almost periodic functions and differential equations

Contents 1 Definition and elementary properties of almost periodic 6 Some simple properties of trajectories 11 Comments and references to the literature 12 2 Harmonic analysis of alm

Almost periodic functions and differential equations

Almost periodic functions

Published by the Press Syndicate of the University of Cambridge,

The Pitt Building, Trumpington Street, Cambridge CB2 1RP

32 East 57th Street, New York, NY 10022, USA

296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia

© Moscow University Publishing House 1978

English edition © Cambridge University Press 1982

Originally published in Russian as Pochti - periodicheskie funktsii

differentsial' nye uravneniya by the Moscow University Publishing House 1978 Assessed by E D Solomentsev and V A Sadovnichii

First published in English, with permission of the Editorial Board of the Moscow

University Publishing House, by Cambridge University Press 1982

Printed in Great Britain at the University Press, Cambridge

Library of Congress catalogue card number: 83 4352

British Library Cataloguing in Publication Data

Contents

1 Definition and elementary properties of almost periodic

6 Some simple properties of trajectories 11

Comments and references to the literature 12

2 Harmonic analysis of almost periodic functions 14

1 Prerequisites about Fourier—Stieltjes integrals 14

3 The mean-value theorem; the Bohr transformation;

Fourier series; the uniqueness theorem 21

5 Almost periodic functions with values in a Hilbert

6 The almost periodic functions of Stepanov 33

Comments and references to the literature 36

2 The connection between the Fourier exponents of a

vi Contents

4 Theorem of the argument for continuous numerical

Comments and references to the literature 51

4 Generalisation of the uniqueness theorem (N-almost

1 Introductory remarks, definition and simplest

properties of N-almost periodic functions 53

2 Fourier series, the approximation theorem, and the

Comments and references to the literature 62

1 Definition and elementary properties of weakly almost

2 Harmonic analysis of weakly almost periodic functions 68

Comments and references to the literature 76

6 A theorem concerning the integral and certain

2 Further theorems concerning the integral 81

3 Information from harmonic analysis 87

4 A spectral condition for almost periodicity 91

5 Harmonic analysis of bounded solutions of linear

Comments and references to the literature 96

7 Stability in the sense of Lyapunov and almost

4 Corollaries of the separation lemma (continued) 107

5 A theorem about almost periodic trajectories 109

6 Proof of the theorem about a zero-dimensional fibre 113

7 Statement of the principle of the stationary point 116

Contents vii

8 Realisation of the principle of the stationary point

9 Realisation of the principle of the stationary point

Comments and references to the literature 123

2 Weak almost periodicity (the case of a uniformly

4 Weak almost periodicity (the general case) 134

5 Problems of compactness and almost periodicity 135

6 Weakening of the stability conditions 140

7 On solvability in the Besicovitch class 142 Comments and references to the literature 147

1 General properties of monotonic operators 149

2 Solvability of the Cauchy problem for an evolution

3 The evolution equation on the entire line: questions

of the boundedness and the compactness of solutions 157

4 Almost periodic solutions of the evolution equation 161 Comments and references to the literature 165

10 Linear equations in a Banach space (questions of

2 The connection between regularity and the

exponential dichotomy on the whole line 170

Comments and references to the literature 181

11 The averaging principle on the whole line for

2 Some properties of parabolic operators 183

viii Contents

Comments and references to the literature 199

Preface

The theory of almost periodic functions was mainly created and

published during 1924-1926 by the Danish mathematician Harald Bohr Bohr's work was preceded by the important investigations of

P Bohl and E Esclangon Subsequently, during the 1920s and

1930s, Bohr's theory was substantially developed by S Bochner, H

Weyl, A Besicovitch,, J Favard, J von Neumann, V V Stepanov,

N N Bogolyubov, and others In particular, the theory of almost periodic functions gave a strong impetus to the development of

harmonic analysis on groups (almost periodic functions, Fourier

series and integrals on groups) In 1933 Bochner published an important article devoted to the extension of the theory of almost periodic functions to vector-valued (abstract) functions with values

in a Banach space

In recent years the theory of almost periodic equations has been developed in connection with problems of differential equations, stability theory, dynamical systems, and so on The circle of applica- tions of the theory has been appreciably extended, and includes not only ordinary differential equations and classical dynamical systems,

but wide classes of partial differential equations and equations in

Banach spaces In this process an important role has been played

by the investigations of L Amerio and his school, which are directed

at extending certain classical results of Favard, Bochner, von Neumann and S L Sobolev to differential equations in Banach spaces

