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A numerical characteristic of a function of several variables, which may be regarded as the multi-dimensional analogue of the [[Variation of a function|variation of a function]] in one unknown. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a0134701.png" /> be a real-valued function given on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a0134702.png" />-dimensional parallelepipedon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a0134703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a0134704.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a0134705.png" /> be the class of all continuous vector functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a0134706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a0134707.png" />, such that each of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a0134708.png" /> is non-decreasing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a0134709.png" />, and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347012.png" />. Then
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{{MSC|26B30|26A45,26}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347013.png" /></td> </tr></table>
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[[Category:Analysis]]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347014.png" /> is an arbitrary system of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347015.png" />. This definition for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347016.png" /> was proposed by C. Arzelà [[#References|[1]]] (see also [[#References|[2]]], p. 543). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347017.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347018.png" /> has bounded (finite) Arzelà variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347019.png" />, and the class of all such functions is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347020.png" />. For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347021.png" /> to belong to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347022.png" /> it is necessary and sufficient that there exists a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347025.png" /> are finite non-decreasing functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347026.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347027.png" /> is called non-decreasing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347028.png" /> if
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347029.png" /></td> </tr></table>
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A generalization to functions of several variables of the [[Variation of a function]] of one variable, proposed by
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C. Arzelà in {{Cite|Ar}} (see also {{Cite|Ha}}, p. 543). However the modern theory of functions of bounded variation uses a different generalization (see [[Function of bounded variation]] and [[Variation of a function]]). Therefore the Arzelà variation is seldomly used nowadays.
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347030.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347031.png" />). The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347032.png" /> contains the class of functions of bounded [[Hardy variation|Hardy variation]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013470/a01347033.png" />.
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Consider a rectangle $R:= [a_1, b_1]\times \ldots \times [a_n, b_n]\subset \mathbb R^n$ and denote by
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* $\Gamma$ the class of continuous $\gamma= (\gamma_1, \ldots, \gamma_n):[0,1]\to R$ such that each component $\gamma_j$ is nondecreasing and maps $[0,1]$ onto $[a_j, b_j]$.
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* $\Pi$ the family of $N+1$-tuples of points $0\leq t_1 < \ldots < t_{N+1}\leq 1$.
  
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'''Definition'''
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The Arzelà variation of a function $f:R\to\mathbb R$ is then defined as
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\[
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V_A (f):= \sup_{\gamma\in \Gamma}\; TV (f\circ \gamma) = \sup_{\gamma\in \Gamma}\;
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\left(\sup \left\{ \sum_{i=1}^N |f(\gamma (t_{i+1})) - f (\gamma (t_i))| : (t_1, \ldots , t_{N+1}\in \Pi)\right\}\right)
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\]
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($TV (f\circ \gamma)$ is then the classical total variation of the real variable function $f\circ \gamma$, see [[Variation of a function]]).
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A function $f$ has finite Arzelà variation if and only if it can be written as the difference of two functions $f^+-f^-$ with the property that
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\[
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f^{\pm} (x_1, \ldots, x_n) \leq f^{\pm} (y_1, \ldots, y_n) \qquad \mbox{if $x_i\leq y_i$ for all $i$}.
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\]
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This statement generalizes the [[Jordan decomposition (of a function)|Jordan decomposition]] of functions of bounded variation of one variable.
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The class of functions with finite Arzelà variation contains the class of functions with finite [[Hardy variation]].
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Arzelà,  ''Rend. Accad. Sci. Bologna'' , '''9''' :  2  (1905)  pp. 100–107</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"H. Hahn,  "Theorie der reellen Funktionen" , '''1''' , Springer  (1921)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Ar}}|| C. Arzelà,  ''Rend. Accad. Sci. Bologna'' , '''9''' :  2  (1905)  pp. 100–107.
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|-
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|valign="top"|{{Ref|Ha}}|| H. Hahn,  "Theorie der reellen Funktionen" , '''1''' , Springer  (1921).
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|-
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|}

Revision as of 10:10, 16 September 2012

2020 Mathematics Subject Classification: Primary: 26B30 Secondary: 26A4526-XX [MSN][ZBL]

A generalization to functions of several variables of the Variation of a function of one variable, proposed by C. Arzelà in [Ar] (see also [Ha], p. 543). However the modern theory of functions of bounded variation uses a different generalization (see Function of bounded variation and Variation of a function). Therefore the Arzelà variation is seldomly used nowadays.

Consider a rectangle $R:= [a_1, b_1]\times \ldots \times [a_n, b_n]\subset \mathbb R^n$ and denote by

  • $\Gamma$ the class of continuous $\gamma= (\gamma_1, \ldots, \gamma_n):[0,1]\to R$ such that each component $\gamma_j$ is nondecreasing and maps $[0,1]$ onto $[a_j, b_j]$.
  • $\Pi$ the family of $N+1$-tuples of points $0\leq t_1 < \ldots < t_{N+1}\leq 1$.

Definition The Arzelà variation of a function $f:R\to\mathbb R$ is then defined as \[ V_A (f):= \sup_{\gamma\in \Gamma}\; TV (f\circ \gamma) = \sup_{\gamma\in \Gamma}\; \left(\sup \left\{ \sum_{i=1}^N |f(\gamma (t_{i+1})) - f (\gamma (t_i))| : (t_1, \ldots , t_{N+1}\in \Pi)\right\}\right) \] ($TV (f\circ \gamma)$ is then the classical total variation of the real variable function $f\circ \gamma$, see Variation of a function).

A function $f$ has finite Arzelà variation if and only if it can be written as the difference of two functions $f^+-f^-$ with the property that \[ f^{\pm} (x_1, \ldots, x_n) \leq f^{\pm} (y_1, \ldots, y_n) \qquad \mbox{if '"`UNIQ-MathJax15-QINU`"' for all '"`UNIQ-MathJax16-QINU`"'}. \] This statement generalizes the Jordan decomposition of functions of bounded variation of one variable.

The class of functions with finite Arzelà variation contains the class of functions with finite Hardy variation.

References

[Ar] C. Arzelà, Rend. Accad. Sci. Bologna , 9 : 2 (1905) pp. 100–107.
[Ha] H. Hahn, "Theorie der reellen Funktionen" , 1 , Springer (1921).
How to Cite This Entry:
Arzelà variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arzel%C3%A0_variation&oldid=27956
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article