On a discrete version of Tanaka's theorem for maximal functions

In this paper we prove a discrete version of Tanaka's Theorem \cite{Ta} for the Hardy-Littlewood maximal operator in dimension $n=1$, both in the non-centered and centered cases. For the discrete non-centered maximal operator $\wM $ we prove that, given a function $f: \Z \to \R$ of bounded variation, $$\Var(\wM f) \leq \Var(f),$$ where $\Var(f)$ represents the total variation of $f$. For the discrete centered maximal operator $M$ we prove that, given a function $f: \Z \to \R$ such that $f \in \ell^1(\Z)$, $$\Var(Mf) \leq C \|f\|_{\ell^1(\Z)}.$$ This provides a positive solution to a question of Haj{\l}asz and Onninen \cite{HO} in the discrete one-dimensional case.