View source for Arzelà variation

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

<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>

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

<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>

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" />.

====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>