View source for Pierpont variation

{{MSC|26B30|26A45}} 

[[Category:Analysis]]

{{TEX|done}}

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.

Consider  a rectangle $R:= [a_1, b_1]\times \ldots \times [a_n, b_n]\subset  \mathbb R^n$ and let
$\Pi_k^m$ be 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====
{|
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|valign="top"|{{Ref|Ha}}|| H. Hahn,   "Theorie der reellen Funktionen" , '''1''' , Springer  (1921).   JFM {{ZBL|48.0261.09}} 
|-
|valign="top"|{{Ref|Pi}}|| J. Pierpont,   "Lectures on the theory of functions of real variables" , '''1''' , Dover (1959).  {{MR|0105467}}  JFM {{ZBL|36.0346.01}} 
|-
|}