverting the contributions. I will check that there are no vague statements in the articles that would make domain of the DDF appear real. Finally, as for your text suggestion, "generally considered" sounds like a pretty toxic term to me, but as I am a new user, let me focus on the fact that your text suggestion is obviously not true in the first place: "although they are <..> considered <...> generalization of real-valued functions of one variable <...> Dirac delta function are not such functions". Well, DDF is a real valued function of an exactly one variable -- of a function. It's a linear functional -- hence the name. You might want to consider consulting the article in question:namely, the definitions section. And as a reminder, please, revert your reverts of my contributions related to the DDF and bring your concerns to the corresp. Talk pages. Thank you. Konstantin Pavlovskii (talk) 20:43, 10 August 2017 (UTC)

This is just a note that I replied at User talk:D.Lazard's user talk page, but it's fine by me if anyone to refactor this discussion so it's all in one place if they want. Sławomir Biały (talk) 13:44, 10 August 2017 (UTC)

Done. Konstantin Pavlovskii (talk) 09:19, 13 August 2017 (UTC)

Just a note. Anyone first rigorously learning the notion of a function in a course on set theory will be very open-minded as to what constitutes a function. So it is good that the article is clear on what is its notion of "function". YohanN7 (talk) 14:03, 10 August 2017 (UTC)

Every article using Dirac delta function or talking about it supposes that the reader has some knowledge of calculus. The the reference to set theory is not relevant here. If one does not clearly state that Dirac delta function, is not really a function, a reader of these articles will surely think that it is a real-valued function of a real variable (the basis of calculus), which it is not. Thus not distinguishing Dirac delta function from usual functions is misleading and confusing. Thus even if, in some cases, it may make sense to consider Dirac delta function as a function, this should be avoided in WP. Moreover, any edit that would imply that Dirac delta is a function should include a reliable source supporting this. D.Lazard (talk) 14:53, 10 August 2017 (UTC)

It is rather strange to read that the reference to the set theory definition is "irrelevant". It is akin to saying that vectors that aren't arrows in ℝ³ aren't vectors (because they aren't encountered in Calculus 101). Note that all I was saying is that it is good that the article now is trying to be precise on what it means by "function". YohanN7 (talk) 08:49, 11 August 2017 (UTC)

D.Lazard, when the article says that the Diract delta function is a function it does at the same time explicitly say that it is not a function of a real variable, and I strongly suggest to add another clarification: "therefore it is not defined for any real argument". There is no source of confusion. However, just saying that "Dirac delta function is not a function" is not true. As simple as that. Saying that it's not a function of a real variable is true, but it begs a question: "Eeeh, but a function of what kind of a variable then? Or is it not a function in principle?". There is no way around making a statement about the functionhood of the Dirac delta (in terms of the concept of a function -- weird to even mention that). And the principal answer to this question can't be "no it's not a function" because it's not true. So to me it looks like there's no other option but to explain straightforwardly to the reader whatever is already in the article plus that additional clarification that I suggested in one form or another. And (perhaps I'm getting too emotional here) there has never been an issue of a "Dirac delta function" being a "misnomer", this issue was born on the pages of the Wikipedia and traces back to at least 2002 when the community for some reason let that (mind slip?) happen that suddenly gave birth to the problem of a "misnomer". It's a pretty well-known fact that any distribution is a function and any measure is a function. Why whould anyone even expect of something defined to be a distribution (or a measure) to suddenly not be a function? This is what would sound wild to anyone (and is not true, above all).Konstantin Pavlovskii (talk) 18:50, 10 August 2017 (UTC)

The Dirac delta function was not defined to be a distribution, though - it was defined to be a real-valued function which is zero everywhere except the origin and integrates to 1 on the real line. Of course this function does not exist. This article has a challenge, therefore, that it is using the name for a non-existing object (the Delta function) to refer to an existing object (the Delta distribution). That is what the phrase "the Dirac delta function is not a function" is meant to suggest. Whether the Delta distribution is "really" a function, under the hood, is not the point. The ambivalence in this article between the Delta function and the Delta distribution has increased over time; older versions such as [1] had their own flaws but were more clear in the lede about the function nature of the Delta function. Also see my comment at [2]. — Carl (CBM · talk) 19:40, 10 August 2017 (UTC)

Hi Carl. I am not getting it, sorry, you say: <<using the name for a non-existing object>>. There is no such thing as the name of an non-existing object. It's math, objects don't appear and then pass away. Any of them either exists or doesn't in general, they are not parts of material world. When an object is non-existing, so are all its properties including the name, definition, etc. It sounds even more confusing to me as I read further: <<That is what the phrase "the Dirac delta function is not a function" is meant to suggest.>> Suggest what? You can't check whether or not an object is a function if it doesn't even exist. This statement doesn't make sense in the absense of the object that it's talking about. Could you please clarify the rest, I think I'm stuck. Thanks. Konstantin Pavlovskii (talk) 23:05, 10 August 2017 (UTC)

