Talk:Bounded variation

Plan of the entry

The first proposal for the plan of the voice can be seen here together with some further discussion Daniele.tampieri 10:15, 11 March 2007 (UTC)

Further additions by several contributors have led me to single out the following structure.

• Informal definition
• History

1. Camille Jordan for one variable.
2. Leonida Tonelli and Lamberto Cesari for several variable.
3. Cacioppoli, De Giorgi, Smoller, Conway, Vol'pert and other for the applications.

• Formal definition

1. One variable
2. Several variables

• Basic properties
• Generalizations

1. Weighted BV functions (tanks to T.J. Sullivan)
2. Special Functions of Bounded Variation, i.e. SBV functions

• Applications

1. In maths
2. In Physics and Engineering

• See also
• References
• Bibliography
• External links

If you have ideas on how to improve this structure, you're welcome. Thank you. :) Daniele.tampieri 10:44, 11 March 2007 (UTC)

An update: after several years of analysis and study on the topic, both the plan of the entry and the content of many of its sections should be improved. I propose the following new plan (the standard sections "See also" etc. are not shown even if the should be present):

• Informal definition
• History

1. Camille Jordan for one variable.
2. Leonida Tonelli and Lamberto Cesari for several variables.
3. De Giorgi, Fichera and Miranda: modern definitions based on the heat kernel and the theory of distributions
4. Cacioppoli, De Giorgi, Smoller, Conway, Vol'pert, Ambrosio and others for the applications.

• Formal definition

1. One variable
2. Several variables
3. Discussion of the difference between the two definitions and its relation to the development of the theory

• Basic properties

1. One variable
2. Several variables

• Generalizations

1. Weighted BV functions (tanks to T.J. Sullivan)
2. Special Functions of Bounded Variation, i.e. SBV functions
3. GBV functions and other classes

• Applications

1. In maths
2. In Physics and Engineering

As usual, ideas on how to improve the entry are most welcome! Daniele.tampieri (talk) 08:05, 1 August 2015 (UTC)

About the removig of the "Algebraic Operations with BV Functions" section

The "Algebraic Operations with BV Functions" section was removed since it violated the WP:COI policy and also it was a (fortunately short) mere collection of trivial facts. First of all, the space of BV functions is a vector space and the proof (for the one variable case) is very simple and compactly presented in standard texts as (Kolmogorov & Fomin 1969, pp. 328–329), therefore it is closed under subtraction and addition. It is an algebra as already stated in the related section of the article, therefore it is also closed under multiplication. The quotient of two functions of such space could not belong to the same space, as the following trivial example shows: consider the two functions $1$ ∈ BV([$-1$,$1$]) and $x$ ∈ BV([$-1$,$1$]). Then $1/x$ ∉ L₁([$-1$,$1$]), i.e. the quotient of such functions of bounded variation is not even integrable on the [$-1$,$1$] interval. Similar elementary examples also show that closure under composition is trivially false: further, the composition of two function belongs to the realm of analytic operations therefore, strictly speaking, it is not an algebraic operation (even if can be studied using the methods of abstract algebra). Daniele.tampieri (talk) 16:21, 31 October 2011 (UTC)

References

• Kolmogorov, Andrej N.; Fomin, Sergej V. (1969), Introductory Real Analysis, New York: Dover Publications, pp. xii+403, ISBN 0486612260, MR 0377445, Zbl 0213.07305.

Algebra

The sentence from the article:

One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose first derivative

doesn't say what is meant. At least it is logically inconsistent, as continuous functions are of bounded variation. 82.75.140.46 (talk) 09:58, 26 January 2012 (UTC)

Is this the sentence you were refering to?

"function of bounded variation are the smallest algebra which has to be embedded in every space of generalized functions preserving the result of multiplication."

it seems that the "smallest" there should be changed to a "largest" since, well, the algebra of functions of bounded variation contains many $\mathbb {C}$-sub-algebras (the constant functions, for instance). On that note, what ring are they algebra over? Zdorovo (talk) 01:46, 1 May 2013 (UTC)

I was also confused about the claim about BV being the smallest algebra. This hasn't been fixed in over three years, and I don't know a citation, so I'm deleting it. If someone has a citation of a correct statement, they should include it. 217.42.152.107 (talk) 20:46, 11 July 2016 (UTC)

Simplify

The notation used in this article is ugly and overcomplicated. Even the very first definition about bounded variation for a 1-variable function is notated in an overly complicated manner. Similarly with the left and right limits being notated as $x_{0^{-}}$ and $x_{0^{+}}$ instead of $x_{0}^{-}$ and $x_{0}^{+}$. What a confusing article. 00:00, 27 January 2012 (UTC) — Preceding unsigned comment added by Angry bee (talk • contribs)

And why is the one variable version defined for a closed subset of R while the multivariable one for an open subset? Angry bee (talk) 23:24, 24 May 2012 (UTC)

The definition for BV in one variable and several variables is not consistent. BV([a,b]) is defined as a set of functions, whereas the general cases is defined as a subset of L^1 (which consists of equivalence classes of functions).

"BV functions have only jump-type discontinuities"

BV function with a removable discontinuity: http://www.encyclopediaofmath.org/index.php/Function_of_bounded_variation#Continuity (Warning 6) — Preceding unsigned comment added by 132.230.12.59 (talk) 11:44, 11 June 2013 (UTC)

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Continuity assumption?

The single-variable case assumes continuity of f. This seems unnecessary and is not done in Wolfram MathWorld's article on BV. And it makes the first property proven (only jump-type and removable discontinuities) trivial. Why was continuity assumed? It appears to be safe to remove it. — Preceding unsigned comment added by 69.253.18.244 (talk) 18:10, 1 June 2020 (UTC)

In fact the only point where continuity was implicitly assumed was the chain of inclusions in the introduction. This chain of inclusions is not very relevant and it is repeated in several other articles, and it could even be removed; I've added "continuous".pma 22:43, 2 November 2020 (UTC)

Examples

The examples start off with two cases which aren't bounded variations. I'm not sure whether that's the best way to present examples. Isn't it better to first show what are bounded variations, before introducing things which aren't? 62.216.5.216 (talk) 19:51, 2 November 2020 (UTC)

That's true, but a large class of examples is already mentioned in the inclusions in the introduction. Maybe this could be recalled in the section of examples pma 22:50, 2 November 2020 (UTC)