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One of the three components in the [[Lebesgue decomposition|Lebesgue decomposition]] of a function of bounded variation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j0544101.png" /> be a [[Function of bounded variation|function of bounded variation]] on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j0544102.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j0544103.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j0544104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j0544105.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j0544106.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j0544107.png" /> is called the jump of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j0544108.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j0544109.png" /> from the left and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441010.png" /> the jump of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441011.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441012.png" /> from the right. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441013.png" />, then
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{{MSC|26A45|28A15}}
  
<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/j/j054/j054410/j05441014.png" /></td> </tr></table>
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[[Category:Analysis]]
  
is called the jump of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441015.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441016.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441017.png" /> be the sequence of all points of discontinuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441019.png" /> and put
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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/j/j054/j054410/j05441020.png" /></td> </tr></table>
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One of the three components in the [[Lebesgue decomposition|Lebesgue decomposition]] of a [[Function of bounded variation|function of bounded variation]] depending on one real variable, rediscovered and extended to functions of several variables by De Giorgi and its school (see {{Cite|AFP}}). According to Lebesgue, if $I\subset\mathbb R$ is an interval, a right-continuous function of bounded variation $f: I\to\mathbb R$ can be decomposed in a canonical way into three functions $f_a+f_j+f_c$. The function $f_j$ is the ''jump part'' of $f$ (or jump function of $f$, using the therminology of Lebesgue {{Cite|Le}}) and it is defined by
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\begin{equation}\label{e:jump}
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f_j (x)= \sum_{y\leq x}\, f (y^+) - f(y^-)\, .
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\end{equation}
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Therefore its [[Generalized derivative|distributional derivative]] is the atomic part of the distributional dertivative of $f$. The jump part can also be characterized as the (right-continuous) function $g$ with smallest variation such that $f-g$ is continuous (cp. with Remark 1 at page 163 of {{Cite|Le}}). Observe therefore that the [[Variation of a function|total variation]] of $f$ is the sum of the total variation of $f_j$ and the total variation of $f-f_j$.  
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441021.png" /> is called the jump function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441022.png" />. Note that the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441023.png" /> is a continuous function of bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441024.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441026.png" /> is the variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441027.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441028.png" /> (cf. [[Variation of a function|Variation of a function]]). Moreover,
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The term ''jump function'' is used also for those functions of bounded variation $f$ such that $f=f_j$, i.e. so that their distributional derivative is a purely atomic measure.  
 
 
<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/j/j054/j054410/j05441029.png" /></td> </tr></table>
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Lebesgue,  "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars  (1928)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,  "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441030.png" /> is also called the saltus function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441031.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441032.png" /> of bounded variation that equals its jump function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054410/j05441033.png" /> is itself often called a jump function.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Szökefalvi-Nagy,  "Introduction to real functions and orthogonal expansions" , Oxford Univ. Press (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>
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===References===
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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|DS}}||      N. Dunford, J.T. Schwartz, "Linear operators. General theory",    '''1''', Interscience (1958)  {{MR|0117523}} {{ZBL|0635.47001}}
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|-
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|valign="top"|{{Ref|Ha}}|| P.R. Halmos,  "Measure theory", v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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|-
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|valign="top"|{{Ref|Le}}||  H. Lebesgue,  "Leçons sur l'intégration et la récherche des fonctions  primitives", Gauthier-Villars  (1928).
 +
|-
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|valign="top"|{{Ref|Na}}|| I.P. Natanson,  "Theorie der Funktionen einer reellen Veränderlichen" ,  H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)
 +
|-
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|valign="top"|{{Ref|Sa}}|| S. Saks,  "Theory of the integral" , Hafner  (1952) {{MR|0167578}} {{ZBL|63.0183.05}}
 +
|-
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|valign="top"|{{Ref|Sz}}|| B. Szökefalvi-Nagy,  "Introduction to real functions and orthogonal expansions" , Oxford Univ. Press (1965)
 +
|-
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|}

Revision as of 08:59, 1 September 2012

2020 Mathematics Subject Classification: Primary: 26A45 Secondary: 28A15 [MSN][ZBL]

One of the three components in the Lebesgue decomposition of a function of bounded variation depending on one real variable, rediscovered and extended to functions of several variables by De Giorgi and its school (see [AFP]). According to Lebesgue, if $I\subset\mathbb R$ is an interval, a right-continuous function of bounded variation $f: I\to\mathbb R$ can be decomposed in a canonical way into three functions $f_a+f_j+f_c$. The function $f_j$ is the jump part of $f$ (or jump function of $f$, using the therminology of Lebesgue [Le]) and it is defined by \begin{equation}\label{e:jump} f_j (x)= \sum_{y\leq x}\, f (y^+) - f(y^-)\, . \end{equation} Therefore its distributional derivative is the atomic part of the distributional dertivative of $f$. The jump part can also be characterized as the (right-continuous) function $g$ with smallest variation such that $f-g$ is continuous (cp. with Remark 1 at page 163 of [Le]). Observe therefore that the total variation of $f$ is the sum of the total variation of $f_j$ and the total variation of $f-f_j$.

The term jump function is used also for those functions of bounded variation $f$ such that $f=f_j$, i.e. so that their distributional derivative is a purely atomic measure.

References

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
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory", 1, Interscience (1958) MR0117523 Zbl 0635.47001
[Ha] P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Le] H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives", Gauthier-Villars (1928).
[Na] I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)
[Sa] S. Saks, "Theory of the integral" , Hafner (1952) MR0167578 Zbl 63.0183.05
[Sz] B. Szökefalvi-Nagy, "Introduction to real functions and orthogonal expansions" , Oxford Univ. Press (1965)
How to Cite This Entry:
Jump function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jump_function&oldid=27826
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article