# Tonelli plane variation

2010 Mathematics Subject Classification: Primary: 26B30 Secondary: 26A45 [MSN][ZBL]

A generalization to functions of two variables of the Variation of a function of one variable, proposed by Tonelli in [To].

Definition Given a measurable function $f: \mathbb R^2\to\mathbb R$ we define the Tonelli variation $f$ as $V_T (f) := \int_{-\infty}^\infty TV (f (\cdot, y))\, dy + \int_{-\infty}^\infty TV (f (x, \cdot))\, dx\,$ where $TV (g)$ denotes the classical total variation of a function of one real variable.

Cesari in [Ce] proposed the following modification of the Tonelli's plane variation, which is sometimes called Tonelli-Cesari variation

Definition Given a measurable function $f: \mathbb R^2\to\mathbb R$ we define the Tonelli-Cesari variation $f$ as $V_{TC} (f) := \inf \left\{ V_T (g) : g = f \;\lambda\text{-a.e.}\right\}\, .$

It turns out that the function $f$ has bounded Tonelli-Cesari variation if and only if it has bounded variation in the modern sense (see Variation of a function and Function of bounded variation). Among the many generalizations of the variation of a function of one variable given in the first half of the twentieth century, the Tonelli-Cesari is therefore the only one equivalent to the modern point of view. For a thorough discussion of the historical aspects of the theory of functions of bounded variation we refer to Section 3.12 of [AFP].

How to Cite This Entry:
Tonelli plane variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tonelli_plane_variation&oldid=30111
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article