Remarks on the effective Jordan decomposition

Abstract

According to the Jordan Decomposition Theorem, for any function f:[0,1]→R of bounded variation there exists a pair of nondecreasing functions f1,f2:[0,1]→R with f=f1−f2. Such a pair is called a Jordan decomposition of f. Ko (1991) and Zheng and Rettinger (2005) have analyzed computability-theoretic and complexity-theoretic aspects of this theorem. We complement their observations by three new results. One of our results says that there exists a polynomial time computable, absolutely continuous function f:[0,1]→R with polynomial modulus of absolute continuity such that, on the one hand, the so-called standard Jordan decomposition of f is computable but, on the other hand, neither the standard Jordan decomposition of f nor any other Jordan decomposition of f can be computed easily, say, in polynomial time, or in exponential time.