Talk:Distribution (mathematical analysis)

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Section sizes

Section size for Distribution (mathematical analysis) (61 sections)
Section name	Byte count		Prose size (words)
Section name	Header	Total	Header	Total
(Top)	5,295	5,295	285	285
History	1,164	1,164	137	137
Notation	2,308	2,308	15	15
Basic idea	1,644	7,065	174	591
Functions and measures as distributions	2,226	2,226	76	76
Adding and multiplying distributions	621	621	74	74
Derivatives of distributions	2,574	2,574	267	267
Definitions of test functions and distributions	5,037	26,398	75	1,208
Topology on C'k(U) markup removed; cannot link (help)	5,312	10,577	196	598
Topology on C'k(K) markup removed; cannot link (help)	774	774	47	47
Trivial extensions and independence of C'k(K)'s topology from U markup removed; cannot link (help)	4,491	4,491	355	355
Canonical LF topology	1,793	1,793	122	122
Distributions	1,285	8,991	94	413
Characterizations of distributions	5,233	5,233	14	14
Topology on the space of distributions and its relation to the weak-* topology	2,473	2,473	305	305
Localization of distributions	590	15,036	75	980
Extensions and restrictions to an open subset	3,359	3,359	248	248
Gluing and distributions that vanish in a set	1,916	1,916	58	58
Support of a distribution	3,188	3,188	277	277
Distributions with compact support	44	2,725	0	80
Support in a point set and Dirac measures	1,187	1,187	80	80
Distribution with compact support	590	590	0	0
Distributions of finite order with support in an open subset	904	904	0	0
Global structure of distributions	977	3,258	131	242
Distributions as sheaves	836	836	0	0
Decomposition of distributions as sums of derivatives of continuous functions	1,445	1,445	111	111
Operations on distributions	2,113	44,840	122	2,701
Preliminaries: transpose of a linear operator	3,324	3,324	294	294
Differential operators	32	15,598	0	908
Differentiation of distributions	2,193	2,193	142	142
Differential operators acting on smooth functions	6,815	6,815	293	293
Multiplication of distributions by smooth functions	3,019	6,558	176	473
Problem of multiplying distributions	3,539	3,539	297	297
Composition with a smooth function	2,447	2,447	86	86
Convolution	1,790	18,255	119	1,254
Translation and symmetry	1,154	1,154	57	57
Convolution of a test function with a distribution	2,767	2,767	230	230
Convolution of a smooth function with a distribution	2,551	2,551	164	164
Convolution of distributions	5,081	5,081	348	348
Convolution versus multiplication	4,912	4,912	336	336
Tensor products of distributions	3,103	3,103	37	37
Spaces of distributions	3,026	22,964	265	1,805
Radon measures	2,504	6,069	114	489
Positive Radon measures	562	562	67	67
Locally integrable functions as distributions	2,223	2,223	226	226
Test functions as distributions	780	780	82	82
Distributions with compact support	1,991	2,407	135	175
Restriction of distributions to compact sets	416	416	40	40
Distributions of finite order	1,701	3,575	31	171
Structure of distributions of finite order	1,874	1,874	140	140
Tempered distributions and Fourier transform	711	7,887	55	705
Schwartz space	2,989	2,989	263	263
Tempered distributions	1,941	1,941	213	213
Fourier transform	1,429	1,429	126	126
Expressing tempered distributions as sums of derivatives	817	817	48	48
Using holomorphic functions as test functions	510	510	62	62
See also	992	992	0	0
Notes	37	37	0	0
References	36	36	0	0
Bibliography	3,687	3,687	0	0
Further reading	1,461	1,461	0	0
Total	131,793	131,793	7,784	7,784

Archives

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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)[reply]

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)\][reply]

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)[reply]

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)[reply]

Tautological sentence

“This leads to the space of (all) distributions on U, usually denoted by D’(U) (note the prime), which by definition is the space of all distributions on U”

What does this even mean? It sounds like a joke. The space of all distribution on U is by definition the space of all distributions on U? This is simply stating “X is by definition X” Siupa (talk) 09:18, 14 May 2025 (UTC)[reply]

Fixed. Hairer (talk) 07:42, 22 September 2025 (UTC)[reply]

Edit deleted by ClueBot NG twice

Hello everyone, I am new to editing Wikipedia and I get some trouble correcting a mistake in the section Tempered distributions.

