2
AHMED
I.
ZAYED
the DePaul Department of Mathematical Sciences, J. Marshall Ash and Roger L.
Jones; Marshall Ash for his 65th birthday and Roger Jones for his retirement after
a long and distinguished career at DePaul.
The aim of this introductory chapter is to give an overview of the monograph
and provide a synopsis of each chapter. This volume contains ten chapters, four
chapters on ergodic theory and six on harmonic analysis, yet some chapters may
fall in either category. The chapters are grouped together in two parts arranged by
topics. The part on ergodic theory begins by an article on the work of Roger Jones
written by J. Rosenblatt and the part on harmonic analysis begins by an article on
J. Marshall Ash's work written by G. Welland.
Although the title of the first chapter is "The Mathematical Work of Roger
Jones," the content tells a different story. The chapter is actually a beautifully
written survey article on some aspects of ergodic theory using the work of Roger
Jones as a mirror to reflect the advances that transpired in the last twenty years
or so on solving a number of problems on the convergence of ergodic averages. To
explain some of these problems, let (X, j3, m) be a nonatomic probability space, and
letT be an invertible, measure-preserving, ergodic transformation of (X, j3, m). Let
f..L
be a probability measure on the integers Z and consider the weighted average
given by
J.Lf(x)
=
LJ.L(k)f(Tkx).
kEZ
Now, given a sequence
(J.Ln)
of probability measures on Z, one wants to answer the
following questions:
(a) When do the averages
f-Lnf
converge in
Lp-norm?
What is the limit oper-
ator?
(b) When do the averages
f-Lnf(x)
converge almost everywhere? For which
Lp-spaces would almost everywhere convergence occur for all
f?
(c) For which Lp-spaces does there exist a subsequence
(J.Ln,J
such that the
averages
f-Lnmf(x)
converge almost everywhere for all
f?
What character-
izes subsequences with this property?
One can also ask similar questions for operators given by convolutions. For a
specific instance of this, Rosenblatt discusses subsequences of convolutions of the
form
(¢n
*f),
where (¢n) is an approximate identity in L1(JR.) and
f
belongs to
Lp(JR.).
He shows that for each positive approximate identity (¢n) there exists a
subsequence
(¢nk)
such that, for each
p
with 1
p
oo and each
f
in
Lp(JR.),
lim
¢nk
*
f
=
f
almost everywhere.
k---+oo
Chapter Two is an article by
Y.
Derriennic and M. Lin on the central limit the-
orem for random walks that investigates the convergence of other types of averages
that converge to a centered normal probability distribution.
Let
11
=
{Pk : k
E
Z} be an ergodic probability distribution on Z, (S,
L:,
m), be
a non-atomic probability space, and
r :
S ____, S
be an ergodic probability preserving
transformation, with induced unitary operator Ton
L
2
(m).
Moreover, let
{Xn}
be
a Markov chain with state space
S
and consider the random walk of law v on the
orbits of
r
with corresponding Markov operator
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