Talk:Distribution (mathematics)

Sobolev

Saaska, 27 Nov 2003 I thought it would be fair to include Sobolev here.

composition of a distribution with a differentiable injective function

Is it possible to define the composition of a distribution with a differentiable injective function? Formally, it should be like

$\left\langle T\circ f,\ \varphi \right\rangle =\left\langle T,\ {\frac {\varphi \circ f^{-1}}{f'\circ f^{-1}}}\right\rangle$

even if f is not injective, but the support of T does not include any critical point of f it should work (summing up for all the values of $f^{-1}$)

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David 18 Dec 2004

I think that:

If u is a distribution in D?(A) and T is a C^00(A) invertible function:

<u o T, g> =<u, g o T^(-1) |det J|>

where g is a test function and J is the Jacobian matrix of T^(-1).

yes, this is the same formula as above, but it may not be general enough

Typo?

This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense:

d/dx (S * T) = (d/dx S) * T + S * (d/dx T).

Is this a typo? Seems to me it should be

d/dx (S * T) = (d/dx S) * T = S * (d/dx T).

Josh Cherry 14:47, 18 Apr 2004 (UTC)

Don't think so. Charles Matthews 15:33, 18 Apr 2004 (UTC)

OK, help me out here. My reasoning is a follows:

• Differentiation corresponds to convolution with the derivative of the delta function. From this and the commutativity and associativity of convolution, my version seems to follow.
• Differentiation corresponds to multiplication by iω in the frequency domain. From this and the convolution theorem, the same result seems easily derived.
• For concreteness, let T be the δ function. Clearly d/dx(S * T) = d/dx S. Clearly (d/dx S) * T = d/dx S. And S * (d/dx T), the convolution of S with the derivative of the δ function, is also d/dx S.

So where have I gone wrong? Josh Cherry 16:10, 18 Apr 2004 (UTC)

I now think you have a point ... Charles Matthews 16:50, 18 Apr 2004 (UTC)

So, this was changed by an anonymous user on 31 January; should be changed back. Charles Matthews 18:06, 18 Apr 2004 (UTC)

OK, I've made the change. Josh Cherry 20:19, 18 Apr 2004 (UTC)

“It is abuse of language to assert that distributions are not functions”

The first paragraph says: “While a distribution, such the Dirac delta function is, indeed a function, there does not exist a function on the space of the real line that is equivalent to the Dirac delta function. Instead, distributions are constructed from alternative domains. Since the formal treatment of distributions is often considerably different from the typical functions of more elementary theories, it is a common abuse of language to assert that distributions are not functions.” This sounds highly dubious. Function is a map from numbers to numbers. Distribution is traditionally defined as a map from functions to numbers. They are linear functionals: if we view functions as vectors (eg. of a Hilbert space), distributions are covectors. So this first paragraph sounds very controversial to me. Michal Grňo (talk) 07:32, 3 December 2019 (UTC)

Distributions, a.k.a. generalised functions, are indeed not functions proper. The cited part of the introductory paragraph has been removed accordingly by me. — JivanP (talk) 06:13, 8 December 2019 (UTC)

The article is far too technical

The series of edits made by the same user since May 2020 has made this article overly technical, and no longer in the spirit of a Wikipedia article on the subject. To see thepoint, compare the Sep 22, 2020 version with the last version prior to the series of edits by the same editor, the April 20, 2020 version.

The April 2020 version opens with a Basic Idea section that illustrates the essence of the subject. The Sep 2020 version introduces a lot of notation and ancillary concepts, much of which does not seem necessary to introduce the definition of a test function.

I would strongly urge that the article be edited back to resemble the April 2020 version, and any advanced material being added after the the simplified introduction to the subject.

Undsoweiter (talk) 19:08, 23 September 2020 (UTC)

The article that you linked to contained false information such as the last sentence of this claim: " The elements of D(U) are the infinitely differentiable functions $\varphi$ : U → R with compact support – also known as bump functions. This is a real vector space. It can be given a topology by defining the limit of a sequence of elements of D(U)."

You can not define the topology using sequences and this is a non-trivial fact that has been proven. The above false statement should not appear in this article. The correct definition of the canonical LF-topology on the space of test functions is unfortunately technical.

Also, the old article was missing important information such as how to extend differential operators to distributions, which is arguably is one of the more important uses of distributions.

