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ALMOST PERIODIC
FUNCTIONS

A. S. BESIOOVITOH
Fellow of Trinity College
Cambridge

CAM·BRIDGE
AT THE UNIVERSITY PRESS

DOVER PUBLICATIONS, INC.

This new Dover edition is a
reiseue of the first edition republished
through epeci sl permission of Cambridge University Press.
Copyright 1954 by Dover Publications, Inc.

Printed and Bound in the United States of America

CONTENTS
page vu

Preface
Introduction
Chapter I.

IX

UNIFORMLY ALMOST PERIODIC FUNCTIONS:

§ 1. Definition and elementary properties
2. Normality of tt.a.p. functions
3. Mean values of u.a.p. functions and their Fourier
series
4. Fundamental theorem of the theory of u.a.p.
functions
5. Polynomial approxima:tion to u.a.p. functions .
6. Limit periodic functions .
7. Base of u.a.p. functions. Connection of u.a.p.
functions with limit periodic functions of
several variables
8. Summation of Fourier series of u.a.p. functions
by partial sums
9. Bochner-Fejer summation of u.a.p. functions .
10. Some particular cases of Fourier series of u.a.p.
functions
11. Aritqmetical nature of translation numbers
12. U.a.p. functions of two variables

Chapter II.

1
10
12
21
29
32

34
38
46
51
52
59

GENERALISATION OF ALMOST PERIODIC
FUNCTIONS:

Introduction

67

§ 1. Auxiliary theorems and formulae

68
70
79
82

2. General closures and general almost periodicity
3. S a.p:functions
4. W a.p. functions

CONTENTS

vi

§ 5. SP a.p. and WP a.p. functions (p > 1)
6.
7.
8.
9.

.page 88
B a.p. functions
91
J3r a.p. functions
100
Algorithm for polynomial approximation
104
Parseval equation and Riesc-Fischer theorem •
109

Appendix. jj a.p.

FUNCTIONS

113

Chapter III. ANALYTIC ALMOST PERIODIC FUNCTIONS:
§ 1. Some auxiliary theorems in the theory of analytic
functions
2. Definition of analytic almost periodic functions
and their elementary properties
3. Dirichlet series
4. Behaviour of u.a.p. functions at u= co
5. On the behaviour of analytic functions outside the
strip of uniform almost periodicity
6. On the behaviour of analytic functions on the
boundary of the strip of uniform almost
periodicity
Memoirs referred to in the text

130
141
147
158
163

169
179

PREFACE
THE theory of almost periodic functions, created by H. Bohr,
has now completed two stages of its development.
Almost periodicity, as a structural property, is a generalisation
of pure periodicity and Bohr's original methods for establishing
the fundamental results of the theory were always based on
reducing the problem to a problem of purely periodic functions.
But though the underlying idea of Bohr's method was clear and
simple, the actual proofs of the main results were very difficult
and complicated. New methods were given by N. Wiener and
H. Weyl, by which the results were arrived at in a much shorter
way. But these methods have lost the elementary character of
Bohr's methods. It was C. de la Vallee Poussin who succeeded
in giving a new proof (based partly on H. Weyl's idea), which
was very short and at the same time based entirely on classical
results in the theory of purely periodic functions.
This represents one stage in the development of the theory
of almost periodic functions.
Bohr's theory of a.p. functions was restricted to the class of
uniformly continuous functions. Then efforts were directed to
generalisations of the theory. Thanks to the work of W. Stepanoff, N. Wiener, H. Weyl, H. Bohr and others, generalisations
may be considered to have reached a certain completeness.
This was the second stage in the development of the theory.
These circumstances suggest that the present moment is not
unfavourable for writing an account of the theory.
In Chapter 1 of this account we develop the fundamental
part of the theory of a.p. functions of a real variable-the theory
of uniformly a.p. functions. In the main problems of this chapter
we adopt the methods of H. Bohr, de la Vallee Poussin, Weyl
and Bochner.
Chapter II is devoted to a systematic investigation of generalisations of the theory.

viii

PREFACE

In Chapter III we develop the theory of analytic a.p. functions,
which in essentials remains unaltered as it was published by
H. Bohr.
This account is not encyclopaedic. Our aim is to give the
fundamental results of the theory, and we have omitted all
discussion of certain special problems. Thus the work of Bohr,
Neugebauer, Walter and Bochner on differential equations and
difference equations has not been considered in this book, nor
has the theory of harmonic a.p. functions developed by J. Favard
in his interesting paper. For all these questions the reader is
referred to the original papers.
A. S.B.

INTRODUCTION
theory of almost periodic functions was created and developed in its main features by H. Bohr during the last decade.
Like many other important math~matical discoveries it is connected with several branches of the modern theory of functions.
On the one hand, almost periodicity as a structural property of
functions is a generalisation of pure periodicity: on the other
hand, the theory of almost periodic functions opens a way of
studying a wide class of trigonometric series of the general type
and of exponential series (Dirichlet series), giving in the latter
case important contributions to the general problems of the
theory of analytic functions.
Almost periodicity is a generalisation of pure periodicity: the
general property can be illustrated by means of the particular
example
f (x) =sin 27rx +sin 2'77'x .V2.

THE

This function is not periodic : there exists no value of 7' which
satisfies the equation
f(x + 7') = f(x)
for all values of x. But we can establish the existence of
numbers for which this equation is approximately satisfied with
an arbitrary degree of accuracy. For given any e > 0 as small as
we please we can always find an integer 7' such that 7' .V2 differs
from another integer by less than e/2'77'. It can be pro.ved that
there exist infinitely many such numbers 7', and that the difference between two consecutive ones is bounded. For each of
these numbers we have
f(x +

7')

=sin 2'77' (x + 7') +sin 27r (x + 7') .V2
... sin 2'77'x +sin (2'77'x .V2 +Be)

(I fJ I~ 1)

=J(x)+B'e.

(18'1~1)

INTRODUCTION

Almost periodicity of a function j(x) in general is defined by
this property :
The equation
j(x+-r)=J(x)
is satisfied with an arbitrary degree of accuracy by infinitely
many values of -r, these values being spread over the whole
range from -co to +co in such a way as not to leave empty
intervals of arbitrarily great length.
Almost periodicity is a deep structural property of functions
which is invariant with respect to the operations of addition
(subtraction) and multiplication, and also in many cases with
respect to division, differentiation, integration and other limiting processes.
To the structural affinity between almost periodic functions
and purely periodic functions may be added an analytical similarity. To any almost periodic function corresponds a "Fourier
series" of the type of a general trigonometric series
00

(1)

~ AneiAnz
n-1

j(x)"""'

(An being real num hers and An real or complex): it is obtained
from the function by the same formal process as in the case of
purely periodic functions (namely, by the method of undetermined coefficients and term-by-term integration). The series (1)
need not converge to f(x), but there is a much closer connection
between the series and the function than we have yet seen. ln
the first place Parseval's equation is true, i.e.

M

{1/(x) 12} =~I An 12,

from which follows at once the uniqueness theorem, according
to which there exists at most one almost periodic function having
a given trigonometric series for its Fourier series. Parseval's
equation constitutes the fundamental theorem of the theory of
almost periodic functions. Further, the series (1) is "summable
to f (x)," in the sense that there exists a sequence of polynomials
00

I

n-1

p~lAneiAnz

(k=1,2, ... )

INTRODUCTION

xi

(where 0 ~ p,. .:i 1, and where for each k only a finite number of
the factors p differ from zero) which
(a) converge to f (x) uniformly in x, and
(b) converge formally to the series (1),
by which is meant that for each n
p~l.-..-1, as k-+oo.
Conversely, any trigonometric polynomial is an almost periodic
function, and so is the uniform limit of a sequence of trigonometric polynomials. It is easily proved that the Fourier series
of such a limit function is the formal limit of the sequence of
trigonometric polynomials. Thus the Class of Fourier series of
almost periodic functions consists of all trigonometric series
of the general type
:$A,.eiA,.o:,
to which correspond uniformly convergent sequences of polynomials of the type
:$p~IA,.eiA,.o:,
(k = 1, 2, ... )
formally convergent to the series. Thus the theory of almost
periodic functions opened up for study a class of general trigonometric series : the extent of this class will be discussed later on.
The first investigations of trigonometric series other than
purely periodic ones were carried out by Bohl. He considered
the class of functions represented by series of the form
:$A~. "•· ... , nz ei (nl"'.+naal+... +nzav "',
(n)

where a 1 , a2 , ••• , a1 are arbitrary real numbers, and A,.1, ,.., ... , ,.1 real
or complex numbers. The necessary and sufficient conditions
that a function is so representable are that it possesses certain
quasi-periodic properties which are at first glance very similar
to almost periodicity; but Bohl's restriction on the exponents
of the trigonometric series places his problem in the class of
those whose solution follows in a more or less natural way from
existing theories rather than of those giving rise to an entirely
new theory.
A quite new way of studying trigonometric series is opened
up by Bohr's theory of almost periodic functions. We indicated

xii

INTRODUCTION

above the class of trigonometric series which correspond to
almost periodic functions: we have not yet indicated how wide
the class is. It is not possible to give any direct test for a series
to be the Fourier series of an almost periodic function, nor can
the similar problem be solved for the class of purely periodic continuous functions. But when the property of almost periodicity
is properly generalised, then the corresponding class of Fourier
series acquires a rather definite character of completeness.
The original work of Bohr was confined to the almost periodicity defined above. Thereafter work was done· in the way of
generalisation of the property by Stepan off, Wiener, W eyl, Bohr
and others. The new types of almost periodic functions were
represented by new classes of trigonometric series
~AneiAn"'.

