4

AHMED

I.

ZAYED

In this chapter, Gundy presents a new development of the probability and er-

godic theory treatment of this problem. It is shown that a trigonometric polynomial

that does not generate a scaling function can be modified in an arbitrarily small

neighborhood of its zeros so that the modified function does generate a scaling

function. This is illustrated by an examination of the Haar scaling function and

the "stretched" Haar function.

The author then treats the general problem where the function

m(~)

is assumed

to be continuous with "forbidden zeros." It is shown that a "defective" filter

m(~),

one that does not generate a scaling function, may be converted to a low-pass filter

by changing its modulus of continuity in a neighborhood its zero set.

Chapter Four introduces a new approach to a very general ergodic theorem on

foliations. What are foliations, anyway? D. Rudolph begins by defining a Borel

foliation of a Polish space

X

by

IRn

or

zn,

which is little different from the geometric

notion of a foliation. He chooses the following formulation:

DEFINITION

0.1. Suppose X is a standard Borel space and let F be either

!Rn

or

zn.

By a (Borel) F-foliation of

X,

we mean a Borel map

q:

X

X

F---+ X

with

the following properties:

( 1) For each fixed x

E

X,

4

restricted to { x} x F is a Borel isomorphism with

4(x,

0)

=

x.

(2) The sets

Fx

=

4( {

x} x

F)

partition

X

in that they are either disjoint or

identical.

(3) For

x'

E

Fx

set

4x,x{v)

=

i1

if

i1

is the unique vector with

4(x, v)

=

4(x',i1).

We require

4x,x'

to be an isometry of

F.

(4) For every r 0 there is an at most countable collection of Borel subsets

Ai

so that

4

restricted to

Ai

x

Br

is a Borel isomorphism and the images

4(Ai

x

Br)

cover all of X.

The sets

Fx

are called the "leaves" of the foliation and the sets

Ai

x

Br

are called

"pancake stacks."

Rudolph next shows how to take any Borel probability measure and diffuse it

onto the leaves of such a foliation, giving Borel leaf measures J-lx on each leaf. The

goal of this plan then is to attempt to prove convergence of averages of the form

JBR

j

dJ-lx

J-lx(BR)

for

f

E

L

1

(J-l) as

R

---+

oo, where BR is a ball of radius

R

in the leaf through x

centered at the origin.

Chapter Five is a short note by G. Weiland in which he reviews the work of

J.

Marshall Ash with focus on Ash's contributions to two main areas: convergence of

trigonometric and Fourier series in the mean and pointwise, in particular in higher

dimensions, and differentiation problems, in particular, the equivalence of different

notion of derivatives, such as Riemann, generalized Riemann, Peano, Schwartz, and

quantum derivatives, and their

LP

analogues.

In Chapter Six,

J.

Marshall Ash and G. Wang give a nice survey of recent

results on two uniqueness questions on multiple trigonometric series. The first

question deals

with trigonometric series that converge to zero and the second deals

with trigonometric series that converge to integrable functions. The first question

goes back to Cantor who showed that