DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

CONVERGENCE OF SERIES OF TRANSLATIONS
BY POWERS OF TWO
GUODONG LI
We will give a necessary and sufficient condition on c = (c n )suchthat ∑ c n f {2 n x} converges
in L 2 -norm for all f ∈ L 0 2 or L 2 .
Copyright © 2006 Guodong Li. 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.
1. Introduction
Let f (x) be a real or complex function defined on the interval [0,1]. Let {x}=x mod (1).
Hence, f {x} means f ({x}). Let L 2 be the space of all square integrable functions on [0,1]
and L 0 2 is the space of all zero-mean functions in L 2 .LetL 2 (2) be the set of all functions
in L 2 with Fourier coefficients supported by powers of 2, that is,
L 2 (2) =
{
f (x) =
∞∑
k=−∞
a k exp ( sign(k)2πi2 |k| x ) : a = ( a n
)
∈ l2
}. (1.1)
Let L 0 2(2) denote the set of functions in L 2 (2) with a 0 = 0andletl2 0 denote the set of
sequences a = (a n ) ∈ l 2 with a 0 = 0.
Diophantine approximations have been well studied by many authors. It is well known
that (1/n) ∑ n
k=1 f {nx} converges in L 2 -norm to ∫ 1
0 f (x)dx. Khinchin conjectured that
(1/n) ∑ n
k=1 f {nx} converges almost everywhere to ∫ 1
0 f (x)dx for all f ∈ L ∞ . But this was
disproved by Marstrand (see [4]). Let c n be a sequence of real or complex numbers. The
consideration of a.e. convergence and L 2 -norm convergence of series ∑ c n f {nx} is another
natural problem in Diophantine approximations. Some sufficient conditions for
this series to converge are given in [5] and category counterexamples can be found in
[2]. There are similar considerations in ergodic theory, too. Several authors have used
Rokhlin’s lemma and inductive constructions to provide counterexamples to the convergence
of this series (see [2, 5, 3, 1]).
The series ∑ c n f {2 n x} is a lacunary series, which is clearly closely related to the Diophantine
series ∑ c n f {nx}. Using the techniques in [5], it is easy to give a sufficient
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Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 667–676