On the Generalisation of the Hahn-Jordan Decomposition for Real C\`adl\`ag Functions
Abstract
For a real c\`adl\`ag function f and a positive constant c we find another c\`adl\`ag 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\`adl\`ag functions. We also present partial results for more general functions.
Turn this paper into a lesson
ArcXiv compiles a structured reading guide from this paper's metadata: plain-English importance, contributions, prerequisite concepts, which sections to read first, flashcards, and a quiz. Grounded in the abstract, never invented.