Vitali variation

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.
[Fr1]	M. Fréchet, "Sur les fonctionelles bilinéaires" Trans. Amer. Math. Soc. , 16 : 3 (1915) pp. 215–234
[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.
[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.
[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.