Talk:Distribution (mathematics)

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)

I believe distributions are common in physics (please correct me if I'm wrong), and I am not sure that physics students will know about Frechet spaces (I am not a physics student myself, so I don't even know if physics undergrads learn any topology) – so perhaps the ultimate goal is to make this article accessible to physicists? It would be helpful if others could give their thoughts on this.

Also, I agree that ideally "a 'typical' reader should not have to go down a rabbit hole of wikipedia links" to understand an article, but I would say that this article is (at least somewhat) an exception to this rule, because it requires a much higher than average amount of prerequisite knowledge (again, please correct me if I am wrong) Joel Brennan (talk) 00:15, 18 November 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)

I recently learned distributions and I found it frustrating the first time I glanced at this article, hoping for an overview of the subject. This issue was not even technicality: I understand Fréchet spaces and topological linear spaces. The article was just not reader-friendly and felt more like a giant list of notation. Compare this to the article on Sobolev space, which starts off with motivation and looks much less cluttered. 74.101.253.193 (talk) 22:11, 16 August 2021 (UTC)

Completely agree about moving discussion of TVS topologies to another article

No one looking for a straightforward explanation of distributions will be able to get through the current article.

As far as the continuity of distributions is concerned, all one needs to say in this article is that a linear functional $T:{\mathcal {D}}(U)\to \mathbf {R}$ (or $\mathbf {C}$) is continuous if and only if it is sequentially continuous i.e., if $\varphi _{n}\to \varphi$ in ${\mathcal {D}}(U)$ (supports of all $\varphi _{n}$ are contained in a compact subset $K$ of $U$ and the sequence and all of its derivatives converge uniformly on $K$), then $T(\varphi _{n})\to T(\varphi )$ in $\mathbf {R}$. Explanations of sequential continuity vs. continuity in TVS can be given elsewhere.

Alternatively, you could say that $T$ is continuous if and only if for every compact subset $K$ of $U$ there exist $C>0$, $n\in \mathbf {N}$ such that $|T(\varphi )|\leq C\sum _{|\alpha |\leq n}\|\partial ^{\alpha }\varphi \|_{\infty }$ for every $\varphi$ supported in $K$. But I think the sequentially definition is more accessible.

However, these changes would require a lot of rewriting of the first part of this article.Reader634 (talk) 07:45, 12 December 2021 (UTC)

Comment

I think the recent changes are absolutely in the right direction. I made a few minor edits, mostly to remove "scare" quotes about the difficulties of defining the topology on the space of distributions (it's enough to refer to the relevant article). Also, I think more of the discussion on topologies can be moved elsewhere, and all (or nearly all?) of the material on $C^{k}$ spaces can be removed from this article, since $C^{\infty }$ test functions are the fundamental ones for distributions. As a minor point, I'd denote the regular distribution associated with multiplication by a locally integrable function $f$ by $T_{f}$ rather than $D_{f}$ (which could be confused with differentiation) but I don't think there's a standard notation here. Reader634 (talk) 09:08, 31 December 2021 (UTC)

Some notations not explained

Same as in Spaces of test functions and distributions: notation $L_{c}^{p}$ first appearing in 6 (Spaces of distributions) seems not to be explained anywhere on this page. Mamuka Jibladze (talk) 17:23, 31 January 2022 (UTC)\]

KINDLY DO NOT ERASE OTHER PEOPLE'S POSTS WHEN YOU POST YOUR OWN. 2601:200:C000:1A0:7097:9C72:4BFB:717D (talk) 00:11, 18 May 2022 (UTC)

Tempered distributions

"Tempered distributions" is used but not defined in this article. I've heard they are "distributions that are upper bounded by polynomials", which I'm not quite sure how to make precise. I've also heard the more and difficult definition in terms of (the extension of?) linear functions on Schwartz function. Needs to be made clear in this article. Jess_Riedel (talk) Jess_Riedel (talk) 19:07, 16 March 2024 (UTC)