Abstract
The Jordan Decomposition Theorem says that any real function of bounded variation can be expressed as a difference of two increasing functions. This paper investigates the effective version of Jordan decomposition and discusses the properties of variation of computable real functions. First we show that the effective version of Jordan decomposition does not hold in general. Then we give a sufficient and necessary condition for computable real functions of bounded variation which have an effective Jordan decomposition. Applying this condition, we construct a computable real function which has a computable modulus of absolute continuity (hence of bounded variation) but is not effectively Jordan decomposable. Finally, we prove a version of this result restricted to polynomial time which answers negatively an open question of Ko in [8].
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Zheng, X., Rettinger, R. Effective Jordan Decomposition. Theory Comput Syst 38, 189–209 (2005). https://doi.org/10.1007/s00224-004-1193-z
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DOI: https://doi.org/10.1007/s00224-004-1193-z