As before, to a series of one of the new classes still corresponds
a convergent series of polynomials
:$p~IA,eiAn"',
(k = 1, 2, ... )
but to each new type of almost periodicity corresponds a different
kind of convergence of this sequence-not uniform convergence,
although the conve-rgence always has some features of uniformity.
In fact there exists a strict reciprocity between the kind of
almost periodicity of a function and the kind of convergence
of the corresponding sequence of polynomials.
When these generalisations are taken into consideration some
answer can be made to the above question of the extent of the
class of trigonometric series which are Fourier series of almost
periodic functions. This answer is given by the Riesc-Fischer
Theorem:
Any trigonometric series ~AneiA,.z, subject to the single condition
tha;t the series ~ I A,.l 2 is convergent, is the Fourier series of an
almost periodic junction.
This is completely analogous to the Riesc-Fischer Theorem for
the case of purely periodic functions. Generalisati"Ons of this
result similar to those for purely periodic functions are possible.
Thus the Fourier series of all almost periodic functions form

INTRODUCTION

xiii

as large a subset of the class of all trigonometric series of the
general type as the Fourier series of purely periodic functions
do of the class of all trigonometric series of the ordinary type.
There is no doubt whatever that a trigonometric series of the
general type
(with no restriction on the coefficients), in general does not
represent a function (is not "summable ") in any natural way,
and it may be that almost periodicity is the decisive test for a
non-artificial summabil i ty.
Almost periodicity is generalised in a natural way to the
class of analytic functions in a strip a< Rz < b by the condition
that the approximate equation
j(z + -ri)=/(z),
(-r real)
must be satisfied in the whole strip. One of the main features
of the theory of analytic almost periodic functions is the existence of the "Dirichlet series" IA 11 eAnz, which corresponds to
the Fourier series of almost periodic functions of a real variable.
The consequence is the same as in the case of almost periodic
functions of a real variable. We get a possibility of enlarging
the class of exponential series accessible to investigation. While
in the case of ordinary Dirichlet series IA,.eAnz the exponents are
subject to the condition of forming a monotone sequence, there
is no restriction of this kind on Dirichlet series of almost periodic
functions. In fact any set of real num hers may form Dirichlet
exponents of an analytic almost periodic function.
The connection between analytic functions and their Dirichlet
series is even deeper than between almost periodic functions of
a real variable and their Fourier series. The behaviour of almost
periodic functions and the character of their singularities at infinity are defined by the natu~e of their Dirichlet series.
The study of harmonic and doubly periodic functions has also
brought interesting and important results.
Applications of almost periodic functions have been made to
linear differential and difference equations, and undoubtedly
further development of the theory will lead to wider applications.

CHAPTER I

UNIFORMLY ALMOST PERIODIC
FUNCTIONS

§ 1. Definition and elementary properties.
1°, A set E of real numbers is said to be relatively dense (r.d.)
exists a number l > 0 such that any interval' of length l
contains at least one number of E. Any such number lis called
an inclusion interval of the set E. Thus the sets of numbers (i) ± n,
(ii) ± Vn, where n takes all the positive integral values, are both
r.d. On the other hand, neither of the sets (iii) of all positive
numbers, (iv) of all prime numbers ± Pn is r.d.

if there

2°. Let f(x) be a real or complex function defined for all real
values of x. A number Tis called a translation number of f(x)
belonging to e ~ 0 if
u.b.
[f(x+ T)-f(x) I~ e.
-OO<z<+ao

From this definition we deduce the following properties of
translation numbers of the same function.
(i) A translation number belonging to e belongs also to
any e' >e.
(ii) If Tis a translation number belonging toe, then so is- T.
(iii) If T 1 , T 2 are translation numbers belonging respectively
to e1 , e11 , then T1 ± Ts is a translation number belonging to e1 + e2.
We denote the set of all the translation numbers of a function
j(x) belonging to eby E {e,f(x)}. We deduce from (i) that
E {e',f(x)} ::> E {e,f(x)}*
for any e' > e, and from (ii) that the set E {e,f(x)} for any e ~ 0
and for any f(x) is syfnmetrical with respect to the point 0.
* The formulae A ) B, A C Bare read respectively: A contains B, A is contained in B.

2

UNIFORMLY ALMOST PERIODIC FUNCTIONS

3°. A continuous function f(x) is called uniformly almost
periodic (u.a.p.) if for any e > 0 the set E {e,f(x)} is r.d.
We shall always denote by l£ an inclusion interval of E {e,f(x)}.
From the definition it follows at once that any continuous purely
periodic functionf(x) is u.a.p., since for any e the set E {e,j(x)}
contains all numbers np (p a period off (x), n integer) and thus
is r.d.
We shall now prove several theorems, which establish elementary properties of u.a.p. functions.
4°. THEOREM. Any u.a.p.function j(x) is bounded.
For, put e = 1 and denote by M the maximum of\ f (x) I in an
interval (0, l 1). It can be easily seen that corresponding to any x
we can define a number T of E {1,f(x)}, such that x + T belongs
to (0, l1), and consequently that
lf(x+T) ~~ M.
But
\f(x+T)-J(x)j~1,
thus
jj(x) I:aM+ 1,
for all values of x, which proves the theorem.
5°. THEOREM • .Any u.a.p.function is uniformly continuous.
For, given e >0, take an l£;3 and letS (0 < S < 1) be a number
such that IJ(x1 )-j(x2)1 < ie for any x1 , x2 belonging to the
interval (0, l~;s + 1) if only Ix1 - x2 1 < S. Let now x', x" be any
two numbers such that I x' -x" I< o. There exists a number T
of the set E {t e, f (x)} such that both the numbers x' + T, x" + T
belong to the interval (0, l£; 3 + 1 ). We have then
IJ(x' + T)- f(x" + T) I< ie.
On the other hand
lf(x+T)-f(x)l < ie
for any x. Thus
IJ(x')-f(x") I~ lf(x')-f(a/ +T) I+ IJ(x' + T)- f(x" + T) I
+ lf(x" + T)-f(x") I< e,
which proves the theorem.

Corollary. If f(x) is u.a.p., then to any e > 0 corresponds a
number S=S(e)>O such that the set E{e,f(x)} contains all
numbers of the interval (- S, + S).

UNIFORMLY ALMOST PERIODIC FUNCTIONS

3

6°. THEOREM. If f(x) is u.a.p., then cf(x) (c any constant),
/(x) (the conjugate function off (x)), {f(x)}2 are also u.a.p.

If (x + T)- f(x) I~ e,
Icf(x + T)- cf(x) I~ I c I €,
1/(x+T)-/(x)l~e,
I{f(x + T)} 2 - {f(x)}21 ~ 2M€,
M=
u.b.
1/(x)l.

For, if
then

where

-OO<:!'<+oo

Thus each of the sets

E {I c I €, cf(x)}, E {€,/(x)}, E {2M€, {f(x)}2},
contains the set E {e,f(x)} and consequently is r.d., which proves
the theorem, since € is arbitrary:
7°.

THEOREM.

Iff (x) is u.a.p. and if
l.b.

-oo<X< +oo

lf(x) I= m > 0,

then 1/f (x) is also u.a.p.
For, if
IJ(x + T)- f(x) I~ €,
then
Thus E

I

I-

1
1
\f(x+ T)-J(x)
f(x+T)- j(x) - f(x+T).j(x)

{~~~· f~x;} contains E {€,f(x)} and

I<
=


m2 •

consequently is r.d.

This being true for an arbitrary e the theorem is proved.

Remark. The last two theorems are particular cases of the
following
THEOREM. If f(x) is u.a.p., then any uniformly continuous
function off (x), F {f (x)} is also u.a.p.

8°. THEOREM. If a sequence of u.a.p. functions Un (x)} converges uniformly in the whole interval - oo < x < + oo to a function
f(x), thenf(x) is also u.a.p.
For, given e, there exists a functionfno(x) such that

If (x) for all values of x. Let now

fno (x) I<
T

i

be a number of E{i€, fn 0 (x)}.

4

UNIFORMLY ALMOST PERIODIC FUNCTIONS

Then
IJ(x+ T)- /(x) I~ IJ(x + T)-/n0 (x+T) I+ IJno(x+T)- /n 0(x) I
+ l/n0 (x)- f(x) I< e,
which shows that
E {e,f(x)} ::> E {ieJno(x)}.
Thus E {e,j (x)} is r.d. This being true for any e we conclude
thatj(x) is u.a.p.
9°. In 6° and 7° we established almost periodicity of functions
of one u.a.p. function. We pass now to functions of more than
one u.a.p. function. We shall first establish some properties of
translation numbers of u.a.p. functions.

Lemma. Given any two positive n1lmbers e1, e2 ( e1 < e2), there
exists a numberS (e1. e2 ) > 0 such that E {e2 ,f(x)) contains every
number whose distance from the set E {e1 ,f(x)} is not greater
than Et. e2).
By the corollary of 5° there exists a number o= S (e2 - e1 ) > 0
such that E {e2 - e1 , f (x)} contains all n urn hers of the interval.

o(

<- s, + o).

