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)

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)

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)