The two first sentences of the last paragraph are completely wrong: take $f(x)=d/dx\cos(e^{x})$ , it is a bona fide tempered distribution while none of its successive derivatives is bounded (or integrable)...

I tried to correct twice this but each time the ClueBot NG removes my edit. Does someone know what it means?

Note also that the third sentence of the last paragraph is slightly wrong, the function is not differentiable at the origin. Besides, I do not understand what is the purpose of giving an example of a Schwartz function in the section dedicated to tempered distribution. It would make more sense to move it in the section Schwartz space. Pumkineater69notused (talk) 02:24, 10 November 2025 (UTC)[reply]

Pumkineater69notused, this (reverts for reference: revert 1, revert 2, revert 3, which you should include for reference so that others know exactly what you are referring to) seems to be something that ClueBot NG should not be reverting, as it is clearly not vandalism. I don't know what ClueBot uses as cues, but I suspect that a lack of prior edit history might be one of them. As to the validity of the edit I cannot comment, as this is a bit beyond my ken, but you should not mark an edit as minor if it introduces a visible change in the rendered page (as you did on the last edit). Maybe since, to the non-expert, it is not at all clear from the text what the real definition of a tempered distribution is, perhaps others are hesitant to support your edits; your edit might even have a feel of possibly of being your own derivation, which would not be permitted. Perhaps making other edits (i.e. establish yourself as an editor) first will lessen the probability of this occurring. —Quondum 18:21, 23 November 2025 (UTC)[reply]

Thank you for your answer, I have edited a fourth time, this time not ticking the "minor edit" box, and it seems like the edit has been accepted.

I too feel like the definition of a tempered distribution is not optimal. More generally, the Wikipedia page on distributions seems to have been wrote primarily by TVS enjoyers (see also the first comment on this page). In my opinion the page should not focus on TVS considerations but rather give practical insights, as done in the other languages pages. After all, distributional calculus was invented to make sense of formal computations, it should be easy. Pumkineater69notused (talk) 23:11, 23 November 2025 (UTC)[reply]

I've reverted the edit. Earlier I added two inline citations in support of the claim you removed. Please be so kind to check those references first and/or initiate a discussion on the talk page.

Kind regards, Roffaduft (talk) 07:38, 24 November 2025 (UTC)[reply]

@Roffaduft

1) The claim I have trouble with is "The tempered distributions can also be characterized as slowly growing, meaning that each derivative of

T

grows at most as fast as some polynomial." I understand that it means: for all

T\in {\mathcal {S}}'(\mathbb {R} ^{n})

and all

\alpha \in \mathbb {N} ^{n}

, the derivative

\partial ^{\alpha }T

is a locally integrable function, bounded above in modulus by a polynomial. Is it what you meant by the quoted sentence?

2) You said you gave references for the part of the text that I erased, which contains the sentence I quoted in 1) above. I cannot manage to see a reference for that in the actual Wikipedia page, could you help me find it? Or at least a reference in a published paper or a textbook?

3) The quoted sentence, if interpreted as I did, is completely wrong. A dummy example is the Dirac delta

T=\delta

, which is a tempered distribution but is not a locally integrable function (thus none of its successive derivative can be a locally integrable function). Do you agree with this?

4) A more subtle example is

f(x)=e^{x}\sin(e^{x})

, which is the prototypical example of a tempered distribution that is (violently) unbounded, not integrable, together with all its derivatives. For this function, could you point me which derivative grows at most as fast as a polynomial?

