TOPICS IN ERGODIC THEORY AND HARMONIC ANALYSIS: AN OVERVIEW
5
THEOREM 0.2 (Cantor). If the trigonometric series I:nE.Z eneinx converges to
zero for each x
E
T
=
[0, 27r),
in the sense that Sn
=
2:~=-n Ckeikx tends to zero
as n
-t
oo, then all Cn must be zero.
In 1905, Lebesgue proved the following more general version. If
lim ( c_ne-inx
+
Cneinx)
=
0, for all
X
E E,
n-+oo
where
E
C
T
has positive measure, then
lim v'ic-nl
2
+
icnl
2
=
0.
n-+oo
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
function
f
is defined as
S [f]
=
L
j
(n) einx,
nE.Z
A
1
!11"
where f (n)
= -
2
f (x) e-mxdx.
7r
-11"
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
S [f]
of that function.
This theorem is called the
£1
uniqueness theorem.
It
should be noted that if
f
E LP(T), 1
p
oo, its Fourier series converges to it almost everywhere. This
is to be contrasted with Kolmogorov example of an
L
1
(T)
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
~d,
x
=
(xb · · · ,xd) E Td, and n,x
=
n1x1
+ · · · +
ndxd. Let
S
=
L
an
=
L
an(x)
=
L
Cnein,x.
nE.Zd nE.Zd nE.Zd
The series
S
can be obtained as the limit of a number of different partial sums.
Five summation methods to add up
S
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
=
}I:;=l
m]
and for each
real number
r,
define
sr
=
2:
an,
lnl::;r
to be the r-th spherical partial sum of
S.
We say that
S
converges spherically to
s if limr-+oo
sr
=
s
For two multi-indices
m
and n, we say
m
~
n iff or all 1 :::; j :::; d,
m
1
~
n1. For
any real number
r,
let
r.
=
(r, · · · , r).
DEFINITION 0.5 (Square Convergence).
Form~
Q, call
Bm
=
L
an
-m::;n::;m
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