Volume 444, 2007
Topics in Ergodic Theory and Harmonic Analysis: An
Ahmed I. Zayed
ABSTRACT. The aim of this introductory chapter is to give an overview of the
monograph and provide a synopsis of each chapter.
There are strong connections between harmonic analysis and ergodic theory
(the study of the long term behavior of dynamical systems). In particular, the
question of convergence is central to both areas. Indeed, using the Calderon transfer
principle, it is often possible to directly translate results from one of these areas to
the other. For example, because of the Calderon transfer principle, the local ergodic
theorem is a consequence of (and implies) the Lebesgue differentiation theorem.
Maximal functions and covering lemmas play a critical role in both areas, and tools
developed in the context of one of the areas will often prove useful to solve problems
in the other.
To illustrate this, observe that techniques developed by Nagel and Stein to gen-
eralize non-tangential convergence of harmonic functions have been used to study
moving ergodic averages. Furthermore, Fourier transform techniques developed to
study singular integrals have been used to study convergence of some non-standard
ergodic averages. In the other direction, techniques developed to study variation
of ergodic averages have been used to obtain similar results about singular integral
operators in harmonic analysis. Thus, each of the two areas enriches the other in
terms of both results and techniques.
A more recent example of the interaction between harmonic analysis and er-
godic theory is the proof of the spectacular result by Terence Tao and Ben Green
that the set of prime numbers contains arbitrarily long arithmetic progressions. The
breakthrough achieved by Tao and Green is attributed to their use and application
of techniques from ergodic theory and harmonic analysis to number theory.
This monograph is based on talks delivered by plenary speakers at a conference
on Harmonic Analysis and Ergodic Theory that took place at DePaul University,
Chicago, December 2-4, 2005. The conference was in honor of two members of
Mathematics Subject Classification.
Primary 42-06, 37-06; Secondary 42-02, 37-02.
Key words and phrases.
Borel foliations, Central Limit Theorem for Random Walks, Ergodic
Averages, Function Interpolation, Low Pass Filters, Nehari's Theorem for Polydisk, Restricted
Weak-type Operators, s-functions, Uniqueness Theorems for Multi-dimensional Trigonometric
Series, Wavelets .
@2007 American Mathematical Society