Distributions are unfortunately an innately technical topic. However, I am fine with simplifying the article but not at the expense of adding false-but-simple information or removing important information. Mgkrupa 22:26, 23 September 2020 (UTC)

My comment about technicality and style of the article does not mean I endorse the accuracy of every claim in the April version I refer to. If there are incorrect statements, they can be corrected without resorting to the drastic changes you have made to the article. By all means, fix errors. But this does not justify the excessively technical approach you have followed in your rewrite.

Undsoweiter (talk) 20:31, 30 September 2020 (UTC)

Over time, I will try to make this article less technical. However, we first need to establish what assumptions can and can not be made about this article and its "typical" reader. According to Wikipedia:Manual of Style/Mathematics, this article should follow the following guidelines (as well as others not listed here). It should be be written "one level down", which means:

• "consider the typical level where the topic is studied (for example, secondary, undergraduate, or postgraduate) and write the article for readers who are at the previous level." Also,
• "articles on undergraduate topics can be aimed at a reader with a secondary school background, and articles on postgraduate topics can be aimed at readers with some undergraduate background."
• "Articles should be as accessible as possible to readers not already familiar with the subject matter."
• "When in doubt, articles should define the notation they use."
• "If an article requires extensive notation, consider introducing the notation as a bulleted list or separating it into a "Notation" section."
• "An article about a mathematical object should provide an exact definition of the object, perhaps in a "Definition" section after section(s) of motivation."
• "Writing one level down also supports our goal to provide a tertiary source on the topic, which readers can use before they begin to read other sources about it."

I think that it is safe to assume that the reader has knowledge of calculus. But before we start editing this article to make it less technical, it's important to know what else we can assume about a "typical" reader of this article. This is important because, for example, whenever it is reasonable and possible to do so, then terminology that a reader is unlikely to be familiar with should be briefly defined/described within this article, instead of just having a link to the article about the term (this is because ideally, a "typical" reader should not have to go down a rabbit hole of Wikipedia links and search through various articles in order to understand something stated in this article about distributions). So we need to agree on the following (non-exhaustive) list of assumptions before we can start rewriting this article:

1. Is it safe to assume that the reader is likely an advanced undergraduate or higher? (I personally think so).
2. Is it safe to assume that the reader is likely a graduate student or higher?
3. Is it safe to assume that the reader is likely a mathematics, physics, or engineering student?
4. Is it safe to assume that the reader has studied metric spaces? (I personally think that it is).
5. Is it safe to assume that the reader has knowledge of general topology (in particular, of non-metrizable topological spaces)?
6. Is it safe to assume that the reader has studied Banach spaces? (If not, then the Fréchet spaces and related notions that are used in this article will need more detailed explanations).
7. Is it safe to assume that the reader has studied Fréchet spaces? My guess is probably not and so the reader should not be assumed to know about Fréchet space. But if they are familiar with the basics of Banach spaces then the required knowledge for Fréchet spaces can be described using Banach space terminology.
8. Is it safe to assume that the reader has studied non-metrizable topological vector spaces? I think that this can not be assumed. However, unfortunately, neither the canonical LF topology nor the topology on the space of distributions is a sequential space so this topology can not be described using sequences (let along a metric). Suggestions about how to define and describe these topologies to readers who are not used to dealing with non-sequential (and also non-metrizable) spaces would be welcome. The current description of these topologies is (unfortunately) technical and I'd like for it to be less technical but I'm not sure how to make it less technical.

Mgkrupa 18:51, 26 October 2020 (UTC)

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I am in favor of creating a separate article for the canonical LF topology (this topology has been studied enough to warrant its own article) and in this way we can simplify the article on distributions by placing some technical details into this new article. Mgkrupa 23:27, 23 September 2020 (UTC)

For instance, we can make the presentation less general by replacing k with ∞. Mgkrupa 23:51, 23 September 2020 (UTC)

Symbols

According to Wikipedia:Manual of Style/Mathematics, the symbols ∀ ∃ and ⇔ should not be used unnecessarily so over time, I will be removing instances of these symbols. Also, words like "we," "recall," and especially "clearly" and "obviously" should be avoided (since something that is clear to you may not be clear to the reader). Mgkrupa 17:52, 26 October 2020 (UTC)