"The smallest odd multiple of 8" is a name for a nonexistent natural number. Here's a long list of nonexistent objects at MathOverflow [3]. So there are many names in mathematics for nonexistent objects. The prototypical "Dirac delta function" is defined as a real valued function on the real line that is zero except at the origin and integrates to 1 on the real line. We can prove that no such real valued function exists, of course. When most people write "the Dirac delta function is not a function", they are simply alluding to this fact, to point out that they are using the term "Dirac delta function" to mean something else. For example, this answer at MathOverflow [4] shows the usage: "δ isn't a function - yet sometimes it behaves like one." — Carl (CBM · talk) 01:21, 11 August 2017 (UTC)

Oh, I see it was a joke. Jesus I thought you were serious :) Of course you can make up whatever name you want and pretend that it's a name for something that's proven to not exist under ZFC, say. But the latter statement (that it's its name) is... could you tell me (a stupid physicist) what's its status in ZFC? As for the Dirac delta function, I think I've said enough, and the historical reasons for the cognitive slip that you think made it grow into a <<fake "misnomer">>... This slip now looks way more dissapointing to me than before. It so hard to avoid those misfortunate congnitive malfunctions big and small... they grow through the years and a couple decades later their falsehood just becomes a part of you. And one day you realize you're not quite the person you've been trying to be, not at all. Konstantin Pavlovskii (talk) 02:40, 11 August 2017 (UTC)

Konstantin Pavlovskii takes formal definitions too seriously. They often mix an idea and its implementation. Often an idea has many implementations. For instance, some sources define a natural number to be a finite ordinal in such a way that it is true that $5\in 6,$ but it is evil to say so ("evil" according to nLab, not to me). See "Equivalent definitions of mathematical structures". Likewise, it is evil to say that delta function is a distribution and therefore a function. After all, in ZFC everything is a set; but it is evil to say "5 is a set". This happens outside math, too; it would be evil to say "the fifth bit of this song is 0" even though this could be true according to a given mp3 file. It is not serious, to bee too much serious; it is rather evil. "Some subjects are so serious that one can only joke about them." (Niels Bohr) Boris Tsirelson (talk) 14:03, 11 August 2017 (UTC)

Hi Boris. The article clearly identifies a particular class of implementations of the Dirac delta function in the very beginning by saying that <<the Dirac delta function, or δ function, is a generalized function, or distribution on the real number line...>>. Seeing this why don't you say that the article itself already "mixes an idea and its implementation". And then the same article goes on to say <<...[is a distribution] on the real number line that is zero everywhere except at zero>>. Jesus one of two things: either this functional operates on a trivial vector space on R (i.e. the R itself -- just {0} doesn't work because in this case it wouldn't talk about functional values "everywhere except zero") or the author doesn't know what they are talking about. Perhaps the latter since the Dirac delta function defined in the "Definitions" section doesn't take any values on the real number line whatsoever. So, a quick summary:

1. The article as of now does mix an idea and its implementation in the top section.

2. The article as of now does talk about the values that "the Dirac delta function", a "distribution on the real number line" takes for real arguments (it doesn't take them in any of the implementations of the Dirac delta described in the article).

3. Then, a few lines below the article states that the values it was just talking about (see #2) don't exist by saying that <<The Dirac delta function is not a function <...> of real variables>>

If the article picks the "distribution" implementation in the first sentence, then what's wrong with sticking to it until the end of the paragraph at least? And let's fix these statements in the top section that contradict each other. Otherwise it's just not a sound article: it discredits the Wikipedia (the best case scenario) or fools the reader into believing them all at once (the worst case scenario). When I suggest how to fix them, my contribution gets reverted, even though it explains things straightforwardly to the reader sticking to the implementation introduced in the beginning of the section. Even saying <<in the usual sense of real variables>> is ambigous. What does it mean? If someone asked what's a "function in the usual sense of real varibles?" I would answer "a real-valued function of a simply ordered set of real variables". Does anyone disagree? Is it what's actually meant by this sentence? Konstantin Pavlovskii (talk) 04:23, 12 August 2017 (UTC)

Hi Konstantin. Yes, the article does mix an idea and its implementation, since this is usual in mathematical literature. Yes, another article "Distribution (mathematics)" treats distributions as linear functionals. But it never calls these functionals "functions" (in spite of the uncontroversial fact that they are functions); this is also usual in mathematical literature. Unless otherwise stated, a reader usually interpret "function" (in this context) as a function on R or Rⁿ.

Now think about a text that defines a natural number to be a finite ordinal in such a way that it is true that in fact 5 belongs to 6. Unless devoted to implementation details, such text never says that 5 belongs to 6 (in spite of the uncontroversial fact that it belongs).

A reader of Wikipedia, being often not a mathematician, usually is not interested in implementation details.

