On strong uniform distribution III
Let a = ( a i 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 ϵ A lim N→∞ 1 N ∑ i=1 N f(<a ix>)= ∫ 0...
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| Veröffentlicht in: | Indagationes mathematicae Jg. 14; H. 2; S. 233 - 240 |
|---|---|
| 1. Verfasser: | |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Elsevier B.V
23.06.2003
|
| ISSN: | 0019-3577, 1872-6100 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | Let
a = (
a
i
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 ϵ
A
lim
N→∞
1
N
∑
i=1
N
f(<a
ix>)=
∫
0
1
f(t)dt,
almost everywhere with respect to Lebesgue measure. Let
a
i=(a
1,i)
∞
i=1, …,a
k=(a
k,i)
∞
i=1,
denote finitely many (
L
1)
∗ sequences, and for a sequence
a, let
G
a(u)=|{i:a
i ≤ u}|,
where for a finite set
A we have used |
A| to denote its cardinality. Also let
a
1 ∘ … ∘
a
k
denote the set
{b
1…b
k : b
1ϵa
k},
counted with multiplicity and ordered by absolute value. Suppose there exists
K > 0 such that for all
u ≥ 1
|G
a
1
(u)|…|G
a
k
(u)|≤|G
a
1
compfn;…compfn;
a
k
(u)|.
Then if log
+|
x| = log max(1,|
x|) we show that
a
1 ∘ … ∘
a
k
is an (
L(
log
+
L)
k−1
)
∗ sequence. |
|---|---|
| ISSN: | 0019-3577 1872-6100 |
| DOI: | 10.1016/S0019-3577(03)90007-3 |