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
Previous Page Next Page