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One of the numerical characteristics of a function of several variables. It may be considered as a multi-dimensional analogue of the [[Variation of a function|variation of a function]] of one variable. Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v0967901.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v0967902.png" /> be defined on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v0967903.png" />-dimensional parallelepipedon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v0967904.png" />. One introduces the following notation:
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{{TEX|done}}
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{{MSC|26B30|26A45}}
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[[Category:Analysis]]
  
<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/v/v096/v096790/v0967905.png" /></td> </tr></table>
 
  
<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/v/v096/v096790/v0967906.png" /></td> </tr></table>
 
  
<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/v/v096/v096790/v0967907.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 Vitali in {{Cite|Vi}} (see also {{Cite|Ha}}). The same definition of variation was subsequently proposed by H. Lebesgue {{Cite|Le}} and M. Fréchet {{Cite|Fr}} and it is sometimes called Fréchet variation. 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 Vitali variation is seldomly used  nowadays.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v0967908.png" /> be an arbitrary subdivision of the parallelepipedon by hyperplanes
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Consider  a rectangle $R:= [a_1, b_1]\times \ldots \times [a_n, b_n]\subset  \mathbb R^n$ and a function $f:R\to \mathbb R$. We define
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\[
 +
\Delta_{h_k} (f, x) := f (x_1, \ldots, x_k+ h_k, \ldots, x_n) - f(x_1, \ldots, x_k, \ldots x_n)
 +
\]
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and, recursively,
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\[
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\Delta_{h_1h_2\ldots h_k} (f, x):= \Delta_{h_k} \left(\Delta_{h_1\ldots h_{k-1}} , x\right)\, .
 +
\]
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Consider  next the collection $\Pi_k$ of finite ordered families $\pi_k$ of  points $t_k^1 < t_k^2< \ldots < t_k^{N_k+1}\in [a_k, b_k]$. For  each such $\pi_k$ we denote by $h^i_k$ the difference $t_k^{i+1}-  t_k^i$.  
  
<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/v/v096/v096790/v0967909.png" /></td> </tr></table>
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'''Definition''' We define the Vitali variation of $f$ to be the supremum over $(\pi_1, \ldots, \pi_n)\in \Pi_1\times \ldots  \times \Pi_n$ of the sums
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\begin{equation}\label{e:v_variation}
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\sum_{i_1=1}^{N_1} \ldots  \sum_{i_n=1}^{N_n} \left|\Delta_{h^{i_1}_1\ldots h^{i_n}_n} \left(f,  \left(x^{i_1}_1, \ldots x^{i_n}_n\right)\right)\right|\,
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\end{equation}
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If the Vitali variation is finite, then one says that $f$ has bounded (finite) Vitali variation.
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$f$ has finite Vitali variation if and only if it can be written as difference of two functions for which all the sums of type \eqref{e:v_variation} are nonnegative. This statement generalizes
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the [[Jordan decomposition|Jordan decomposition]] of a [[Function of bounded variation|function of bounded variation]] of one variable.  
  
<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/v/v096/v096790/v09679010.png" /></td> </tr></table>
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The class of functions with finite Vitali variation may be used to introduce the multi-dimensional [[Stieltjes integral|Stieltjes integral]], as was observed in {{Cite|Fr}}. Fréchet also used it in {{Cite|Fr1}} to study continuous bilinear functionals on the space of continuous functions of two variables of the form $(x_1,x_2)\mapsto \phi (x_1)\phi (x_2)$.  
  
into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679011.png" />-dimensional parallelepipeda. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679012.png" /> be the least upper bound of sums of the type
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The classical [[Jordan criterion]] for the convergence of Fourier series can be extended to functions which have finite Vitali variation (see {{Cite|MT}}). In particular, if a function $f$ on $[0,2\pi]^n$ has finite Vitali variation, then the rectangular partial sums of its Fourier series converges at every $x=(x_1, \ldots x_n)$ to the value
 
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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/v/v096/v096790/v09679013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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\frac{1}{2^n} \sum f(x_1^\pm, \ldots, x_n^\pm)\, .
 
