Moving averages in the plane

Differentiating along rectangles in lacunary directions

“…The next result specifies the announced Theorem 1. It is mainly a consequence of the preceding proposition and some standard techniques as developed in Moonens and Rosenblatt [6]. Theorem 6.…”

Differentiating Orlicz spaces with rare bases of rectangles

“…Using the previous proposition, we can, using standard techniques developed e.g. in a previous work by the second and third authors [12] or in a paper by the current authors [3], obtain negative differentiation results in a range of Orlicz spaces for some differentiation bases of rectangles associated to various sets θ.…”

Averaging on $n$-dimensional rectangles

“…Following Stokolos [11] (and using the terminology introduced in Moonens and Rosenblatt [9]), we say that a family of standard dyadic rectangles in R n has finite width in case it is a finite union of families of rectangles totally ordered by inclusion, and that it has infinite width otherwise. Il follows from a general result by Dilworth [4] that a family of rectangles in R n has infinite width if and only if it contains families of incomparable (with respect to inclusion) rectangles having arbitrary large (finite) cardinality.…”

“…In order to prove Theorem 7, it is sufficient, according to Proposition 3, to prove the following lemma, improving on Stokolos techniques in [11] and [13], and using Rademacher functions as in [9]. Lemma 10.…”