5) I cannot find a reference for this function to be a tempered distribution, but this is a simple consequence that a bounded function is a tempered distribution and the derivative of a tempered distribution is again a tempered distribution (see Rudin, Functionnal Analysis, Theorem 7.13, p. 192). Do you agree with this? Pumkineater69notused (talk) 09:16, 24 November 2025 (UTC)[reply]

My apologies, I wasn't accurate when I was talking about the "wrong claim". What I was referring to was the changes you made to the before-last paragraph. Those statements are correct (i.e. all apples are fruit, not every fruit is an apple) and supported by the inline citations.

I agree the main issue is the somewhat ambiguously phrased:

The tempered distributions can also be characterized as slowly growing,...

to which I like to refer to Remark 3 of Terrence Tao's notes:

https://terrytao.wordpress.com/2009/04/19/245c-notes-3-distributions/

I think this remark would be a great premise for an extra, final, paragraph.

Kind regards, Roffaduft (talk) 11:04, 24 November 2025 (UTC)[reply]

@Roffaduft

1) Although I did erase the content of the before-last paragraph, I gave a generalization in the last paragraph of the version I proposed. Could you please explain me what is the problem?

2) As far as I understand, a characterization is an "if and only if", see e.g. Characterization (mathematics). If you do not want that I edit your text, then please make that clearer.

3) Please note that Tao's example is exactly the one I gave.

4) I believe you got mixed up with the terms "rapidly falling" and "slowly growing". I have never seen them used in the context of tempered distributions. These terms do not appear in the book of Rudin you cited. The concept of a function "slowly growing at the infinity" is used by Treves to coin Schwartz multipliers (see Definition 25.3, p. 275), which is not exactly the same as a tempered distribution. A multiplier is something you can multiply a tempered distribution with, so that the result makes sense as a distribution, and is again a distribution. You should not mark them as a general example of tempered distribution, although this is true it is not instructive and, again, it is a mere application of the criterion I gave.

5) The terminology "rapidly falling" and "slowly growing" is more commonly used in the theory of hyperfunctions (see Kaneko, Introduction to the Theory of Hyperfunctions, Chapter 8). Pumkineater69notused (talk) 12:19, 24 November 2025 (UTC)[reply]

First of all, skip the unnecessary boldface, it comes across as demanding.

1) By doing so, you turned a perfectly fine, structured and factually accurate paragraph into borderline WP:NOR

2) It's not "my text"

3) That was exactly the point, given that you were unable to find a reference

4) Lots of unwarranted claims about my motives here. Instead, see point 1.

Again: I think Remark 3 from Terrence Tao's notes would form a great premise for an additional paragraph addressing the issue.

Kind regards, Roffaduft (talk) 12:55, 24 November 2025 (UTC)[reply]

@Roffaduft

"The tempered distributions can also be characterized as slowly growing, meaning that each derivative of grows at most as fast as some polynomial." cannot be true. A characterization is an "if and only if", see Characterization (mathematics). ~2025-35995-98 (talk) 13:39, 24 November 2025 (UTC)[reply]

Did I claim otherwise? Roffaduft (talk) 13:42, 24 November 2025 (UTC)[reply]

Pumkineater69notused, ~~I suspect that the use of the term "characterization" here is incorrect (it seems that it was almost certainly not intended as an if-and-only-if, but merely as a description).~~ The lead of Schwartz space suggests that the space of tempered distributions is defined as the members of the (continuous) dual space of the Schwarz space, which has the flavour of a definitive definition. This is stated again at continuous dual space. —Quondum 18:55, 24 November 2025 (UTC)[reply]

Looking a little more closely, the statement "The tempered distributions can also be characterized as slowly growing, meaning that each derivative of T grows at most as fast as some polynomial" should be challenged as potentially invalid. It may not need an additional paragraph at this point, but maybe that sentence should be deleted. —Quondum 19:21, 24 November 2025 (UTC)[reply]

Regular distributions

Apart from a paragraph in the lead starting with Most commonly encountered functions (and not going into details that this is supposed to be $L_{\text{loc}}^{1}$ etc.), the article does not seem to mention at all how functions are considered as distributions, which is quite central to the basic understanding of the concept. I feel like the article's organisation is kind of a mess in general but it would be good if this could be included. 1234qwer 1234qwer 4 00:23, 6 December 2025 (UTC)[reply]