Vitali variation
2010 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}\tag{1} \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 (1) 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 |
Fréchet variation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_variation&oldid=28291