We survey briefly the contents of our book In the first three chapters we present the general properties of almost periodic func- tions, including the fundamental approximation theorem From the

x Preface

very beginning we consider functions with values in a metric or

Banach space, but do not single out the case of a finite-dimensional

Banach space and, in particular, the case of the usual numerical almost periodic functions Of the known proofs of the approximation theorem we present just one: a proof based on an idea of Bogolyubov

However, it should be noted that another instructive proof due to Weyl and based on the theory of compact operators in a Hilbert space appears in many textbooks on functional analysis

Chapter 4 is devoted to the theory of N-almost periodic functions

In comparison with the corresponding chapter of the book

Almost-Periodic Functions by B M Levitan (Gostekhizdat, Moscow (1953)),

we have added a proof of the fundamental lemma of Bogolyubov

about the structure of a relatively dense set

Chapter 5 is concerned with the theory of weakly almost periodic functions developed mainly by Amerio

Chapter 6 contains, as well as traditionally fundamental questions (the theorem of Bohl—Bohr about the integral, and Favard's theorem about the integral), more refined ones', for instance, the theorem of

M I Kadets about the integral

We mention especially Chapter 7 whose title is Stability in the sense of Lyapunov and almost periodicity The two chapters that follow it are formally based on it Actually, we use only the simplest results, and when there is a need to refer to more difficult propositions

we give independent proofs Therefore, Chapters 6-11 can be read independently of one another

Chapter 8 contains Favard theory, by which we mean the theory

of almost periodic solutions of linear equations in a Banach space

In Chapter 9 the results from the theory of monotonic operators are applied to the problem of the almost periodicity of solutions of functional equations In Chapter 10 we give another approach to the problem of almost periodicity Finally, Chapter 11 is slightly outside the framework of the main theme of our book In it we give one of the possible abstract versions of the classical averaging principle of Bogolyubov

Chapters 1-5 were written mainly by B M Levitan, and Chapters

6-11 by V V Zhikov

The authors thank K V Valikov for his assistance with the reading

of the typescript

Translator's note

This translation has been approved by Professor Zhikov, to whom

I am grateful for correcting my mistranslations and some misprints

in the original Russian version

Professor Zhikov has asked me to mention that the theory of

Besicovitch almost periodic functions is not reflected fully enough

in the book, since this theory has recently been applied in spectral

theory and in the theory of homogenisation of partial differential equations with almost periodic coefficients The additional references are, in the main, concerned with this theme

L W Longdon

1 Almost periodic functions in metric

Definition 1 A set E c J of real numbers is called relatively dense

if there exists a number 1>0 such that any interval (a, a +1) c f of

length 1 contains at least one number from E

Definition 2 A number T is called an e-almost period of f :1 -* X if

teJ

Definition 3 A continuous function f : j -+ X is called almost periodic

if it has a relatively dense set of 6-almost periods for each 6 >0, that

is, if there is a number 1 =1(e)> 0 such that each interval (a, a +1)c J

contains at least one number T -= T E satisfying (1)

Every periodic function is also almost periodic For if f is periodic

of period T, then all numbers of the form nT (n = E1, ±2, ) are also periods of f, and so they are almost periods of f for any e >0

Finally, the set of numbers nT is relatively dense It is easy to produce examples of almost periodic functions that are not periodic, for instance, f(t)= cos t + cos t.s12

2 Almost periodic functions in metric spaces

We prove some of the simplest properties of almost periodic functions; these are straight-forward consequences of the definition

Property 1 An almost periodic function f :J *X is compact in the sense that the set is compact