But by (iii) of 2° the sum of any two numbers taken one from
each of the sets E {et,f(x)), E {e2 - Et, f(x)} and a fortiori from
each of the sets E { e1 , f (x)), (- o, + S) belongs to the set
E {e2 ,f(x)}, which proves the lemma.
10°. Lemma. If e, S a;re two arbitrary positive numbers
and f 1 ( x ), f 2 ( x) two u.a. p. functions, then the set of numbers of
E {e, f 1 (x)} whose distances from the set E {e, f 2 (x)} are less than S,
is r.d.
Consider the sets E {!e, / 1(x)} and E e, /2 (x)}, and let a
number l = kS (k integer) be an inclusion interval of each of
these sets. Divide the whole interval (- oo, + oo) into intervals
{(n- 1) l, nl), where n takes all integral values. Inside every
interval {(n- 1) l, nl} we can find numbers T1 «n>, T2«n> belonging
respectively toE {i-e,f1 (x)), E f!e,fs (x)). We shall have

a

- l < Tt(n) - Ts(n) < + l.

UNIFORMLY ALMOST PERIODIC FUNCTIONS

5

Denot'e by A.i the interval (i - 1) S;;; x < iS, so that the difference
T1<n>- Ts<n> always belongs to one of the intervals A.i (i = - k + 1,
... , k). Obviously there exists an integer n 0 such that to any
integer n corresponds an integer n' (- n0 ;;; n' ~ n0 ) for which
the difference T11n'>- Ts<n'l belongs to the same interval A.i as
the difference T1<n>- r 2<n>. Thus we have
T1(n)- Ts(n) = 7'1(n')- Ts(n') + BS, (- 1 < B< + 1)
or
T1<n>- T11n'l = rs<n>- T21n') + Bo.
But by (iii) of 2° the numbers T11nl- T11n'>, rs<n>- T21n') belong
respectively to E {e, / 1 (x)}, E {e, / 2 (x)). Thus the distance
of every number T1<n>- r 1<n'> (- ~ < n < + oo) from the set
E {e, fs (x)} is less than
The modulus of the difference of
the values of T1<n>- r 1<n'> for any two consecutive values of n
is easily seen to be less than (2n0 + 3) l, and thus the set
{r1<n>- T11n'>) (- oo < n < + oo) is r.d., which proves the lemma.

o.

11°. THEOREM. For any e > 0 and for any two u.a.p. Junctions
jl(x) and fs (x) the set* E {e, fl(x)}. E {e,Js (x)} is relatively dense.
Take a positive number .e1 < e. By the lemma of go there
exists a S > 0 such that all numbers whose distance from the
setE{e11 / 1 (x)) is less than S, belong to E{e,f1 (a:)). Now by
the preceding lemma the set G of the numbers of E {it,fs(x)}
whose distances from the set E {e1, / 1 (x)} are less than S, and
which consequently belong to E {e,j1 (x)}, is r.d., i.e. the set
E {€,/1 (x)}. E {e1,Js(x)} is r.d., and a fortiori so is the set
E {e, / 1 (x)) . E {e,/2 (x)).
12°. THEOREM. The sum of two u.a.p. functions fl(x), j 2 (x)

is 1t.a.p.
For, taking an arbitrary e > 0, letT be any number of the set
E {!e,/1 (x)}. E {!€./s(x)}.
Then
l/1 (x+ T) + fs (x + T)- /1 (x)- Is (x) I~ e,
which shows that T belongs toE {e,f1 (x) + fs (x)). Thus
E {e, / 1 (x) + / 2 (x)} ::> E {te,/1 (x)}. E {i-e, / 2 (x)),
so that E {e,f1 (x) +fs (x)} is r.d., which proves the theorem.
* A and B being two sets we denote by A. B (or by Ax B) the set of all
common elements of A and B.

6

UNIFORMLY ALMOST PERIODIC FUNCTIONS

The theorem can be generalised immediately to the case of
the sum 'of any finite number of u.a.p. functions.
co

Corollary. A uniformly convergent series I aneiA,x, where
n=l

are real, is a u.a.p.function.
For, each term of the series being a u.a.p. function (in fact a
purely periodic function), the sum of the first n terms Sn (x) is
a u.a.p. function, and the sum s (x) of the series being the
uniform limit of Sn (x) is also a u.a.p. function (on account of
the theorem of 8°).

}t.1 , }t.2 , •••

13°. THEOREM. The product of two u.a.p. functi:ons ft (x), !2 (x)
is a u.a.p. Junction.
By the preceding theorem the functions 11 (x) +12 (x) and
1t (x)-1 2(x) are u.a.p., and by the theorem of 6° th.eir squares
are also u.a.p. functions. Thus 11 (x) .12.(x) can be represented
as the sum of two u.a.p. functions
1t (x) ·12 (x) =! Ut (x) + f2 (x)} 2 -! {11 (x)-12 (x)} 2,

which proves the theorem.

Corollary. If jt(x), !2 (x) are u.a.p. and if l.b.iJ2 (x) I is
positive, then .h (x)/J2 (x) is u.a.p.
For by the theorem of7° l/J2 (x) is u.a.p., and thus
1

1t (x)/!2 (x) = 1t (x) . (x)
12
is the product of two u.a.p. functions.
14°. We shall now establish, under certain conditions, the
almost periodicity of the derivative and of an integral of a u.a.p.
function.
BocHNER's THEOREM. If the derivative of a u.a.p. function
j(x) is uniformly contimwus, then it is u.a.p.

Take a sequence {h:n} of real numbers converging to zero.
We write
1(x+ h,.)/(x) -1' (
h
x+ eh·n.)
n

UNIFORMLY ALMOST PERIODIC FUNCTIONS

7

The expression on the left being the difference of two u.a.p.
functions is u.a.p. Thus f' (x + Bhn) (n = 1, 2, ... ) form a
sequence of u.a.p. functions uniformly convergent to f' (x) (on
account of the uniform continuity off' (x)). Consequently f' (x)
is u.a.p.
15°. THEOREM. Ifanindejiniteintegralofau.a.p.functionf(x)
is bounded, then it is a u.a.p. function.
Obviously we may assumef(a;) to be real. We write

F(x) = (f(x) da;,
where a is a constant, and assume F(x) to be bounded, so that
both of the two numbers
(1)

are finite. We have to prove that F (x) is u.a.p., i.e. that the
set E {e, F (x)} is r.d. for any e > 0. The numbers T of this set
are defined by the condition that they satisfy the inequality
(2)

I F(x+ T)- F(x) I=

I

J:+.,.j(x) dx

I~ e

for all x.
Now given an 1J > 0 we can always define numbers x1 and x 2
to satisfy the inequalities
(3)
F (x1) < k1 + TJ, F (x2) > ks- "1·
Take now an arbitrary ec > 0 and let T 1 be any number of
E {e', f (x)}. Writing I x1 - x2 1 = d we shall have

IJ::::f (x) dx- .Cf(x) dx I= IJ::{J(x +

1

T

)-

j(x)} da;

1.e.
IF (xs + T ) - F (x1 + T ) - F (x2) + F (x1) I ~ e'd,
whence
F (x1 + T 1 ) ~ F (x2 + T 1 ) - {F (x2)- F (x1)} + e' d.
But by (1) and (3)
F (x~ + T1 ) ~ k2, F (xs)- F (a~l) > k2- k1- 2,,
thus
(4)
F(x1 + T') < k1 + 2, + e'd.
1

1

I~ e' d,

8

UNIFORMLY ALMOST PERIODIC FUNCTIONS

Take now an e" > 0 which will be defined later on. Let -r"
be any number of E {€'', j(x)}. Analogously to (4) we have
(observing that -r' + -r" C E {e' + e",f(x)])
(5)

F (x1 + -r' + -r'') < k1 + 211 + (e' + e") d.

We shall now consider the integral
a -r' satisfying the inequality

x+-r"
x

I

f (x) da;.

Defining

(6)
we write
x+T"j(x) daJ= Jx,+-r'+""f(x) dx+ Jx,+-r'{f(x)-/(x+-r")} d~.

J

X

X

X1+T'

By (1), (4), (5),
Jx,+-r'+-r"J(x) dx I
x1 +-r'
=I F(x1 + -r' +-r")-F(xl + -r')l < 211 +(e' + e") d,
and by (6)
-1

x +-r'

Hence

(7)

IJx'

I

{J(x)-j(x+-r")}dx <e"lc.

x+-r"
x
f(x)dx

If

I<211+(e'+e")d+e''lc.

Given an e > 0 we put
11

;J ;

= ~, e' = 6~, e" = miri. ( e', 3

we obtain from (7)
(8)
for all x and for all -r" C E {e",J (x)}. Hence by (2)

E {e, F(x)} ::> E {e",f(x)},
which shows that E {e, F(x)} is r.d.
16°. Some elementary properties of sets E {e, f(x)} can be
obtained by considering S. Bochner's translation function.

UNIFORMLY ALMOST PERIODIC FUNCTIONS

9

Given a u.a.p. functionf(x) we define the translation function
v,('r) of f(x) by the equation
VJ(T)=

u.b.

+co

-oo<x<

lf(x+T)-j(x)l.

