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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.
If you have ideas on how to improve this structure, you're welcome. Thank you. :) Daniele.tampieri 10:44, 11 March 2007 (UTC)
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 ∈ BV([,]) and ∈ BV([,]). Then ∉ L1([,]), i.e. the quotient of such functions of bounded variation is not even integrable on the [,] 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)
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 -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)
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 and instead of and . 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 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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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)
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)
This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
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.
If you have ideas on how to improve this structure, you're welcome. Thank you. :) Daniele.tampieri 10:44, 11 March 2007 (UTC)
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 ∈ BV([,]) and ∈ BV([,]). Then ∉ L1([,]), i.e. the quotient of such functions of bounded variation is not even integrable on the [,] 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)
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 -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)
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 and instead of and . 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 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)
Hello fellow Wikipedians,
I have just modified one external link on Bounded variation. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
This message was posted before February 2018.
After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than
regular verification using the archive tool instructions below. Editors
have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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source check}}
(last update: 5 June 2024).
Cheers.— InternetArchiveBot ( Report bug) 02:02, 24 July 2017 (UTC)
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)
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)