Proof It is sufficient to prove that for any e > 0, Rf contains a finite e-net for Rf Let 1=1(e) be the length in Definition 3 corresponding

to a given E We set

Rfa = e f: X = Pt), //21

From the continuity of f it follows that the set Rhi is compact; we

show that it is an E-net for the set RI Let to € J be chosen arbitrarily,

and take an e-almost period 7 = re such that to+ T 1/2, that is,

Remark For numerical almost periodic functions (that is, when

X = R 1) and for almost periodic functions with values in a dimensional Banach space, Property 1 reduces to the following: if

finite-f is an almost periodic function, then R f is bounded

Property 2 Let f :J > X be a continuous almost periodic function Then f is uniformly continuous on J

Proof We take an arbitrary e >0 and set E l = e/3 and 1 = 1(0 The

function f is uniformly continuous in the closed interval [—I, 1+1], that is, there is a positive number 8 = 3(E 1 ) (without loss of generality

we may assume that 8 < 1) such that

whenever Is"— s'l< 8, s', s"EJ Now let t', t" be any numbers from J

for which It' — CI< 8 We take a 7 = TEI with 0 /, that is,

tT El —t'+1 Then t'' + 7E, e [-1, 1+1] We set s' = t'+ re , and

s"= t" + rE, From (1), (2) and the triangle inequality we have

PU(t"), fit'))=P(Pt"), fis"))+P(Ps"),Ps1))

+P(fis'), f(e))< E

Definition and elementary properties 3

Property 3 Let fn :J - X, n = 0, 1, 2, , be a sequence of continuous almost periodic functions that converges uniformly on J to a function

f Then f is almost periodic

Proof We take an arbitrary e > 0 and let n = n e be such that

te j

Let T = T[ file ] denote an (e/3)-almost period of the function fne Then

it follows from (1), (3), and the triangle inequality that

Proof Since the set 22 c is compact and the function g(x) is

con-tinuous on -f , g(x) is uniformly continuous on 0.-/f Therefore, for all 6 >0 there exists a 8 = 8(e)> 0 such that for all x', x"E with

Corollary Let f be a continuous almost periodic function with values

in a Banach space X Then l[f(t)li k is a continuous numerical almost periodic function for all k >0

Property 5 Suppose that fis an almost periodic function with values

in a Banach space X If the (strong) derivative f' exists and it is uniformly continuous on J, then f' is an almost periodic function

Proof The proof uses the concept of an integral of a vector-valued function In the case of continuous functions this is very simple because the Riemann integral exists with the usual fundamental

4 Almost periodic functions in metric spaces

properties (see, for example, G E Shilov, Mathematical Analysis

Functions of a Single Variable, Part 3, Ch 12, § 12.5) By hypothesis,

the derivative f' is uniformly continuous, and so for all s >0 there

is a ô = 8(e)>0 such that It(e) — f (r)ii< s whenever le— t"1 < S

l[f (t +ri)—f(t)11 dri <E

Consequently, the sequence of almost periodic functions On (t)=

n[f(t +1/n) —f(t)] converges uniformly on J to f(t) Now we only

need to use Property 3

2 Bochner's criterion

The main results of this section are also valid for almost

periodic functions with values in an arbitrary metric space X But

for simplicity we shall assume that X is a Banach space We shall

use the following notation:

X denotes a complex Banach space; x, y, z, are elements of X,

and iix11 is the norm of x EX C(X) denotes the Banach space of

continuous bounded functions f: J 2Y with the norm

iifit)ilc(x) = sup

teJ

and 0(X) is the subspace of C(X) consisting of almost periodic

functions Let us note that the spaces C(X) and O(X) are invariant

under translations, that is, C(X) (0(X)) contains together with

f = f(s) the function f (s)= f(s +t) for all t E J

1 Bochner's theorem Let f:J -+X be a continuous function For f

to be almost periodic it is necessary and sufficient that the family

of functions H = = {fit + h)}, —co < h <co, is compact in C(X)