Evidently the set E{e,f(x)} is identical with the set
of values of T for which v1 ( T) ~ e.

E{v1 (T)~

e}

The function v (T) = v1 ( T) satisfies the following conditions :

(a)

v(T)~O,

v(O)=O,

(b) v(-T)=v(T),
(c)

V ('1'1

+ Tg):!

V

(T1 ) +

'IJ

(Tg),

(d) v (T) is u.a.p.
The properties (a), (b), (c) are obvious. To prove (d) we write
'/J
V

(X) ~

'IJ (X

whence

(a;+ T)
+ T) +

~V

'IJ ( -

(a;)+ V (T),
T) =

V (X

+ T) +

V ( T),

I v (x + T)- v(x) I:! v(T).

For x = 0 we have
lv(T)-v(O)I=v(T),
thus
(1)

u.b.

lv(x+T)-v(x)l=v(T),

-oo<X<+oo

I.e.

u.b.l f (x + T)-f(x) I= u.b.l v (x + T)- v (xH.

From this equation we conclude that v (x) is a uniformly continuous function, and that

E {e, v (x)} = E {e,f(x)}
for all e, which proves (d).
The converse can be easily proved, that
Any function v (T) satisfying the conditions (a), (b), (c), (d) is
a translation function of a u.a.p. function.

In fact the equation (1) follows from (a), (b), (c), and it shows
that v ( T) is its own translation function.

1

10

UNIFORMLY ALMOST PERIODIC FUNCTIONS

§ 2. Normality of u.a.p. functions.
1°. S. Bochner has introduced a certain property (called
normality), based on the convergence of sequences of functions,
derived in a special way from a given function, and has proved
that this property belongs to all u.a.p. functions and to no
others, so that u.a.p. functions may be defined as functions
possessing this property.
DEFINITION. A continuous function f(x) is called normal, if,
given any sequence {hi} of real numbers, there exists a subsequence
{hni} such that the sequence of functions {f(x + hni)} is uniformly
convergent.
We shall prove that the class of normal functions and the
class of u.a.p. functions are identical.

2°. Lemma. Given a u.a.p. function f (x) and a sequence

{h,;} of real numbers, then to any e > 0 there corresponds a subseqnence {h"i} such that the modulus of the d~fference of any pair
of functions f(x + hni) is less than e.
Any h,; can be represented in the form
h,; = T,; + r,;,
where T,; belongs to E {le, f(x)} and r,; satisfies the inequality
0 .;a r,; ~ lfi/oi· For each hi we consider only one representation in
this form. Let r be a limit point of the set of all r,. Define
a number•o such that
I/(x")- f(x') I< e/2,
if only
Ix"- x' I < 2o.
Then the set of all h,; for which r- o < r,; < r + o satisfies the
condition of the lemma. For let h1 , h,. be two such values. We
have
u.b.if(x + hj)- f(x + h,.) I= u.b.lf(x + Tj-TTr. +rj- r,.)- f(x) I
.
~ u.b.lf(x + Tj- ,.,. + rj- r,.)- f(x + rj- rk) I
+ u.b.lf(x + ri- r,.)- f(x) I;
each of the last two terms is less than e/2, since Tj- ,.,. is
a translation number belonging to e/2 and lrJ- r,. I <. 2o. Thus

IJ(x+hi)- j(x + hk)l< e.

UNIFORMLY ALMOST PERIODIC FUNCTIONS

3°.

THEOREM.

11

Any u.a.p.functionf(x) is normal.

For, let {hi} be a sequence of real numbers. By the preceding
lemma w;e can choose a subsequence {hn•;} such that for any two
positive integers j, k and for all x
IJ(x + hn•1 ) - j(x + hn·~c) I < 1.
Similarly we can choose a subsequence {h71;2l} of the sequence
{hn·,} such that for any two positive integers j, k and for all x
IJ(x + hnfl)- f(x

+ hn~l) I < i·

We then choose a subsequence {hnjS)) of the sequence {h71;2l}
such that
I f(x + hnjl)- f(x+hn~Sl) I<

z,

and so on. Take now the sequence of functions
(1)

j(x + h711-), j(x + h.f;l), j(x + h71 ~3l),

I.. et j, k (j < k) be two positive integers. We obviously have
IJ(x + hnY1) - J(x+ hn~l) I<

1
J,

which shows that the sequence (1) is uniformly convergent. As
the sequence {hi} was arbitrary, the theorem is proved.

4°. CoNVERSE THEOREM. Any normalfunotionf(x) is u.a.p.
For, assume the contrary: supposef(x)is not a u.a.p. function.
Then there exists an e >0 such that theE {e,f(x)} is not r.d.
Take now an arbitrary real number h1 and let (a 2 , b2) be an
interval of length > 21 h1 1 which does not contain any number
of E {e, j(x)}. Denote by h2 the centre of· this interval. Evidently h8 - ht belongs to the interval (a 2 , b2 ), and a fortiori does
not belong to E {e, f(x)}. Define now an interval (aa, b3 ) of
length > 2 (I h1 1+I h.a I) which does not contain any number of
E {e, f(x)}. Let h3 be the centre of (a3 , b3). For the same
reason as before the numbers ha- h1 , ha- h2 do not belong to
E{e,f(x)}. In a similar way we define h4 , h5 , ••• so that none
of the numbers hi- hj belongs to E{e,f(x)}. Thus for any i,j
u.b.lf(x + h;)- f(x

+ hj) I= u.b.lf(x +hi- hj)- f(x) I> e.

12

UNIFORMLY ALMOST PERIODIC FUNCTIONS

Thus we arrive at the conclusion that no subsequence of the
sequence {j(x +hi)} is uniformly convergent, which is in con·
tradiction with the assumption that f(x) is normal, and the
theorem is therefore proved.
The proofs given in this section are due to J. Favard.

§ 3. Mean values of u.a.p. functions and their Fourier
series.
1°. U.ap. functions, like purely periodic functions, can be
" represented by their Fourier series." To establish this we have
to consider certain limiting expressions-mean values of u.a.p.
functions. Let j(x) be a real u.a.p. function. We consider the
integral
1
X of(x)dx,

jx

and we call its upper limit and its lower limit, as X-++ oo, the
upper and the lower mean value of the junction f(x), and we
denote them by
iJ {j(x)}, M. {f(x)}.
If they are equal we call their common value the mean value
of j(x), and we denote it by M{f(x)}. It is obvious that the
mean value of any integrable purely periodic function exists.
If instead of j(x) we have a function of two or more variables,
then we indicate the variable with respect to which the mean
value is taken by a suffix: we write for instance Mx {f(x, y)}.
In the case of a complex function j(x) we define only the
mean value M {f(x)} by the limit of

~ J;f(x) dx,

as X-++ oo .

2°. THEOREM. The mean value of any u.a.p. function f(x)
exists.
Let T, e be two positive numbers and n a positive integer.
We write
1 JnT

k-n-1 1 J<k+l)T
-T
f(x) dx.
n
kT

-T
j(x) dx = ~
n
o
k-o

UNIFORMLY ALMOST PERIODIC FUNCTIONS

13

Denote as usual by lE an inclusion interval of E {e, f(x)}, and
let TJ; be a number of E {t>,f(x)} included in the interval
(kT, kT+ l,). We have
(lc+l) T

f

f(x) dx ==

J(k+l) T-Tk

kT

f(x + TTr.) dx

kT-Tk

= J:t(x) dx

+

+ J:{f(x + Tk)-f(x)} dx

0

J

kT-Tk

f(x + T,.)dx +

=l1 +I2+l3 + I 4 •
Evidently II 2 1< e'l'. Now writing
A=
u.b.

-a:><X<+a:>

J

(k+l) T-Tk

T

f(x +Tk) dx

1/(x) 1,

and observing that the length of the range of integration in I 3
and I 4 is less than lfi, we have I Ial < Al,, I I 4 1 < Alfi. Hence
(k+llT

J
and thus

k2'

1
nT

(1)

f(x)dx=

~T

f(x)dx+()(eT+ 2Al,),

(1()1~

1)

0

fnTf(x)dx=T1 JTj(x)dx+B (e+---ry
2Al•)* .
0

0

Let now"' be a positive number as small as we please. Taking
in the above formula
16Alfi
T >'1} - ,
we shall have
1 JnT
1 (T
nT o f(x) dx = T Jo f(x) dx + () ~.

(2)

Corresponding to any positive number X, define the integer n
by the condition
(2.1)
nT ~X< (n+ 1) T.
From the boundedness of f(x) we conclude

lim{~ J:f(x)dx-n~f:Tf(x)da}=O,as

X-oo.

* In the course of proof 8 denotes different values satisfying the conditions

Bl ~ 1.

