Pierpont variation

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).