Proof (a) Necessity We assume that f is an almost periodic

func-tion (see § 1, Definition 3) We denote by {r} the set of all rational

points on J and let If hnl= {fit + h n )} be an arbitrary sequence of

functions from H By using Property 1 and applying the diagonal

process, we can select from the sequence {Pt + hn )} a subsequence

(we denote it again by {f(t + hn)}) which converges for any r E {r}

We prove that the sequence {fit + hn )} converges in C(X) We take

an arbitrary E >0 and let 1= l be the corresponding length Let

Bochner' s criterion 5

8 = 8(s) be chosen in accordance with Property 2 We subdivide the segment [0, 1] into p segments ilk (k = 1, 2, , p) of length not greater than 8, and in each Ai( we choose a rational point rk Suppose

that n = nE is chosen so that

for n, m._.- nE and k = 1, 2, , p For every to e J we find a r = ro such that

0. -to +r .5/

Suppose that the number t'o = to + r falls in the interval 41(0 and that

rko E Ak o is the rational point chosen earlier Then by our choice of

8 we have

lifle o + hn) — firk o + hn)li< 6,

ilf(t 10+ hm) — firk o + h„,)il< E

It follows from (4) and (5) that

lif(to + hn) — f(to+ h )1I

=ii.f.(to+ hn) — Pt' o + hn)II±ii.f(t1 o + hn) — Erko + h,, )II

+11f(rko+hn)—f(rko + LAI ±iif(ko+ hm) — Re 0 + hm)li +Mt' o+ h.) — fit° + LAI <5E

Since to E j was chosen arbitrarily, the last inequality implies that

the sequence {f(t + hn)} converges in C(X), that is, the set H is

compact in C (X)

(b) Sufficiency We assume that the family {f(t + h )}, —oo < h <cc,

is compact in C(X) and prove that f(t) is almost periodic (in the

sense of Definition 3, § 1) First of all we show that f is a bounded function For if this were not the case, then we could find a sequence

of numbers hn for which Ilf(hn)11-*co But then neither the sequence

If (t + hn)} nor any subsequence of it would be convergent at t =0 From the boundedness of f it follows that the family of functions {P} = { f(t + h)}, —co < h <cc can be regarded as a set in C(X)

By a criterion of Hausdorff, for all E >0 there are numbers

h l , h2, , h„ such that for all h € J there is a k = k(h) such that

6 Almost periodic functions in metric spaces

that is, the numbers h — h k (h) (k =1, 2, , p) are E-almost periods

for f(t) Now we only need to prove that the set of numbers h — hk

is relatively dense We set

L= max Ihki

1 .1c çp

Then

h—L 5h—h k -h+L,

and since h is arbitrary this inequality implies that every interval

of length 2L contains an 6-almost period for f

2 Now we are going to deduce further properties of almost periodic functions that are obtained more simply from Bochner's criterion than from our definition

Property 6 The sum f(t)+g(t) of two almost periodic functions is almost periodic The product of an almost periodic function f(t) and

a numerical almost periodic function 0 (t) is almost periodic

Proof Let {hn } be an arbitrary sequence of real numbers Firstly

we extract from it a subsequence {h' n } such that the sequence of

functions If(t + h'n )} converges, and then a subsequence {h"n } of { h'n}

for which the subsequence of functions {g(t + h"n )} is convergent Then, clearly, the subsequence { f(t + h"n )+g(t + h" n )} is convergent

Similarly, the product can be proved to be an almost periodic function

Let X 1 , X2, , Xn be Banach spaces, and let X = nkXk be their cartesian product, that is, the Banach space with elements x = (x 1 , x 2, , Xn) and the norm

11x11 = kill lixkil

It follows easily from Bochner's criterion that if fi (t), f2(t), , fn (t)

are almost periodic functions from J into X 1 , X2, , Xn, then the

function f(t)= (Mt), f2 (t), , fn (t)) is an almost periodic function from J into X The next property is easily deduced from this remark

Property 7 Let fi(t), f2(t), , fn(t) be almost periodic functions from

J into Banach spaces X 1 , X2, , X n, respectively Then for every

s > 0, all the functions Mt), f2(t), , fn (t) have a common relatively dense set of E -almost periods

