ON A DISCRETE VERSION OF TANAKA'S THEOREM FOR MAXIMAL FUNCTIONS.

In this paper we prove a discrete version of Tanaka's theorem for the Hardy-Littlewood maximal operator in dimension n = 1, both in the noncentered and centered cases. For the non-centered maximal operator Μ M we prove that, given a function f : Z → R of bounded variation, Var(Mf) ≤ Var(f), where Var(f) represents the total variation of f. For the centered maximal operator M we prove that, given a function f : Z → R such that f ϵ Μ1(Z), Var(Mf) ≤ C‖f‖Μе1(Z). This provides a positive solution to a question of HajΜlasz and Onninen in the discrete one-dimensional case. [ABSTRACT FROM AUTHOR]