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