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| − | One of the numerical characteristics of a function of several variables that can be regarded as a multi-dimensional analogue of the [[Variation of a function|variation of a function]] of a single variable. Suppose that a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f0414001.png" /> is given on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f0414002.png" />-dimensional parallelopipedon
| + | #REDIRECT[[Vitali variation]] |
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| − | <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/f/f041/f041400/f0414003.png" /></td> </tr></table>
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| − | and introduce the notation
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| − | <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/f/f041/f041400/f0414004.png" /></td> </tr></table>
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| − | <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/f/f041/f041400/f0414005.png" /></td> </tr></table>
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| − | <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/f/f041/f041400/f0414006.png" /></td> </tr></table>
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| − | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f0414007.png" /> be an arbitrary partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f0414008.png" /> by hyperplanes
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| − | <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/f/f041/f041400/f0414009.png" /></td> </tr></table>
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| − | <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/f/f041/f041400/f04140010.png" /></td> </tr></table>
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| − | <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/f/f041/f041400/f04140011.png" /></td> </tr></table>
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| − | into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140012.png" />-dimensional parallelopipeda, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140013.png" /> take the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140014.png" /> in an arbitrary way. The Fréchet variation is defined as follows:
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| − | <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/f/f041/f041400/f04140015.png" /></td> </tr></table>
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| − | <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/f/f041/f041400/f04140016.png" /></td> </tr></table>
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| − | <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/f/f041/f041400/f04140017.png" /></td> </tr></table>
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| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140018.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140019.png" /> has bounded (finite) Fréchet variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140020.png" />, and the class of all such functions is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140021.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140022.png" />, this class was introduced by M. Fréchet [[#References|[1]]] in connection with the investigation of the general form of a bilinear continuous functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140023.png" /> on the space of functions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140024.png" /> that are continuous on the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140025.png" />. He proved that every such functional can be represented in the form
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| − | <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/f/f041/f041400/f04140026.png" /></td> </tr></table>
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140028.png" />.
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| − | Analogues of many of the classical criteria for the convergence of Fourier series are valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140029.png" />-periodic functions in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140030.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140031.png" />, see [[#References|[2]]]). For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140033.png" /> then the rectangular partial sums of the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140034.png" /> converge at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140035.png" /> to the number
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| − | <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/f/f041/f041400/f04140036.png" /></td> </tr></table>
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| − | where the summation is taken over all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140037.png" /> possible combinations of the signs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041400/f04140038.png" />. Here, if the function is continuous, the convergence is uniform (an analogue of the Jordan criterion).
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| − | ====References====
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Fréchet, "Sur les fonctionelles bilinéaires" ''Trans. Amer. Math. Soc.'' , '''16''' : 3 (1915) pp. 215–234</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Morse, W. Transue, "The Fréchet variation and the convergence of multiple Fourier series" ''Proc. Nat. Acad. Sci. USA'' , '''35''' : 7 (1949) pp. 395–399</TD></TR></table>
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