14

UNIFORMLY ALMOST PERIODIC FUNCTIONS

Consequently there exists a number X 0 > 0 such that
' 1
I-

(2.2)

1X

jxj(x)dx--,1 jnTf(x) dx I < '!1
o

4

nT o

.

for all X> X 0 •
Let X', X'' be any two numbers greater than X 0 and n', n"
the corresponding values of n in (2.1). By (2), (2.2)
1

IX'

JX'f(x) dx- X"1 JX"f(x) dx I
1 Jn'T
1 Jn"T
I
~ln'T o f(x)dx-n"T o f(x)dx
0

0

1

1 JX'
1 Jn'T
+II X'
o f(x)dx- n'T
o j(x)dx I·
1 JX"
1 fn"T
+ I X"
o j(x)dx- n"T
o j(x)dx I< TJ,

which proves that the limit of

~ J:J(x)dx
exists, as X - oo •
Thus the theorem is proved.
Taking the limit of the left-hand side of (1) we obtain
(3)

M {j(x)} =

2

~ J:J(x)dx + () (e+ ~,l''),

which is true for any e > 0, T> 0. The second term on the
right-hand side is the error of the representation of M {j(x)}
1
by the integral T
j(x) dx and depends on e, A, l" and T, of

JT

·

0

which e, Tare independent ofj(x) and l", A are defined by f(x).
Evidently for the functionf(x +a), where a is an arbitrary real
number, the numbers l,, A are the same as for f(x). Consequently we may write

1
M{f(x+a)}=T

JTf(x+a)dx+() (e +2.Al")
T.
0

UNIFORMLY ALMOST PERIODIC FUNCTIONS

15

But evidently
M {f(x+ a)}= M {f(x)},
hence
(4)

M{j(x)} =

1
i.e. the integral T

~

J:+

Tf(x)dx + B (e+

Jet+T
a
f(x) dx

2

~l"),

tends to M {f(x)} uniformly

in a, even if a varies together with T, so that we can write, for
instance,
1 f+(T/2)

M{f(x)}=lim -T
T-+cx:>

- (T/2)

f(x)dx.

:3°. Consider now the product f(t+ x)j(t). It is a u.a.p.
function oft. Taking an arbitrary e' > 0 and denoting by lcr• an
inclusion interval of E{e',f(t+x)J(t)} and by .A the upper
bound of 1/(t+x)/(t) !, we applyformula(4)of 2° toj(t+x)/(t):
Mt {f(t+ x)/(t)} =~ f:J(t+x)f(t)dt+ B (e' +

2

~l"').

Obviously .A~ A 2• Putting e' = 2Ae and observing that

E {e',f(t + x)j(t)} 'J E {e,f (t)},
we see that l, is an inclusion interval of E{e',f(t+x)/(t)}.
Thus we may put l•. = l,. We then write
(1) Mt{f(t+x)j(t)} =

~ J: f(t+x)f(t)dt+ B( 2.A.e+ 2A;l").

Writing
Mt{f(t+x)j(t)}=g(x),

~ J:J(t+x)/(t)dt=G(x),

we have
(2)

2
g (x) = G (x) + B ( 2Ae + A;l").

Taking in this equation e sufficiently small, and T sufficiently
large, we can make the second term of the right-hand side as
small as we please.
Thus G (x) tends to g (x) uniformly in x, as T- oo.

16

UNIFORMLY ALMOST PERIODIC FUNCTIONS

Let ,. be any number of the set E {e, f (t)}; we have
1
I G(x+T)- G(x) I~ T
~.A

i.e.

u.b.

JT lf(t + x + T)- f(t+x) l·lf(t)
- Idt
0

-«><t<+oo

lf(t+,-)-f(t)I~Ae,

E {Ae, G (x)} ::> E {e,f(t)},

which shows that G (x) is a u.a.p. function. In the same way
(or by 8°, § 1) we see that g (x) is a u.a.p. function. The
functions G (x), g (x) will play an important part in the proof
of the fundamental theorem on u.a.p. functions.
PROBLEM. Prove that
Mx[Mt {f(t + x)/(t)e-iA<t}] = Mt [Mx {f(t +x)j(t)e-iAx}],
wheref(x) is u.a.p. and :X. is real.
4°. We shall now consider approximation to u.a.p. functions
by trigonometrical (exponential) polynomials. Observe first that
for any real :X., e>A<t is a purely periodic (and a fortiori u.a.p.)
function, and that
M {e'A<t} = 0 for :X. ~ 0,
= 1 for :X. = 0.

Let now f(x) be a u.a.p. function. For any real :X. the function
f(x) e-tA<t, being the product of two u.a.p. functions, is a u.a.p.
function and consequently the mean value

M {j(x) e-iA<t} =a (:X.)
exists.
THEOREM. Let f(x) be a u.a.p. function, :X.ll :X.2 , ... , :X.N
N arbitrary real numbers different from one another, and
bl> b:a, ... , bN N arbitrary complex (or real) numbers. Then
(1) M { IJ(a:)- n~l bnei~xn
N

=M{Ij(x)l 2} - I

n-1

N

la(:X.n)l 2 + I

n-1

lbn-a(:X.n)l 2•

UNIFORMLY ALMOST PERIODIC FUNCTIONS

17

We write

M { jl(x)- n~
=

2

bnei~rt 1 }
1

M [ {f(x)- n~
-

1

b,.e>~x}. {/(x)- n~ 1 bne-i~x} J
N-

= M {f(x)j(x)}- ~ bnM {f(x) e-i~<~~}
n-1

As M {e' ,~,-An.l<~~} differs from nought (and is equal to 1) only
N

for n1 = nt the last sum reduces to the sum I

n=1

I b,. j2•

Thus

The equation (1) is called the equation of approximation in
mean.
5°. From the last theorem we see that the polynomial
N

:i

n-1

bne'~a~

with fixed exponents Xn gives the best approxima-

tion in mean to f (x) (i.e. the least value of the right-hand side
of (1) 4°) if bn- a(:X.n) for all n, in which case we have

18

UNIFORMLY ALMOST PERIODIC FUNCTIONS

The left-hand side of this equation being non-negative, we
conclude that
N

(1)

~

la(A..)I2~M\If(x)l 2 ).

n=l

This inequality being true for an arbitrary number N of real
numbers A,., we conclude that to any positive e corresponds at
most a finite number of values of :X. for which I a (:X.) I >e. All
a (:X.) different from nought satisfy one of the enumerable set of
inequalities
1

1

ia(:X.)i >1, -~la(:X.)i > - . (n=1, 2, ... )
n
n+ 1
Each of these inequalities being satisfied by at most a finite
number of values of :X., we deduce the
THEOREM. There exists at most an enumerably infinite set of
values of :X. for which a (:X.) differs from nought.

6°. Denote these values of :X. by A~> A2 , ••• , and write
a(An)=.An for all n. We call the numbers A1 , A2 , ... Fourier
exponents and the numbers .A 1 , A 2 , ••• Fourier coefficients of the
function f (x). The formal series
.Al.ei~x

+ .A2eiAzx + ... = ~.AneiAnx

is called the Fourier series of the function f (x), and we write
j(x)- ~A,eiAnx,
By (1) we have
(2)

This is the Bessel inequality for u.a.p. functions.
7°. Let f(x) be a purely periodic function with period 271".
+oo
Its Fourier series in the ordinary sense is defined by l: Aneinx,
-co

where
(1)

and these coefficients satisfy the Parseval equation
(2)

UNIFORMLY ALMOST PERIODIC FUNCTIONS

19

But for the case of a purely periodic function

__!_ J21Tf(x) e-inz dx = M {f(x) e-inzl,
27T

0

2~ J:1TJf(x) l

2

dx = M {lf(x) j 2}.

Thus the coefficients (1) (which are different from nought) are
also Fourier coefficients of f(x) in the new sense, and from (2)
we see that there cannot be any other Fourier coefficient in the
new sense. Thus in the case of purely periodic functions the
ordinary definition of the Fourier series coincides with the new one.

8°. THEOREM. The Fourier series of a u.a.p. function represented by the sum of a uniformly convergent trigonometric series
00

f (x) =

~ anei"n"'
n=l

coincides with this series.

00

The series ! anei ("n-A.)"' is uniformly convergent for any
n=l

real A. Therefore
00

M {f(x) e-i;>..z} = ! anM {ei("n-A.>z}.
n=l

Hence
for A different from all An., and
M {f(x) e-i"n"'} =an
for any n, which proves the theorem.
9°. In 6° we showed that for u.a.p. functions the Bessel
inequality holds. In fact it will be shown in § 4 that the
Parseval equation

! JAn j2 = M {Jf(x)J2}
is true for any u.a.p. function. This is Bohr's Fundamental
Theorem. By means of this theorem it will be shown that two
different u.a.p. functions cannot have the same Fourier series.
Thus not only does a u.a.p. function define its Fourier series
but also a Fourier series defines uniquely a u.a.p. function.

20

UNIFORMLY ALMOST PERIODIC FUNCTIONS

10°. We shall now prove a theorem, which will be required
for the proof of the fundamental theorem.
THEOREM. If the mean value of a real non-negative u.a.p.
functionf(x) is nought, thenf(x) = 0 for all x.
Suppose that the theorem is not true and that at some point x 0

f

o

= m > 0.

Then there exists a > 0 such that
j(ll'l) > fm
in the interval (xo- o, Xo + o).
Take now an ltm > 2o. Any interval of length ltm contains
at least one number of the form x 0 + T, where T belongs to
E {!m,j(ll'l)} and a fortiori at least one of the intervals
(xo+T-o,x0 +T), (xo+T,xo+T+o).
In each of these intervals, which are the intervals (x0 x 0),
(Xo, Xo + 0) translated through T, we have
(x0)

o,

j(x)

m

>a·

Thus in each interval of length lm;3 there exists a sub-interval
of length o, where f (x) > m/3. Consequently
a+lm!3
a
f(x)

J

mo

dx > 3

for any a. We have now

mo

1 Jnlmf3
0 = M {f(x)J == lim-lf(x)dx ~ - l ,
3 m/3
n m/8 0
and thus we have arrived at a contradiction, which proves the
theorem.

