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.