Talk:Kronecker delta

Kronecker Delta as a sampling of the Dirac Delta

The article states, "It is important to note that the Kronecker delta is not the result of sampling the Dirac delta function." Isn't this incorrect? If we use the ideal lowpass to limit the bandwidth for sampling, then we will be sampling a sinc function at the middle peak and then zero-crossings. I won't change anything until I cook up a proof or find a ref to add here, or until someone (myself included) shows I'm wrong. Any thoughts? Herr Lip (talk) 19:50, 31 March 2011 (UTC)

I went ahead and put something in, albeit without a proof or ref, for now. I also changed "It is important to note that the Kronecker delta is not the result of sampling the Dirac delta function." to "It is important to note that the Kronecker delta is not the result of directly sampling the Dirac delta function." Adding the "directly" to clarify (what I assume is) the point of the original statement, and help compare to my addition. Herr Lip (talk) 01:16, 30 April 2011 (UTC)

I think what this statement means is that sampling a Dirac delta function at x=0 (for a sufficiently short sample "time" around x=0) must yield a value greater than 1, the expected discrete Kronecker delta value for [0,0]. The reason is that the value of a Dirac delta at exactly x=0 (with infinitely small sample time) must be infinite for the total integral over the number line to be 1. Does anyone agree? David Spector (talk) 22:01, 23 July 2013 (UTC)

what is the discrete convolution of sampled Dirac delta function with discrete sinc function? 1*delta(x)= delta(x), but what about 0*delta(x)=?, since 0*infinity is undetermined or undefined, so the lowpass filter idea is questionable. the idea indeed seems to implicitly take continuous convolution of sampled Dirac delta function with discrete sinc function first and then sample the convolution and get the Kronecker delta, that's no problem — Preceding unsigned comment added by 123.119.83.143 (talk)

Regarding the section "algebraic expression"

I agree that it may look like original research, but it is nonetheless a true algebraic representation, as long as it is used only on integers. I have my doubts about how useful it is, though, as all practical programming languages interprets boolean truth as 1 if converted to integer, which means that the Kronecker delta function can simply be represented as something like: int(i==j) —Preceding unsigned comment added by 77.40.128.194 (talk) 14:27, 6 December 2009 (UTC)

I've removed it because it is original research, is a horribly clunky fragile definition, and I'm not sure it lends any insight. To whoever is using IP addresses to revert this, please stop until you've discussed. Oli Filth^{(talk|contribs)} 13:29, 22 April 2010 (UTC)

Just to clarify the rationale for including the arithmetic expression for $\delta _{ij}$:

• Prunescu, Mihai; Sauras-Altuzarra, Lorenzo; Shunia, Joseph M. (2025), "A Minimal Substitution Basis for the Kalmar Elementary Functions", arXiv:2505.23787 [math.LO], show that elementary recursive functions are exactly those generated by addition, remainder, and base-2 exponentiation.
• The Kronecker delta is a standard example of an elementary recursive function.

Given these two published facts, writing down an explicit term using the allowed operations is a routine derivation, which WP:NOR explicitly permits. The formula does not introduce any new mathematical claims; it simply instantiates the published generating set, and its correctness follows from elementary modular arithmetic, which I provided.

The intention is not to propose a new theory, but to provide a concrete example illustrating the cited result within the context of elementary recursive functions. I hope this helps explain why the definition is not original research and why it is relevant to the section.

Marc Schroeder (talk) 15:13, 24 November 2025 (UTC)

Alas, that paper is from arXiv (not generally notable), has no citations (generally a very bad sign for notability), and is too new (WP:TOONEW) for Wikipedia. —Quantling (talk | contribs) 18:29, 24 November 2025 (UTC)

Prunescu, Sauras-Altuzarra and Shunia explicitly refer to the result in S. S. Marchenkov, Superpositions of Elementary Arithmetic Functions, Journal of Applied and Industrial Mathematics 1(3):351–360, 2007. ISSN 1990-4789. URL https://doi.org/10.1134/S1990478907030106. In particular, Marchenkov shows that

$$\langle x+y,\;x{\bmod {y}},\;x^{2},\;2^{x}\rangle ={\mathcal {E}}^{3}$$

So the algebraic expression you were concerned about is primarily grounded in Marchenkov’s 2007 result. The contribution of Prunescu et al. (2025) is that they removed the redundant $x^{2}$ as a requirement; it doesn’t replace the earlier source.

Given this, the concern that the citation relies on a “not generally notable source that is too new” doesn’t really apply. The key result is well-established in the earlier 2007 publication.

Could you please restore the section? I’m happy to add a clarifying note indicating that the algebraic definition is based on Marchenkov (2007).

I’d also appreciate it if we could discuss any further changes on the talk page before making substantial removals.

Thanks!

Marc Schroeder (talk) 20:47, 24 November 2025 (UTC)

Following the doi link, the paper indicates 9 citations, which is substantially better than 0, though still not generally indicative of the general audience that Wikipedia aims for. But that's not the only criterion that comes into play, and further discussion here could prove fruitful. So, yes, let's discuss this further on the talk page before any substantial insertions of new text to the article.

