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Effectively Absolute Continuity and Effective Jordan Decomposability (2002)
BibTeX
@MISC{Zheng02effectivelyabsolute,
author = {Xizhong Zheng and Robert Rettinger and Burchard von Braunmühl},
title = { Effectively Absolute Continuity and Effective Jordan Decomposability},
year = {2002}
}
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Abstract
Classically, any absolute continuous real function is of bounded variation and hence can always be expressed as a difference of two increasing continuous functions (socalled Jordan decomposition). The effective version of this result is not true. In this paper we give a sufficient and necessary condition for computable real functions which can be expressed as two computable increasing functions (effectively Jordan decomposable, or EJD for short). Using this condition, we prove further that there is a computable real function which has a computable modulus of absolute continuity but is not EJD. The polynomial time version of this result holds accordingly too and this gives a negative answer to an open question of Ko in [6].
