|
|
Line 1: |
Line 1: |
− | 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:
| + | {{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/v/v096/v096790/v0967905.png" /></td> </tr></table>
| + | [[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/v0967906.png" /></td> </tr></table>
| + | {{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/v/v096/v096790/v0967907.png" /></td> </tr></table>
| + | 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}}. 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
| + | 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 $t_k^{i+1}- t_k^i$. |
| | | |
− | <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>
| + | '''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|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>
| + | 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}}. |
− | | |
− | 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
| |
− | | |
− | <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>
| |
− | | |
− | 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>
| + | {| |
| + | |- |
| + | |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. |
| + | |- |
| + | |valign="top"|{{Ref|Ha}}|| H. Hahn, "Theorie der reellen Funktionen" , '''1''' , Springer (1921). |
| + | |- |
| + | |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|Ri}}|| F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955). |
| + | |- |
| + | |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. |
| + | |- |
| + | |} |
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 Vitali in [Vi] (see also [Ha]). The same definition of variation was subsequently proposed by H. Lebesgue [Le] and M. Fréchet [Fr]. 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].
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.
|
[Ha] |
H. Hahn, "Theorie der reellen Funktionen" , 1 , Springer (1921).
|
[Le] |
H. Lebesgue, "Sur l'intégration des fonctions discontinues" Ann. Sci. École Norm. Sup. (3) , 27 (1910) pp. 361–450.
|
[Ri] |
F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955).
|
[Vi] |
G. Vitali, "Sui gruppi di punti e sulle funzioni di variabili reali" Atti Accad. Sci. Torino , 43 (1908) pp. 75–92.
|