R. Nair

Copyright © 2005 R. Nair. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let a=(ai)i=1∞ be a strictly increasing sequence of natural numbers and let 𝒜 be a space of Lebesgue measurable functions defined on [0,1). Let {y} denote the fractional part of the real number y. We say that a is an 𝒜∗ sequence if for each f∈𝒜 we set AN(f,x)=(1/N)∑i=1Nf({aix})(N=1,2,…), then limN→∞  AN(f,x)=∫01f(t)dt, almost everywhere with respect to Lebesgue measure. Let Vq(f,x)=(∑N=1∞|AN+1(f,x)−AN(f,x)|q)1/q(q≥1). In this paper, we show that if a is an (Lp)∗ for p>1, then there exists Dq>0 such that if ‖f‖p denotes (∫01|f(x)|pdx)1/p, ‖Vq(f,·)‖q≤Dq‖f‖p(q>1). We also show that for any (L1)∗ sequence a and any nonconstant integrable function f on the interval [0,1), V1(f,x)=∞, almost everywhere with respect to Lebesgue measure.