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\]
taken over all possible subdivisions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679015.png" />, one says that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679016.png" /> has bounded (finite) Vitali variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679017.png" />, while the class of all such functions is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679018.png" /> or simply by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679019.png" />. The class was defined by G. Vitali [[#References|[1]]]. The same definition of variation was subsequently proposed by H. Lebesgue [[#References|[2]]] and M. Fréchet [[#References|[3]]]. A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679020.png" />, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679021.png" />, belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679022.png" /> if and only if it can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679023.png" />, where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679025.png" /> are such that, for each of them, the sums of the type (*), taken without the modulus sign, are non-negative [[#References|[4]]] (the analogue of the [[Jordan decomposition|Jordan decomposition]] of a function of bounded variation of one variable). The functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679026.png" /> may be used to introduce the multi-dimensional [[Stieltjes integral|Stieltjes integral]]. In particular, for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679027.png" /> which is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679028.png" /> and any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679029.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679030.png" /> the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096790/v09679031.png" /> exists [[#References|[3]]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Vitali,   "Sui gruppi di punti e sulle funzioni di variabili reali"  ''Atti Accad. Sci. Torino'' , '''43'''  (1908)  pp. 75–92</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Lebesgue,  "Sur l'intégration des fonctions discontinues"  ''Ann. Sci. École Norm. Sup. (3)'' , '''27'''  (1910)  pp. 361–450</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"M. Fréchet,  "Extension au cas d'intégrales multiples d'une définition de l'intégrale due à Stieltjes"  ''Nouv. Ann. Math. ser. 4'' , '''10'''  (1910)  pp. 241–256</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Hahn,  "Theorie der reellen Funktionen" , '''1''' , Springer  (1921)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Fr}}||  M. Fréchet,  "Extension au cas d'intégrales multiples d'une définition  de l'intégrale due à Stieltjes"  ''Nouv. Ann. Math. ser. 4'' , '''10'''  (1910)  pp. 241–256.  JFM {{ZBL|41.0333.02}}
 +
|-
 +
|valign="top"|{{Ref|Fr1}}|| M. Fréchet, "Sur les fonctionelles bilinéaires"  ''Trans. Amer. Math. Soc.'' , '''16''' :  3 (1915)  pp. 215–234  JFM {{ZBL|45.0546.01}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}|| H. Hahn,  "Theorie der reellen Funktionen" , '''1''' , Springer  (1921).  JFM {{ZBL|48.0261.09}}
 +
|-
 +
|valign="top"|{{Ref|Le}}|| H. Lebesgue,  "Sur l'intégration des fonctions discontinues"  ''Ann.   Sci. École Norm. Sup. (3)'' , '''27'''  (1910)  pp. 361–450
 +
|-
 +
|valign="top"|{{Ref|MT}}|| M. Morse,  W. Transue,  "The Fréchet variation and the convergence of  multiple Fourier series"  ''Proc. Nat. Acad. Sci. USA'' , '''35''' : (1949)  pp. 395–399.  {{MR|0030587}} {{ZBL|0033.35901}}
 +
|-
 +
|valign="top"|{{Ref|Ri}}|| F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955).  {{MR|0071727}}  {{ZBL|0732.47001}} {{ZBL|0070.10902}} {{ZBL|0046.33103}}
 +
|-
 +
|valign="top"|{{Ref|Vi}}|| G. Vitali,  "Sui gruppi di punti e sulle funzioni di variabili reali"  ''Atti Accad. Sci. Torino'' , '''43''' (1908) pp. 75–92.  JFM {{ZBL|39.0101.05}}
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|-
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|}

