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A numerical characteristic of a function of two variables, by means of which one defines the class of functions of bounded variation in the sense of Tonelli. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t0929901.png" /> is a function given on the rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t0929902.png" />. Assume that the functions
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{{MSC|26B30|26A45}}
  
<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/t/t092/t092990/t0929903.png" /></td> </tr></table>
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[[Category:Analysis]]
  
and
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{{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/t/t092/t092990/t0929904.png" /></td> </tr></table>
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A generalization to functions of two variables of the [[Variation of a function]] of one variable, proposed by Tonelli in {{Cite|To}}.  
  
are Lebesgue measurable (the first on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t0929905.png" />, the second on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t0929906.png" />). If
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'''Definition'''
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Given a measurable function $f: \mathbb R^2\to\mathbb R$ we define the Tonelli variation $f$ as
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\[
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V_T (f) := \int_{-\infty}^\infty TV (f (\cdot, y))\, dy + \int_{-\infty}^\infty TV (f (x, \cdot))\, dx\,  
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\]
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where $TV (g)$ denotes the classical total variation of a function of one real 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/t/t092/t092990/t0929907.png" /></td> </tr></table>
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Cesari in  {{Cite|Ce}} proposed the following modification of the [[Tonelli plane variation|Tonelli's  plane variation]], which is sometimes called Tonelli-Cesari variation
  
then one says that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t0929908.png" /> has bounded (finite, or: is of bounded (finite)) Tonelli plane variation on the rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t0929909.png" />, and the class of all such functions is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t09299010.png" />. This definition was proposed by L. Tonelli (cf. [[#References|[1]]], [[#References|[2]]]). For continuous functions, however, another characterization (in terms of the [[Banach indicatrix|Banach indicatrix]]) of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t09299011.png" /> can be found in an earlier paper of S. Banach [[#References|[4]]]. If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t09299012.png" /> is continuous on the rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t09299013.png" />, then the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t09299014.png" /> has finite area if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t09299015.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092990/t09299016.png" /> (cf. [[Tonelli theorem|Tonelli theorem]]).
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'''Definition'''
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Given a measurable function $f: \mathbb R^2\to\mathbb R$ we define the Tonelli-Cesari variation $f$ as
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\[
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V_{TC} (f) := \inf \left\{ V_T (g) : g = f \;\lambda\text{-a.e.}\right\}\, .
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\]
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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 {{Cite|AFP}}.  
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"L. Tonelli,   "Sur la quadrature des surfaces" ''C.R. Acad. Sci. Paris'' , '''182''' (1926) pp. 1198–1200</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"L. Tonelli,   "Sulla quadratura delle superficie" ''Atti Accad. Naz. Lincei'' , '''3'''  (1926)  pp. 357–363; 445–450; 633–658</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.G. Vitushkin,  "On higher-dimensional variations" , Moscow (1955) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Banach,  "Sur les lignes rectifiables et les surfaces dont l'aire est finie"  ''Fund. Math.'' , '''7''' (1925)  pp. 225–236</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S. Saks,  "Theory of the integral" , Hafner  (1952)  pp. 169  (Translated from French)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|AFP}}||      L. Ambrosio, N.  Fusco, D.  Pallara, "Functions of bounded  variations  and  free  discontinuity  problems". Oxford Mathematical  Monographs. The    Clarendon Press,  Oxford University Press, New York,  2000.      {{MR|1857292}}{{ZBL|0957.49001}}
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|-
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|valign="top"|{{Ref|Ce}}|| L. Cesari, "Sulle  funzioni a variazione limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), '''5''' (1936) pp. 299-313.
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|-
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|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
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|-
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|valign="top"|{{Ref|EG}}||    L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of    functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL,    1992. {{MR|1158660}} {{ZBL|0804.2800}}
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|-
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|valign="top"|{{Ref|Gi}}|| E. Giusti, "Minimal surfaces and functions of bounded variation", Birkhäuser, 1994.
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|-
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|valign="top"|{{Ref|Ro}}|| H.L. Royden,  "Real analysis" , Macmillan (1969). {{MR|0151555}} {{ZBL|0197.03501}}
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|-
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|valign="top"|{{Ref|To}}||  L. Tonelli, "Sulle funzioni di due variabili generalmente a variazione   limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), '''5''' (1936)  pp.  315-320. {{ZBL|0014.29606}}
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|-
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|}

Latest revision as of 10:54, 16 March 2023

2020 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].

References

[AFP] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Ce] L. Cesari, "Sulle funzioni a variazione limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), 5 (1936) pp. 299-313.
[Co] D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Gi] E. Giusti, "Minimal surfaces and functions of bounded variation", Birkhäuser, 1994.
[Ro] H.L. Royden, "Real analysis" , Macmillan (1969). MR0151555 Zbl 0197.03501
[To] L. Tonelli, "Sulle funzioni di due variabili generalmente a variazione limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), 5 (1936) pp. 315-320. Zbl 0014.29606
How to Cite This Entry:
Tonelli plane variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tonelli_plane_variation&oldid=16914
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article