TOPICS IN ERGODIC THEORY AND HARMONIC ANALYSIS: AN OVERVIEW
THEOREM 0.2 (Cantor). If the trigonometric series I:nE.Z eneinx converges to
zero for each x
in the sense that Sn
2:~=-n Ckeikx tends to zero
oo, then all Cn must be zero.
In 1905, Lebesgue proved the following more general version. If
lim ( c_ne-inx
0, for all
has positive measure, then
The second question started with a theorem of de la Vallee-Poussin that can be
stated as follows. But first, recall that the Fourier series of a Lebesgue integrable
is defined as
where f (n)
f (x) e-mxdx.
THEOREM 0.3 (de la Vallee-Poussin). If a one dimensional trigonometric series
I:nE.Z eneinx converges to an everywhere finite, Lebesgue integrable function f, then
it is the Fourier series
of that function.
This theorem is called the
should be noted that if
E LP(T), 1
oo, its Fourier series converges to it almost everywhere. This
is to be contrasted with Kolmogorov example of an
function whose Fourier
series diverges everywhere.
Generalizing these classical results to higher dimensions raises a number of
different interpretations and possibilities. First, let us introduce the following no-
tation and definitions. Let an= an(x)
Cnein,x, where n
(n1, · · · , nd) E
(xb · · · ,xd) E Td, and n,x
+ · · · +
nE.Zd nE.Zd nE.Zd
can be obtained as the limit of a number of different partial sums.
Five summation methods to add up
are considered, Spherical, Square Conver-
gence, Unrestrictedly Rectangular Convergence, Restrictedly Rectangular Conver-
gence, and One-Way Iterative Convergence. Different methods of summation may
lead to different answers. The first three methods are defined as follows.
DEFINITION 0.4 (Spherical Summation). Let lml
and for each
to be the r-th spherical partial sum of
We say that
converges spherically to
s if limr-+oo
For two multi-indices
and n, we say
n iff or all 1 :::; j :::; d,
any real number
(r, · · · , r).
DEFINITION 0.5 (Square Convergence).