11°. The Fourier series of a u.a.p. function f(x) has been
defined as the aggregate of all the " terms"
a(~) ei.u:,
(a (X)= M {f (w) e-i.l.a:})
where a(~)=!= 0. Sometimes it is convenient to add to this
aggregate some terms for which a (X)= 0. This enlarged aggregate is still called the Fourier series of the function. We never
add more than an enumerable set of terms with coefficients
equal to zero.

UNIFORMLY ALMOST PERIODIC FUNCTIONS

21

Let {fk (x)} be an enumerable set of u.a.p. functions. The
set of the Fourier exponents of all the functions of the set
is enumerable. Denote them by An (n = 1, 2, ... ). Then the
Fourier series of each function of the set may be represented in
the form
fk (X),...., !An (k) eiAnx.
THEOREM.

(1)

If a sequence
fk(x)-!An(k)eiJI.nx

(k = 1, 2, ... )

of u.a.p. functions converges uniformly to a ftmction j(x), then
the Fourier series off(x) is given by the equation

(2)
where An= lim An(kl, ask-+ oo ,for all n; in other words the series
(2) is "the formal limit" of the series ( 1), as k-+ oo •

For, we have for any X

IM {/k(x)e-ihx}-M{f(x)e-ihx}l ~

u.b.

lfk(x)- f(x) 1-+0,

-co<x<+co

as k-+oo.

§4. Fundamental theorem of the theory of u.a.p.
functions.
1°. If for a u.a.p. function f (x)

a (X)= M {f(x)e-ihx} = 0
for all real values of "A, we say that the Fourier series of f(x) is
equal to nought.
We shall first prove the fundamental theorem for such functions, and then it will be easy to extend it to the general case.
Irl the case of a function f(x) with Fourier series equal to
nought the Parseval equation takes the form
M{l/(x)l 2 } =0.
We prove this result by considering an auxiliary purely periodic
function F (x) with period T equal to f(x) for 0 ~ x < T. We

22

UNIFORMLY ALMOST PERIODIC FUNCTIONS

shall first prove that for large Tall Fourier coefficients of F(x)
are small. We prove this proposition by proving a more general

Lemma. Iff (x) is a u.a.p. function with Fourier Qeries equal
to nought, then

~ f:J(x)e-iAzdx-+0,
as T-+ oo , uniformly in A., i.e. given an e > 0 there exists a T 0 > 0
such that for all T > T 0 and for all real X

I~ J:J(x)e-iAzdx\< e.
We shall first prove the following lemmas.
2°. Lemma. If f(x) is a u.a.p.function, then

cp (X, T) = ~ J:J(x) e-iAzdx-+0,
as X-++ oo, or X-+- oo, uniformly in 1 ~ T < oo.
We write
/.1.)
7T') e-iAZdaJ
cp (X, T) = - -1
j ( X __
T tr/A
X

JT+( ..

= -

~

T

IT - ! f

0

0

T

-

1</A

! JT+(1r/.l.)
T

T

'

and thus

Writing
A=

u.b.

-co<:o<+co

lf(x)l,

w(o)=

u.b.

-co<z<+co

1/(x+o)-f(x)l,

UNIFORMLY ALMOST PERIODIC FUNCTIONS

23

Hence

which proves the lemma.
3°. Lemma. If M {f (x)} = 0, then given e > 0 there exist two
positive numbers o, T 0 , such that

I~ {Tf(x) e-ih~dx j < e
for all T > T 0 and for all X in the interval(-

o, + o).

By (4) 2°, §3, there exists a number To> 0 such that

11J;t(x+s)dxj<~,

(I)

for all s and for all H > ! To. Now any T > T 0 can be represented
by T = nH, where n is an integer and H satisfies the condition
tTo<H <To.
Now define
(2)

for

oby the condition that
Ie-ih~- II < e/2A

IXI < oand for 0 :i x ~ T0 (and a fortiori for 0 :i x :i H).

We write

IJTf(x)e-iA~dx=-!
I n-1 e-ihkH JHf(x+kH)e-i~dx.
(3) .,-;
'1. o
nH k=O
o ,
By (1), (2)

11

J: f(x + kH) e-ih~dx

~

11

I

J:J(x+kH) dx!

+I~ f:J(x+kH)(e-•~-I)dxl

24

UNIFORMLY ALMOST PERIODIC FUNCTIONS

Hence by (3)

I~ J;J(x) e-•~dx
for T > T0 and I XI <

I<

e

o, wh;ch proves the lemma.

4°. Lemma. If M {f(x)e-i!'x} = 0, then given e > 0 there exist
numbers TIL>O and oiL >0, such that

~~ (f(x)e-iAxdxl < e
forT >T,. and for p.- o,. <X< p. + o,..
This lemma reduces to the preceding one applied to the
functionf(x)e-il'x; which is also u.a.p.
5°. We can now prove the lemma of I 0 • We have to prove
that given e > 0 there exists T0 > 0, such that

(I)

I~ J:J(x) e-iAxdx \ < e

for all X and for all T > T 0 •
By the lemma of 2° there exists a Xo > 0 such that (I) is
satisfied for T >I and for I XI > Ao·
By the lemma of 4° to any p. of the interval - X0 ~ p. ~ + Xo
corresponds a TIL > 0 and an interval (p.- OIL, p. + o,.), such that
(I) is satisfied for any T >TIL and for any X of this interval. By
the Heine-Bore! theorem there exists a finite number of points
p.1 , p.2 , ... , f.l-n, such that the intervals corresponding to these
points cover the whole interval - X0 ~ p. ~ + X0 • For X in each
of the intervals (p.,.- o,.k, P.k + o,.,.), (I) is true for T >TIL,.· Thus,
taking T0 >max. (I, T,.l' T,. 2 , ... , TILn), we see that for T > T 0 the
inequality (1) will be satisfied for I X\> X0 and for X bclonging
to any of the intervals (P.Tc- oiLk' Jl-k + o,.k), i.e. for all values of X.
6°. We now pass to the proof of the Parseval equation

(1)
M{lf(x)l 2 }=0
for a functionf(x) with Fourier series nought. We consider the
purely periodic function F(x) introduced in 1°. We write
+co
. 2>rkz
F(x)- I .A.,.e'7'.
k=-CO

UNIFORMLY ALMOST PERIODIC FUNCTIONS

25

For F(x) the Parseval equation holds:

11T

T-

(2)

+oo

IF(x)l2dx=!I.Aki2~.A_z.
-oo

0

But

~ J:l F(x) l dx= ~ J~lf(x)l 2 dx-+M{/f(x)J 2 },
2

as T-+oo. The equation (1) would be proved if we could prove
that
II.A.k/ 2 -+0,
as T-+oo. We cannot prove this, but we can easily see that
(3)
as T-+- oo. For, by the preceding lemma to any e > 0 corresponds
a T0 > 0, such that for any T > T0 , all I Ak I are less than e. By (2)

I I Akl4< e2I I Akl2~ e2 .A_2,
which proves (3). We shall prove (1) by expressing the sum

! IAkl 4 in terms of j(x).

We introduce two new purely periodic functions

1fTF(x+t)F(t)dt,
-'

(4)

G(x)=T

0

and
1
H (x) = T

(5)

JT G(x +t) G(t)
- dt,
0

and we define the Fourier series.
Inverting the order of integration in the formula for the
Fourier coefficients of G (x), we shall have
1

T

IT G (x) e -t----;y- dx
1IT_
1IT
=p F(t)e TdtT F(x+t)e
.2lTkX

0

i2wkt

-i2,.k(x+t)

· - T - dx.
0
0
On account of the periodicity ofF (x), we have
1
-i2,.k(x+t)
1
-i~~
T F(x+t)e - T
dx=T F(x)e
T dx=Ak.

IT

IT

0

0

26

UNIFORMLY ALMOST PERIODIC FUNCTIONS

Further it is obvious that
1
i2"kt
T F (t)e -T dt =.Ak,

IT_
0

and thus
Thus the Fourier series of G (x) is
i2"kx

G(x)-IJ.AtJ 2 e

T.

As the function H (x) was formed from G (x) in the same way
as G(x) from F(x), we see that the Fourier series of H(x) is
given by the equation
.,2.!!!EE
H (x)-I J.Ak J4e T.
But it is a well-known theorem that if the Fourier series of
a continuous function is uniformly convergent, then its sum is
equal to the function. Thus
2,;kx

and

H (x) =!I .At

j4 ei

H (0) =! j.Ak

4
j •

T-,

limH(O)= 0,

By (3)
as T-+oo, and by (5)

1
lim 'I'

(6)
as T-+oo.
Take a sequence

IT G(x)
1

2
j dx=0,

0

0 < Tt < T2 < ... ,

Tn-+ oo ,

such that, for any n, Tn is a translation number ofj(x) belonging
1
to-. By(6)
n
(7)

Observe that the integrand in this formula varies with n, as
the definition of G (x) depends upon T.
Putting in (4) T= Tn, we shall have for 0 ~ x ~ Tn

UNIFORMLY ALMOST PERIODIC FUNCTIONS

(8) G(x)=

27

~n {f:n-xj(x+t)j(t)dt

+ JTn

- } 1 fTn
8.A.
Tn-x f(x+t-Tn)f(t)dt =-T j(x+t)j(t)dt+-.
n
nO

Take now the u.a.p. function
g (x) =

Mt {j(x + t)j(t)}.

