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Difference between revisions of "Banach indicatrix"

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N(y,f) = +\infty,
 
N(y,f) = +\infty,
 
$$
 
$$
<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/b/b015/b015150/b0151507.png" /></td> </tr></table>
 
  
 
and if it has no roots, then
 
and if it has no roots, then
  
<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/b/b015/b015150/b0151508.png" /></td> </tr></table>
+
$$
 +
N(y,f) = 0.
 +
$$
  
 
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b0151509.png" /> was defined by S. Banach [[#References|[1]]] (see also [[#References|[2]]]). He proved that the indicatrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515010.png" /> of any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515011.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515012.png" /> is a function of Baire class not higher than 2, and
 
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b0151509.png" /> was defined by S. Banach [[#References|[1]]] (see also [[#References|[2]]]). He proved that the indicatrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515010.png" /> of any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515011.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515012.png" /> is a function of Baire class not higher than 2, and

Revision as of 05:07, 15 May 2015

multiplicity function, of a continuous function $y=f(x)$, $a\leq x\leq b$

An integer-valued function $N(y,f)$, $-\infty < y < \infty$, equal to the number of roots of the equation $f(x)=y$. If, for a given value of $y$, this equation has an infinite number of roots, then $$ N(y,f) = +\infty, $$

and if it has no roots, then

$$ N(y,f) = 0. $$

The function was defined by S. Banach [1] (see also [2]). He proved that the indicatrix of any continuous function in the interval is a function of Baire class not higher than 2, and

(*)

where is the variation of on . Thus, equation (*) can be considered as the definition of the variation of a continuous function . The Banach indicatrix is also defined (preserving equation (*)) for functions with discontinuities of the first kind [3]. The concept of a Banach indicatrix was employed to define the variation of functions in several variables [4], [5].

References

[1] S. Banach, "Sur les lignes rectifiables et les surfaces dont l'aire est finie" Fund. Math. , 7 (1925) pp. 225–236
[2] I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)
[3] S.M. Lozinskii, "On the Banach indicatrix" Vestnik Leningrad. Univ. Math. Mekh. Astr. , 7 : 2 pp. 70–87 (In Russian)
[4] A.S. Kronrod, "On functions of two variables" Uspekhi Mat. Nauk , 5 : 1 (1950) pp. 24–134 (In Russian)
[5] A.G. Vitushkin, "On higher-dimensional variations" , Moscow (1955) (In Russian)


Comments

More generally, for any mapping define analogously. Then, let be a separable metric space and let be -measurable for all Borel subsets of . Let for and let be the measure on defined by the Carathéodory construction from . Then

for every Borel set . Cf. [a1], p. 176 ff. For significant extension of (*), cf. [a2].

References

[a1] H. Federer, "Geometric measure theory" , Springer (1969)
[a2] H. Federer, "An analytic characterization of distributions whose partial derivatives are representable by measures" Bull. Amer. Math. Soc. , 60 (1954) pp. 339
How to Cite This Entry:
Banach indicatrix. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Banach_indicatrix&oldid=36407
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article