Namespaces
Variants
Actions

Difference between revisions of "Pierpont variation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
Line 1: Line 1:
One of the numerical characteristics of a function of several variables which can be considered as a multi-dimensional analogue of the [[Variation of a function|variation of a function]] of one variable. Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p0727201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p0727202.png" /> be given on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p0727203.png" />-dimensional parallelopipedon
+
{{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/p/p072/p072720/p0727204.png" /></td> </tr></table>
+
[[Category:Analysis]]
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p0727205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p0727206.png" />, be a subdivision of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p0727207.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p0727208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p0727209.png" /> equal segments by points
+
{{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/p/p072/p072720/p07272010.png" /></td> </tr></table>
+
A  generalization to functions of several variables of the [[Variation of a  function]] of one variable, proposed by Pierpont in {{Cite|Pi}}. 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 Pierpont variation is seldomly used  nowadays.
  
<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/p/p072/p072720/p07272011.png" /></td> </tr></table>
+
Consider  a rectangle $R:= [a_1, b_1]\times \ldots \times [a_n, b_n]\subset  \mathbb R^n$ and let
 +
$\Pi_k^m$ the set of points $a_k = a_k^0 < \ldots < a_k^{m+1}$ which subdivides $[a_k, b_k]$ in $m$ segments of equal length. These subdivisions generate a subdivision $\Sigma^m$ of the rectangle $R$ into $2^m$ closed rectangles $R^m_1, \ldots, R^m_{2^m}$ having equal side lengths.
  
These subdivisions generate a subdivision
+
'''Definition'''
 +
The Pierpont variation of a function $f:R\to \mathbb R$ is defined as
 +
\[
 +
\sup_m\; \frac{1}{m^{n-1}} \sum_{i=1}^{2^m} \omega \left(f, R^m_i\right)
 +
\]
 +
where $\omega (f, E)$ denotes the oscillation of the function $f$ over the set $E$, namely
 +
\[
 +
\omega (f, E) := \sup_E\; f - \inf_E\; f\, .
 +
\]
 +
If the Pierpont variation of $f$ is finite then one says that $f$ has bounded (finite) Pierpont variation.
  
<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/p/p072/p072720/p07272012.png" /></td> </tr></table>
+
If a function $f$ has bounded [[Arzela variation]] then it has also bounded Pierpont variation.
  
of the parallelopipedon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272013.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272014.png" /> parallelopipeda <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272015.png" /> with edges parallel to the coordinate axes.
 
 
Let
 
 
<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/p/p072/p072720/p07272016.png" /></td> </tr></table>
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272017.png" /> is the oscillation of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272019.png" /> (cf. [[Oscillation of a function|Oscillation of a function]]). 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/p/p072/p072720/p07272020.png" /></td> </tr></table>
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272021.png" />, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272022.png" /> is said to be of bounded (finite) Pierpont variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272023.png" />, and the class of all such functions is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272024.png" />. This definition was suggested by J. Pierpont [[#References|[1]]]. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272025.png" /> contains as a subset the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272026.png" /> of all functions of bounded [[Arzelà variation|Arzelà variation]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072720/p07272027.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Pierpont,  "Lectures on the theory of functions of real variables" , '''1''' , Dover, reprint (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Hahn,  "Reellen Funktionen" , '''1''' , Chelsea, reprint  (1948)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Ha}}|| H. Hahn,  "Theorie der reellen Funktionen" , '''1''' , Springer (1921).
 +
|-
 +
|valign="top"|{{Ref|Pi}}|| J. Pierpont,  "Lectures on the theory of functions of real variables" , '''1''' , Dover (1959).
 +
|-
 +
|}

Revision as of 12:28, 16 September 2012

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 Pierpont in [Pi]. 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 Pierpont 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 let $\Pi_k^m$ the set of points $a_k = a_k^0 < \ldots < a_k^{m+1}$ which subdivides $[a_k, b_k]$ in $m$ segments of equal length. These subdivisions generate a subdivision $\Sigma^m$ of the rectangle $R$ into $2^m$ closed rectangles $R^m_1, \ldots, R^m_{2^m}$ having equal side lengths.

Definition The Pierpont variation of a function $f:R\to \mathbb R$ is defined as \[ \sup_m\; \frac{1}{m^{n-1}} \sum_{i=1}^{2^m} \omega \left(f, R^m_i\right) \] where $\omega (f, E)$ denotes the oscillation of the function $f$ over the set $E$, namely \[ \omega (f, E) := \sup_E\; f - \inf_E\; f\, . \] If the Pierpont variation of $f$ is finite then one says that $f$ has bounded (finite) Pierpont variation.

If a function $f$ has bounded Arzela variation then it has also bounded Pierpont variation.


References

[Ha] H. Hahn, "Theorie der reellen Funktionen" , 1 , Springer (1921).
[Pi] J. Pierpont, "Lectures on the theory of functions of real variables" , 1 , Dover (1959).
How to Cite This Entry:
Pierpont variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pierpont_variation&oldid=13447
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article