Think also about an equivalent definition of the space of distributions as the completion of the space of test functions w.r.t an appropriate topology. Now implementation of completion matters; usually you get that a distribution is an equivalence class of Cauchy sequences (or filters, etc). Now a distribution is not a function!

Well, frankly I am not sure it is not; it depends on very subtle implementation details; apriori it may happen that a single set from the ZFC universe is both a function (from something to something) and an equivalence class (of something). But you surely feel that this is a stupid matter. Who bothers whether a finite sequence of bits can be a legitimate sound file in one operation system and a legitimate executable in another system? Well, maybe some expert in computer crime could... but surely not a typical user of a computer.

We do not hide implementation details from the reader, but we give them DUE WEIGHT, and not more. Boris Tsirelson (talk) 04:50, 12 August 2017 (UTC)

On your first paragraph: OK. You understand that "unless otherwise stated" bit at least. Because the question of the functionhood of the Dirac delta function is controversial enough (not in Russia though) to ignite fierce discussions and is a well-known public misconception, I suggest simply clearing up this question in the top section once and for all, at the same time clearly stating the nature of the DDF functionhood right in the same sentence or right next to it. If you prefer a "light version" of the same fix that's fine with me personally (I'd call this an editing consensus, although see the note below), then just stating that the DDF is not a function of a real variable is enough I think, it is exactly the change that was introduced right after I brought up this topic. I didn't have major concerns about the new way it was worded <<The DDF is not a function, <an addition about a real variable> >>. However, this light version is not extremely appealing to me as to a person who frequently has to clear the air in terms of whether the DDF is a function (fullstop). A first-year engineering student is very likely (in my experience) to interpret "The DDF is not a function, <blah-blah-blah> " as "The DDF is not a function, I understand this part, I don't care about what's after the comma". This is because students don't know much about the functions that are not functions of a real variable, so they take that little piece after the comma as an additional clarification of whatever is before the comma rather then a (very serious!) reduction of scope of the statement. That's why saying "The DDF is not a function of a real variable" is much better in my view.

About the ordinals, there's no wide misconception among the public (with serious consequences) on whether or not in the implementation they use 5 is an element of 6. It's not a practical matter, so mentioning this explicitly is not needed, certainly not in the top section of an article that talks about a set of natural numbers. But if in the impelementation you describe 5 is indeed an element of 6, then saying explictly that it's not is not an option, it would be like selling your soul to the devil. Because it's not true and you would be confusing the more advanced readers. So just not mentioning this fact (even though it might be true) is totally acceptable and is not confusing to anyone.

I don't insist on specifying the details to the reader. I would merely want the top section of the article to not contradict itself and to clear up the question of the DDF functionhood to those students/engineers/physicists who are exposed to the publicly accepted view of the DDF as of a misnomer.

About the alternative implementations of a distribution. The Wikipedia is not trying to write about everything in the world and certainly not at once. So, in my view, picking a particular implementation (of the DDF and of a distribution and of a natural number and ...) is neccessary in the quick description of the DDF in the top section. Unneccessary generalizations and "meta-language" of ideas is confusing enough even for experienced readers. I remember reading Canadian Criminal Code that has a list of definitions of what "a public nuisance" or "a public intoxication" (or smth like that I don't recall) mean in this particular section. It's not an option for an article in the Wikipedia to resemble a text of a legislation, it would scare off the readers. It is for a reason that ideas and implementations get mixed in the mathematical literature and in any literature. For example, talking about the Soviet past, a Russian text would say "the Communist party", however the implementations of a communist party are numerous. Speaking about a relatively complicated mathematical concept in the "meta-language" of ideas won't let you use the terminology appropriate to the common implementation of the concept.

So selecting a common implementation and sticking to it in the top section is a must. Whoever is interested in the alternative implementations are advanced enough to not get confused with a top section written like that, it does no harm to them, at the same time, less experienced readers benefit from it because this way it is more understandable and uses the terms that they are used to.

So the program that I suggest is:

a) (a must) fix the mutually contradicting and vague (like <<in the usual sense of real variables>>) statements in the top section while sticking to the common implementations of the mathematical concepts throughout the top section, then

b) clarify the issue of the functionhood of the DDF in the top section with straightforward and unambigious sentences that preferably don't use the "the DDF is not a function <comma>" word pattern, replacing it with "the DDF is not a function of a real variable". The latter being done with the sole intent of benefiting your average engineer or physicist who is not used to the fact that commas in math routinely and greatly reduce the scope of the statements. Konstantin Pavlovskii (talk) 06:46, 12 August 2017 (UTC)

I have edited the lead in order to avoid excessive formalism introduced by Konstantin.pavlovskii, and to keep the important functional property of Delta function. IMO, this new lead is much more informative for readers that have never heard of distributions, and, nevertheless, remains mathematically correct. D.Lazard (talk) 10:42, 12 August 2017 (UTC)

Excellent! Thank you Daniel and Sławomir. Now it looks like a decent top section to me. Konstantin Pavlovskii (talk) 16:09, 12 August 2017 (UTC)