On strong uniform distribution IV
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Let a = (ai )∞ i=1 be a strictly increasing sequence of natural numbers and let Ꮽ 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 Ꮽ∗ sequence if for each f ∈ Ꮽ we set AN ( f ,x) = 1 (1/N) Ni=1 f ({ai x}) (N = 1,2,...), then limN →∞ A N ( f ,x) = 0 f (t)dt, almost everywhere q 1/q (q ≥ with respect to Lebesgue measure. Let Vq ( f ,x) = ( ∞ N =1 |AN+1 ( f ,x) − AN ( f ,x)| ) p ∗ 1). In this paper, we show that if a is an (L ) for p > 1, then there exists Dq > 0 such that 1 if f p denotes ( 0 | f (x)| p dx)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. 1. Introduction Let a = (ai )∞ i=1 be a strictly increasing sequence of natural numbers and let Ꮽ be a space of Lebesgue measurable functions defined on [0,1). Let { y } denote the fractional part of the real number y. Following Marstrand [3] we say that a is an Ꮽ∗ sequence if for each f ∈ Ꮽ we set AN ( f ,x) =
N 1 f ai x N i=1
(N = 1,2,...),
(1.1)
then lim AN ( f ,x) =
N →∞
1 0
f (t)dt,
(1.2)
almost everywhere with respect to Lebesgue measure. We know that any strictly increas∗ ing sequence of integers (an )∞ n=1 is a C sequence where C denotes the space of continuous functions on [0,1). This is because of Weyl’s theorem [9] that for any strictly increasing sequence of integers (an ), the fractional parts ({an x})∞ n=1 are uniformly distributed modulo one for almost all x with Lebesgue measure. On the other hand as shown in [3], the sequence an = n (n = 1,2,...) is not an (L∞ )∗ . There are however examples of sequences of Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:3 (2005) 319–327 DOI: 10.1155/JIA.2005.319
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On strong uniform distribution IV
integers that are (L p )∗ p ≥ 1 and indeed (L1 (logL)k )∗ . These are constructed by primarily ergodic means [3, 4, 5, 6, 8]. Here of course L p denotes the space of functions f such that 1 the norm f p = 0 | f (x)| p dx is finite and L1 (log+ L)k denotes the space of L1 functions 1 such that 0 | f |(log+ | f |)k−1 (x)dx is finite. As usual log+ x denotes log max(1,x). While it is possible to pose many of the questions considered in this subject and indeed this paper for many Banach spaces of measurable functions Ꮽ, they are perhaps primarily of interest in the context of L p spaces and perhaps L1 (log+ L)k . Note that
Span ∪ p>1 L p ⊆ L log+ L
d
⊆ L1 ,
(1.3)
where the inclusions are strict in both cases for each d ≥ 1. Here Span(A) denotes the linear space spanned by the set A. A natural question which arises is whether if (1.2) is known for a particular sequence a = (an )∞ n=1 and a particular function f , anything can 1 be said about the rate at which the averages (AN ( f ,x))∞ N =1 converge to 0 f (t)dt almost everywhere. Using [1, Theorem 1] and the Denjoy-Koks
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