Latest revision as of 12:29, 10 February 2020

2020 Mathematics Subject Classification: Primary: 26B30 Secondary: 26A45 [MSN][ZBL]


A generalization to functions of several variables of the Variation of a function of one variable, proposed by Vitali in [Vi] (see also [Ha]). The same definition of variation was subsequently proposed by H. Lebesgue [Le] and M. Fréchet [Fr] and it is sometimes called Fréchet variation. 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 Vitali 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 a function $f:R\to \mathbb R$. We define \[ \Delta_{h_k} (f, x) := f (x_1, \ldots, x_k+ h_k, \ldots, x_n) - f(x_1, \ldots, x_k, \ldots x_n) \] and, recursively, \[ \Delta_{h_1h_2\ldots h_k} (f, x):= \Delta_{h_k} \left(\Delta_{h_1\ldots h_{k-1}} , x\right)\, . \] Consider next the collection $\Pi_k$ of finite ordered families $\pi_k$ of points $t_k^1 < t_k^2< \ldots < t_k^{N_k+1}\in [a_k, b_k]$. For each such $\pi_k$ we denote by $h^i_k$ the difference $t_k^{i+1}- t_k^i$.

Definition We define the Vitali variation of $f$ to be the supremum over $(\pi_1, \ldots, \pi_n)\in \Pi_1\times \ldots \times \Pi_n$ of the sums \begin{equation}\label{e:v_variation} \sum_{i_1=1}^{N_1} \ldots \sum_{i_n=1}^{N_n} \left|\Delta_{h^{i_1}_1\ldots h^{i_n}_n} \left(f, \left(x^{i_1}_1, \ldots x^{i_n}_n\right)\right)\right|\, \end{equation} If the Vitali variation is finite, then one says that $f$ has bounded (finite) Vitali variation. $f$ has finite Vitali variation if and only if it can be written as difference of two functions for which all the sums of type \eqref{e:v_variation} are nonnegative. This statement generalizes the Jordan decomposition of a function of bounded variation of one variable.

The class of functions with finite Vitali variation may be used to introduce the multi-dimensional Stieltjes integral, as was observed in [Fr]. Fréchet also used it in [Fr1] to study continuous bilinear functionals on the space of continuous functions of two variables of the form $(x_1,x_2)\mapsto \phi (x_1)\phi (x_2)$.

The classical Jordan criterion for the convergence of Fourier series can be extended to functions which have finite Vitali variation (see [MT]). In particular, if a function $f$ on $[0,2\pi]^n$ has finite Vitali variation, then the rectangular partial sums of its Fourier series converges at every $x=(x_1, \ldots x_n)$ to the value \[ \frac{1}{2^n} \sum f(x_1^\pm, \ldots, x_n^\pm)\, . \]

References

[Fr] M. Fréchet, "Extension au cas d'intégrales multiples d'une définition de l'intégrale due à Stieltjes" Nouv. Ann. Math. ser. 4 , 10 (1910) pp. 241–256. JFM Zbl 41.0333.02
[Fr1] M. Fréchet, "Sur les fonctionelles bilinéaires" Trans. Amer. Math. Soc. , 16 : 3 (1915) pp. 215–234 JFM Zbl 45.0546.01
[Ha] H. Hahn, "Theorie der reellen Funktionen" , 1 , Springer (1921). JFM Zbl 48.0261.09
[Le] H. Lebesgue, "Sur l'intégration des fonctions discontinues" Ann. Sci. École Norm. Sup. (3) , 27 (1910) pp. 361–450.
[MT] 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. MR0030587 Zbl 0033.35901
[Ri] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955). MR0071727 Zbl 0732.47001 Zbl 0070.10902 Zbl 0046.33103
[Vi] G. Vitali, "Sui gruppi di punti e sulle funzioni di variabili reali" Atti Accad. Sci. Torino , 43 (1908) pp. 75–92. JFM Zbl 39.0101.05
How to Cite This Entry:
Vitali variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vitali_variation&oldid=16159
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article