On the Generalisation of the Hahn-Jordan Decomposition for Real C\`adl\`ag Functions

TL;DR

This paper extends the Hahn-Jordan decomposition to real càdlàg functions by introducing truncated variations that allow for minimal total variation approximations, even when the original function has infinite variation.

Contribution

It generalizes the Hahn-Jordan decomposition for càdlàg functions using truncated variations, providing a method for minimal variation approximation with finite measures.

Findings

Introduces truncated variation concepts for càdlàg functions.

Provides explicit solutions for minimal variation approximation.

Extends the decomposition to more general functions.

Abstract

For a real c\`{a}dl\`{a}g function f and a positive constant c we find another c\`{a}dl\`{a}g function, which has the smallest total variation pos- sible among all functions uniformly approximating f with accuracy c/2. The solution is expressed with the truncated variation, upward truncated variation and downward truncated variation introduced in [L1] and [L2]. They are always finite even if the total variation of f is infinite, and they may be viewed as the generalisation of the Hahn-Jordan decomposition for real c\`{a}dl\`{a}g functions. We also present partial results for more general functions.