By 3°, §3,
(9)
n~ oo

as

~t.b.
-oo<x<+co

I

I-T
n

fTnf(x+t)j(t)dt-g(x)
I==en~o,
0

,

. By (8), (9)

G(x)=g(x)+B(en+~)·

(10)

Observing that jg(x) I~ .A. 2, I G(x) j ~ .A. 2 for all x, we have

Ii.J:nl G(x)

2
j

dx- ~n

en

2

lg(x)l dx

I< 2.A.2 (en+~)·

Hence, by (7),
1.e.

By the theorem of 10°,
all x, and thus

§3, we conclude that I g (x) 12 = 0 for

g(O)=M {/f(x) /2 1= 0,
which is the Parseval equation. The application of the same
theorem to the function /f(x) j 2 leads us to the conclusion that
f (x) = 0 for all values of x. Thus there exists only one function
f(x) whose Fourier series is nought, and it is identically equal
to nought.

7°. UNIQUENESS THEOREM. If two u.a.p. functions have the
sarne Fourier series, then they are identical.
The theorem follows from the fact that the difference of such
two functions is a u.a.p. function with Fourier series nought.
~

28

UNIFORMLY ALMOST PERIODIC FUNCTIONS

8°. Parseval equation for the general case.

J(x),...., "'
::E AneiAnx

Let

n=l

be any u.a.p. function with its Fourier series. We take again
the function
(I)
g (x)= Mt {j(x + t)j(t)}.
By an argument similar to one employed in 6° for G (x) we see,
on account of Problem of 3°, § 3, that the Fourier series of g (x) is
co

IAn j2 eiAnx.

!

g (x),....,

n=l

By the Bessel inequality (6°,

1.e. the series !"'

I

§3)

An 12 is convergent. Then the series

n=l
co

cf>(x)= !

IAnl2eiAnx

n=l

is uniformly convergent. Thus by the corollary of 12°, § 1, cp(x)
is a u.a.p. function and by the theorem of 8°, §3, the Fourier
series of </> (x) coincides with the series by which it is represented. Consequently the functions g (x) and </> (x) have the
same Fourier series. By the uniqueness theorem (7°)
g (x) =

cp (x),

co

I.e.

g(x)= !

IAnl2eiAnx.

n=l

Putting x = 0, we have

g (0) = "'
::E

n=l

By (1)

I

An 12•

g(O)=M{If(t)l 2 ),

and thus
which is the Parseval equation for the functionf(x).

UNIFORMLY ALMOST PERIODIC FUNCTIONS

29

§ 5. Polynomial approximation to u.a.p. functions.
1°. It has been proved that the limit of a uniformly convergent sequence of u.a.p. functions is a u.a.p. function (8°, § 1).
Finite trigonometrical polynomials being u.a.p. functions we
conclude that the limit of a uniformly convergent sequence of
trigonometric polynomials is a u.a.p. function. The converse is
also true: any u.a.p. function is the limit of a uniformly convergent
sequence of trigonometrical polynomials. This is the main result
of H. Bohr's second paper. For the proof of this theorem H. Bohr
had to develop the theory of purely periodic functions of infinitely
many variables.
S. Bochner arrived at the same result by extending in an
elegant way the Fejer summation to the class of u.a.p. functions.
H. W eyl afterwards gave a new proof of this theorem as a
direct corollary of the Parseval equation. We shall now give
Weyl's proof, but later on we shall consider also Bochner's
method. Weyl's method gives the error of the polynomial
approximation. Bochner does not give the error, but the advantage of his method lies in the fact that it gives a definite
algorithm for the polynomial approximation.
2°.

Given a u.a.p.function

APPROXIMATION THEOREM.
00

j(x),..., ! AneiAnx,
n-1

and a positive number €, there exists a trigonometric polynomial
P (x), whose exponents are Fourier exponents of f(x), and which
satisfies the inequality
IJ(x)-P(x)l<€

for all values of x.

We write

p

.

j(x)- ! Ane•Anx=fp(x).
n-1

By the Parseval equation

30

UNIFORMLY ALMOST PERIODIC FUNCTIONS

Given any 7J > 0, we can evidently find a p such that

11f{ljp(x)I 2}<7J,

(1)
00

since the series ::£

I

A, 12 is convergent,

n=l

On the other hand, by ( 4) 2°, § 3, there exists T0 > 0, such that
(2)

~~J;ifv(x+s)l 2 dx-M{I/v(x)l 2 }1<77

for all T > T0 , and for all real s. Thus, by (1 ), (2),

T1 JTo lfv(x+s)l2dx< 277.

(3)

Let now l 6; 3 be an inclusion interval of E {e/3, f(x)}. Take
a T= N (le;a + 1) > T0 , where N is an integer, and in each of the
intervals {k (l£;3 + 1), k (ZE/a + 1) + l£;3} (k = 0, 1, ... , N -1) take
a '~"k belonging toE {e/3,f(x)}. Taking then a o(0 < o< 1) such
that
lf(x')-/(x") I< e/3,
if only Ix' - x" I < we define the function e (x) in the interval
(0, T) by the equations
e(x)=1 in all intervals h:, '~"k+o), (k=O, 1, ... , N-1)
e (x) = 0 at all other points.
(The intervals ( '~"k• '~"k + o) do not overlap.)
By the Schwarz inequality

o,

(3.1)

u;

2

fv <x +s) e (x) ax 1

:a

J:

l!v <x + s) l2 dx

I:

{e (x)} 2 dx.

Observing that
1

I

Tfv(x+s)e(x)dx=

0

7

Ni J k~fv(x+s)dx
k-0

~

N-1

I:

and that

= ~

k-0

Is fv(x+ '~"k+ s)dx,
0

2

{e (x)} dx =No,

and taking into account (3), we conclude from (3.1)
1

1

Ni
1 k-0

I

6

0

fv (x + '~"k + s) dw

I< ,,271TNo,

UNIFORMLY ALMOST PERIODIC FUNCTIONS
1

or

I

N-

1

f

6

I

No k:o oiP (x + ,.k + s)da: <

31

j271
(le;s + 1)
~-o-- ·

€20

Taking

71 < 18 (lt;s + 1),

we obtain

~~o:~:J;Ip(x+,-k+8)dxl<i·

(4)
Now

~0 J:IP (x+ Tk +8) dx = ~0 J:l(x + Tk + 8) dx- Pk(8),
6

where

Pk(s) == ~!_ J

No

=1-

No

£

on-1

AneiAnC<ll+~k+Bl dx

p
~
eiAns

16

AneiA,tCx+Tkl

dx.

o
Thus Pk (s) is a trigonometric polynomial with the exponents
belonging to Fourier exponents ofl(x).
We now can write (4) in the form
(5)

n-1

I{~o :~:J:I(x+ ,.k+8)dx} -P(8) I< i·
N..:.l

where P(s) =

~
k=O

Pk(8).

Now the value of ohas been chosen in such a way that
.
e
ll<x + Tk + 8)-l(,.k+ 8) I< 3
for 0 ~ x ~ o. Consequently

I~

J:l(x + Tk + s) dx- I (-rk + s)

combining this with

11(,-k+s)-j(s)l~
1

i•

2e .
81 J[0I (x + Tk + 8) dx- I (8) I< "3
6

we obtain

I<~;

We can now replace (5) by

ll(s)- P(8) I< e,
which proves the theorem.

32

UNIFORMLY ALMOST PERIODIC FUNCTIONS

§ 6. Limit periodic functions.
1°. The approximation theorem of the last section enables
us to investigate a class of u.a.p. functions-the class of limit
periodic functions-which is the nearest class, from the point
of view of structure, to the class of purely periodic functions:
DEFINITION. A function f (x) is called limit periodic if it is the
limit of a uniformly convergent sequence {fk (x)}, (k = 1, 2, ... ) of
continuous purely periodic functions.
Thus limit periodic functions are u.a.p.

A u.a.p.function
f (r.c),..., ~AneiAnx,
for which the ratio of any pair of Fourier exponents is rational, is
limit periodic.
We can obviously write in this case An=rnq (n= 1, 2, ... ),
where all rn are rational. By the approximation theorem there
exists a sequence of finite sums
sk (r.c) = ~b~leirnqx
2°.

THEOREM.

uniformly convergent to f(r.c), which proves the theorem, since
the sums sk (r.c) are obviously purely periodic functions.
3°. CoNVERSE THEOREM. The Fourier series of any limit
periodic function f (x} can be represented in the form
j(r.c)-- ~Aneirnqx,

where all rn are rational.
Let jic(r.c), (k = 1, 2, ... )be a sequence of continuous pu~ely
periodic functions, uniformly convergent to the function f (a:),
and let
+co
/k (r.c),..., ~ a~lei!'lm/w•x,
(1)
v==-co

where qk is the period of /k (r.c). The Fourier exponents of all
the functions (1) can be written as a simple sequence {An},
(n = 1, 2, ... ) and then all the series (1) can be written in the
form
(2)

UNIFORMLY ALMOST PERIODIC FUNCTIONS

33

where Al:'l can differ from nought only if A, is a multiple of
2
71". We now write the Fourier series of j(x)

qk

J (x)......, ~A,.eiAnx.

