TOPICS IN ERGODIC THEORY AND HARMONIC ANALYSIS: AN OVERVIEW 3
The authors show that under the assumption that the expected value
E(v)
0, iff
E (I-
P)
112
L2(m)
=
JI- PL2(m),
then the sequence
Jn
L::~=l
f(Xk)
converges in distribution to a normal distribution with variance independent of
x.
The result is then extended to the non-centered case, i.e.,
E(v)
=f.
0, when
f
E
JI- PLr(m)
for some
r
2, and to a contraction operators in reflexive
Banach spaces.
Chapter Three by Gundy is a good example of how to combine techniques
from ergodic theory and harmonic analysis to produce interesting results. The
construction of an orthonormal wavelet basis of L
2
(JR)
of the form
{
'l/Jm,n(x)
=
Tm/2'¢ (Tmx-
n)} ,
m,nE~
in the setting of multiresolution analysis, is based on the construction of a scal-
ing function ¢(x) E L 2
(JR),
whose translates {¢(x-
n)}nE~
form an orthonormal
basis of a subspace of L
2
(JR).
This condition on ¢ is equivalent to requiring that
LkE~ 1¢(~
+
k)j2
=
1 a.e., where c,b is the Fourier transform of¢. The function¢
is required to satisfy the self similarity relation
¢(x)
=
J2
L
hn¢(2x- n),
nE~
which, when we apply the Fourier transform to it, yields
where
¢(~)
=
m(~/2)¢(~/2),
m(~) =
_1
I:
hne-ine.
yl2nE~
The function
m,
which is periodic with period
21r,
is called a low-pass filter. More-
over,
¢(
~) is required to satisfy the following continuity condition at zero:
lim
l¢(~/2jW
=
1
a.e.
J---+00
When, for a given m(~), a solution ¢(~) exists with these two properties, we
say that
m(~)
generates the scaling function ¢(x). To construct a wavelet with
compact support, it suffices to construct a compactly supported scaling function¢.
It is then easy to see that only finitely many
hn
are non zero; hence,
m
becomes a
trigonometric polynomial.
A basic question to ask is this: What are the necessary and sufficient conditions
for
m(~)
to be a low-pass filter? In general, whether a candidate
m(~)
generates a
scaling function depends crucially on the position of its zeros, and on the smoothness
of
m(~)
in a neighborhood of these zeros. Theorems of Cohen and Lawton answer
this question completely when
m(~)
is a trigonometric polynomial. In the years
following the publication of their results, claims were made to the effect that their
criteria were necessary and sufficient for the case when
m(~)
was merely continuous.
However, in a recent publication, Dobric, Gundy and Hitczenko showed that
the Cohen-Lawton condition failed to be necessary in the larger class of continuous
functions. The construction of the counterexample involved a probabilistic calcu-
lation and a coding of
JR1
into a one-sided sequence space. The use of probability
and ergodic theory to solve this problem is not accidental and it is something that
is not apparent in the Cohen-Lawton theorem.
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