The play operator, the truncated variation and the generalisation of the Jordan decomposition

摘要

The play operator minimalizes the total variation on intervals, [0,T],T>0, of functions approximating uniformly given regulated function with given accuracy and starting from a given point. In this article, we link the play operator with the so-called truncated variation functionals, introduced recently by the second-named author, and provide a semi-explicit expression for the play operator in terms of these functionals. Generalisation for time-dependent boundaries is also considered. This gives the best possible lower bounds for the total variation of the outputs of the play operator and its Jordan-like decomposition. Copyright (c) 2014 John Wiley & Sons, Ltd.