(3)

By the theorem of ll 0 , § 3,

A,.=

(4)
ask-+oo.

limA~l,

(n=l,2, ... )

As the theorem is obvious when f (x) is equal to a constant,
we may assume j(x) to be different from a constant. Then
there exists n 0 such that A,.0 =f 0, An0 =f 0. Corresponding to
this n0 there exists k 0 such that
1/~c(x)-j(x)I<IAnol

for all k > k 0 • We have

I A~- An0 I= I M [{fk (x)- j (x)} e-iAnox] I
~ M { l/~c (x)- f(x) I}< I Ano I
for k > k 0 • Thus A~~ =f 0 for all k > k0 , and consequently A,. 0
is a multiple of

71"

qk

for all k > k 0 • We clearly have

- rv(klA no•
-27r v-

(5)

where

2

r:l

qk

(- oo < v < + oo , k > k0 )

is rational. By the same argument we see that any
2
A,., to which corresponds A,=fO, is a multiple of 71" for suffiq~c

ciently large k, and consequently by (5) is a rational multiple
vf A,.0 • Thus we can write
A,.=r,.A,'o,

(r.,.=rational)

for all n for which A,. =f 0. Writing q = Ano we have the theorem.

Corollary. If a sequence {fk (x)} of continuous purely periodic
functions is uniformly convergent to a function which is not a
constant, then there exists an integer ko such that the periods of all
the functions fk (x) for k > ko are rational multiples of the same
number.
4°. By theorems of 2° and 3° we have proved the following:

34

UNIFORMLY ALMOST PERIODIC FUNCTIONS

THEOREM. The class of all limit periodic functions is identical
with the class of all u.a.p. functions all whose Fourier exponents
are rational multiples of the same number.

§ _7. Base of u.a.p. functions. Connection of u.a.p.
functions with limit periodic functions of several
variables.
1°, We shall now study the arithmetic nature of Fourier
exponents of u.a.p. functions ; this will lead us to important
new results concerning u.a.p. functions.
DEFINITION 1. .A set ai (i = 1, 2, ... ) of real numbers is called
linearly independent if for any n the only rational values of
r1. r 2 , ... , r,. satisfying the equation
(1)
are r1 = r2 = ... = rn = 0.
Let
(2)
j(x),..., ~AneiA,o:
be a u.a.p. function with its Fourier series.

2. .A finite or enumerably infinite set ai (i = 1, 2, ... )
of linearly independent numbers is called a base of the function f (x)
(or of the Fourier exponents off (x)) if every Am can be represented
as a finite linear form of a's with rational coefficients, i.e. if for
any m we can write
(3)
Am== ?'~mlal + r2m 1aa +· ... + rln~a"m,
where all r' s are rational.
Evidently every u.a.p. function has a base, and there may be
many different bases of the same function, but for a fixed base
a representation in the form (3) is always unique.
If a base consists of infinitely many terms it is called an
infinite base, otherwise a finite base. If all r's in (3) are integral
for all m the base is called an integral base.
The result of the preceding section can now be expressed in
the form:
The class of all limit periodic functions is identical with the
class of u.a.p. functions with one-term base.
DEFINITION

UNIFORMLY ALMOST PERIODIC FUNCTIONS

35

Evidently a ?f.a.p. function with one-term integral base is
purely periodic.
2°. DEFINITION 1. If a sequence of continuous purely periodic
functions Fk(Xt, x2 , ... , Xm) converges uniformly to a function
F (:11., x2 , ••• , Xm), then this function is called a limit periodic
function of m variables.
If all the functions F~c (x1 , x2 , ... , Xm) have common periods
with respect to each variable, then clearly the function
F (x11 Xa, ••• , Xm)
is purely periodic.
Suppose we have a sequence of continuous purely periodic
functions Fk (x1 , x2 , ••• , Xm 1.) such that lim m~c = oo. Then we
write for any k
<l>k(xl> Xa, ••• ) = Fk (Xt, X2, ... , Xmk),
so that <l>~c(x~> x 2 , ••• ) is constant with respect to each of the
variables XmHl• Xrn.k+2• • • • • If the sequence <l>~c (xi> x 2 , ... ) converges to a function F(xt. x 2 , ... ),then we say that the sequence
Fk(Xt. x 2 , ... , Xmk) converges to the function F(x1 , x 2 , ... ) of
infinitely many variables.
DEFINITION 2. If a sequence Fk (Xt, x., ... , Xmk) (lim m~c = oo)
of continuous purely periodic functions converges uniformly to
a function F (xt, x., ... ), then we call the function F (Xt, x., ... )
a limit periodic function of infinitely many variables.
If all the functions F~c (x1 , x2 , ... , Xmk) have common periods
with respect to each of the variable8 x1 , x2 , ••• , then clearly the
function F(x1 , x 2 , ... ) is purely periodic.

3°. We shall now quote a theorem on Diophantine approximations.
KRONECKER'S THEOREM. If x1°, :ta0, ... , Xm0 are any real
numbers and qc1, q2-1, ... , qm-l any real linearly independent
numbers, then to any number o> 0 correspond a number E and
m integers k1o k2 , ... , km such that the inequalities
IE-xi0 -kiqil<o,
(i=l, 2, ... ,m)
are satisfied.

36

UNIFORMLY ALMOST PERIODIC FUNCTIONS

4°. Let F(ah, x2 , ••• , xm) be a continuous ,purely periodic
function with periods q1 , q2 , ••• , qm with respect to the variables
x1, .x2, ... , Xm· We assume q1-I, q2- 1 , ... , q:;;.' to be linearly independent, and we introduce "a diagonal function" f(x) of
F (x1 , x2 , ••• , Xm) by the equation
f(x)=F(x, a:, ... , a:).

The aggregate of all the values of the diagonal
functionf(x) is everywhere dense in the aggregate of all the values
ofF (x11 x2 , ••• , Xm)·
To prove the theorem we have to prove that, given an arbitrary
point (x1°, x2°, ... , Xm0) and an arbitrary positive number €, there
exists a real number g such that
THEOREM.

(1)

The function F(x1, x 2 , ••• , x,,.), being continuous and purely
periodic, is also uniformly continuous. Consequently there exists
a number o> 0 such that

IF( XlI ' X2I ' .. •' Xrn I) - F(Xl" ' a;2" ' ... 'Xm")I < €,
if only Ix/ - x/' I < o for all i = 1, 2, ... , m. By the preceding
theorem we can define a number g and integers k1 , k 2 , ••• , km
to satisfy the inequalities
(2)

Ig- xP- kiqi I< o.

By (2)

! F(x1°+k1q1, X2°+ k2q2, ... , Xm0 + kmqm) -F(g, g, ···,g) I< €,
which proves (1).

Corollary.
Y.b.

-m<~<+oo

IF(xl, x2 ,

••• ,

Xm)l=

u.b.

-oo<~<+m

1/(x)l.

-m<xm<+oo

5°. THEOREM. .Any u.a.p. function is the diagonal function of
a limit periodic function of a finite or an infinite number of
variables.

UNIFORMLY ALMOST PERIODIC FUNCTIONS

Let

37

J (x) ~ ~AmeiAmx
....

A

,-.w ~..Cl.m

ei(r~m)a!+T~m)a2+ ... +r~)an
m

m

)x

be a u.a.p. function with a base
By the approximation theorem there exists a sequence
s~c(x)

= ~ bV;,,>eiAmx
•f (m) + (m)
(m)
\
= ..,..4 b~>e~,rl
a! r2 a• + ... + rn,.
a.,,mlx,

(k= 1, 2, ... )

of finite trigonometric polynomials uniformly convergent to the
functionf(x). Every sk (x) is the diagonalfunction of the purely
periodic function
v (x1, WI'
,.. ••• , wllk
,.. )-""'bCklei(r~mlalx
xn,) '
1 +r~m)a,x2 + ... +r~m)an
L'k
_., m
m m
whose periods q1 ,q2 , ... , q.k are integral multiples of the numbers
27T 27T
27T
- , -, ... , "-.
~
aa
a.k
The reciprocals of these periods are evidently linearly independent. By the same argument the reciprocals of the periods
of any difference
F1c· (x1, Xz, ••• , Xv~c•)- Fk" (xv x2 , ••• , x•k")
are linearly independent. By the corollary of 4°
u.b.J F~c, (x1, x2 , ••• , Xv,e)- Fk" (x11 X 2 , ••• , Xv~c") j
= u.b.! sk' (x)- sk" (x) [.
The sequence {s~c (x)}, being uniformly convergent, we see that
the sequence {F~c (x1 , x 2 , ••• , x.k)} is also uniformly convergent.
Let F(xl> x2 , ... ) be its limit. It is a limit periodic function.
Writing
f(x)=lims~c(x)=limF~c(x, x, ... , x),
we conclude that
f(x)=F(x, x, ... ),
which proves the theorem.
Evidently if the functionf(x) has an integral base av a 2 , ... ,
then the function F(x1 , x2 , ... ) will be purely periodic, and if
the base is finite, then F (x1 , x 2 , ... ) is a limit periodic function
of a finite number of variables.


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