We do have a page about Elementary recursive functions, which is a page that is likely to be visited by readers who are more sophisticated mathematically than those who visit this page. Perhaps the way to go is to add the relationship between the Kronecker delta and Elementary recursive functions to the latter. That way, readers reading about something a little more esoteric will be able to connect it to something a little more common. Within, the present article we'd limit ourselves to single sentence with a link to that longer discussion. Perhaps, The Kronecker delta function is an example of an elementary recursive function. How does that sound? —Quantling (talk | contribs) 18:00, 25 November 2025 (UTC)

Agreed. Marc Schroeder (talk) 13:36, 10 January 2026 (UTC)

Agree with removal. Mathematics must be rigorous. Most programming languages are explicitly non-rigorous. David Spector (talk) 22:05, 23 July 2013 (UTC)

I see no consensus to go forward with this modification. In summary the Kronecker delta is much more common than the concept of an elementary recursive function, so any tie between them should be most in the article on the latter. A mention in the present article would be okay, but not so up front and central ... up front and central will confuse the reader with esoterics rather than tying the Kronecker delta to something they might already have intuition for. —Quantling (talk | contribs) 02:01, 11 January 2026 (UTC)

You wrote yourself:

Within, the present article we'd limit ourselves to single sentence with a link to that longer discussion. Perhaps, The Kronecker delta function is an example of an elementary recursive function. How does that sound?

That sounded good to me. Now it is a matter of consistency. Can **you** please add the single sentence that you suggested, or at least indicate at which spot in the article you meant to insert it. Don't worry, I will add it verbatim.

(PS: I am not sure that the computational complexity of a function is an esoteric branch of computer science.)

Marc Schroeder (talk) 03:20, 11 January 2026 (UTC)

I made an edit to the article to add this as a "notable property". What do you think? —Quantling (talk | contribs) 21:29, 11 January 2026 (UTC)

This is fine. TYSM.

Marc Schroeder (talk) 03:19, 12 January 2026 (UTC)

Unit Impulse

Isn't the section on the unit Impusle Misleading? Shouldn't the value at 0 for the impulse be '+inf', not '1'? (I know thats not actually correct either, but I'm saying its not right as is) --143.107.106.100 (talk) 22:50, 14 April 2011 (UTC)

There is no value for the Dirac delta at x=0, and it is not a well-behaved function. See my comment above under "Kronecker Delta as a sampling of the Dirac Delta". The proper delta is used in the proper context. David Spector (talk) 22:11, 23 July 2013 (UTC)

Conflicting relationship of the generalized Kronecker delta and Levi-Civita symbol in Gamma matrices

I notice that in this article, the relationship with the Levi-Civita symbol differs by a factor of n!. Here it is given as

$\delta _{\nu _{1}\dots \nu _{n}}^{\mu _{1}\dots \mu _{n}}=\varepsilon ^{\mu _{1}\dots \mu _{n}}\varepsilon _{\nu _{1}\dots \nu _{n}}$

whereas in Gamma matrices#The fifth gamma matrix, γ5 (open the "Proof" box to see it) it is given as

$\delta _{\mu \nu \varrho \sigma }^{\alpha \beta \gamma \delta }={\frac {1}{4!}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon _{\mu \nu \varrho \sigma }$.

This discrepancy is possibly a matter of definition of the generalized Kronecker delta, and should be clarified. — Quondum 04:52, 29 October 2012 (UTC)

Are you sure that the formula in the Gamma matrices article is based on a reliable source rather than derived from an earlier, internally inconsistent, version of this article? JRSpriggs (talk) 07:36, 29 October 2012 (UTC)

No, I am not suggesting that this article is at fault; I was simply pointing out a discrepancy between the two articles that I'd noticed, and hoping someone with more familiarity with it would know. The relevance to this article is that if there is a another accepted definition, this is where it should be noted. The only sources that I've found (and there are not many that I've gone through) support this article's version. The other article's version is "better" in a useful sense (it seems that it represents a projection (being antisymmetrization, an idempotent operation), and unlike this article's version, would mostly eliminate the factorial factors that abound). However, the criterion to be used is notability, not utility. — Quondum 09:14, 29 October 2012 (UTC)

Christopher Pope (Geometry and Group Theory, 2008, http://people.physics.tamu.edu/pope/geom-group.pdf) uses a version of the generalized Kronecker δ that differs with a factor of n! from the one in this article. See his equations (1.239) and (1.240). It seems my supervisor uses the same convention as Pope, and if it is a common definition it should be noted in this article. Would have saved me some trouble... — Preceding unsigned comment added by 129.16.200.109 (talk) 13:11, 21 August 2013 (UTC)

infinity

The definition of Dirac delta should not be that it is infinity at x=0, but just that its area is one. That is, it is a special infinity, or a limit value that tends to infinity. Gah4 (talk) 07:01, 20 February 2022 (UTC)