3 The next property gives a condition for the compactness of a set

of functions from 0(X), and is known as Lyusternik's theorem

Lyusternik's theorem A set M c 0(X) is compact if and only if the

following three conditions are satisfied:

(1) For every fixed to E J the set

Eto = {x eX: x = f(to)JEM}C X

is compact

(2) The set M is equicontinuous, that is, for every £ >0 there is a

8 = 8(e) such that lif(e)— f(r)ii<e whenever It' — ri< 8 for all f e M

(3) The set M is equi-almost periodic, that is, for every 6 > 0 there

is an 1 = lE such that every interval (a, a + 1)c f contains a common

e-almost period for all f€ M

Proof (a) Sufficiency The proof is exactly the same as that of the

necessity for the conditions in Bochner's theorem

(b) Necessity By the criterion of Hausdorff, for every e >0 M

contains a finite e-net: fi , f2, , fn Therefore, for all f € M there is

a ko, 1 -_ ko n, such that

te i

For any to e J, from (7) we obtain

Ilf(to)—fko(to)11< 8,

and so the finite set of elements fi(to), f2(t0), , fn(to) forms a finite

e-net for the set Etc, Consequently, Etc, is compact in X, that is, condition (1) of Lyusternik's theorem holds Condition (2) follows from the uniform continuity of each fk (t) (k = 1, 2, , n) on J and

from (7) Finally, condition (3) follows from (7) and Property 7

Remark For numerical almost periodic functions, condition (1) of Lyusternik's theorem can be restated as follows: the set Eto is bounded

8 Almost periodic functions in metric spaces

3 The connection with stable dynamical systems

Suppose that we are given a 1-parameter group of morphisms of a metric space X, S(t) : X -› X (t E J) If for any x E X

homeo-the corresponding trajectory x t = S(t)x is a continuous function J -+ X

we shall call S(t) a dynamical system or flow

A flow S(t) is called two -sidedly stable or equicontinuous if the transformations S(t) (t EJ) are equicontinuous on every compact set from X

The next property is obtained from Bochner's criterion

Property 8 Every compact trajectory of a two -sidedly stable flow is

an almost periodic function

Proof We set f(t)= S(t)x Since a trajectory is compact, we can extract from any sequence If(tn )1 a fundamental subsequence {Min»

The transformations S(t) are equicontinuous on the set f, and so

sup P(.f(t + t i.), Pt + en )) B

teJ

whenever p(f(e.), fit' n)) 5 8, that is, Bochner's criterion holds The converse holds in a certain sense: with each almost periodic function f :J - - > X can be associated a compact trajectory of a two-sidedly stable dynamical system For if we consider in C(X) a system

of translates, then the trajectory ft = f(s +t) is compact Since the distance between two elements of C(X) is invariant under a transla-tion, we have an isometric and so two-sidedly stable flow It is worth noting that the difference between isometry and two-sided stability

is essentially insignificant; if a two-sidedly stable flow is defined on

a compact space X, then it can be made isometric by choosing the following metric

d(xl, x2) = sup p(S(t)xi, S(t)x2)

teJ

It is easy to see that the metric d is invariant under translation and

topologically equivalent to the original metric p

Letfj - Xbe an almost periodic function We denote by k = Alf)

the closure of the trajectory f = f(s + t) in C(X), and are going to show that ge is minimal in the sense that any trajectory is everywhere

Stable dynamical systems 9

The minimal property of an almost periodic function proved

in the last section is in fact a very simple property of abstract trajectories

1 Let X be a Hausdorff topological space

We shall call a 1-parameter semigroup of continuous operators

S(t) : X - X (t 0) simply a semigroup, and shall use the symbols x t, x(t) to denote the semitrajectory S(t)x (x EX, t 0) A function x(t)

is called a trajectory of a semigroup S(t) if x(t + r) (t is a

semitrajectory for every T El A set Xo c X is called invariant if through each of its points passes at least one trajectory that is entirely contained in Xo An example of a closed invariant set is the closure

Proof Let X1 denote the closure of a compact semitrajectory

Obviously, the set nt_,0s(t)x, is compact and invariant We order the compact invariant sets by inclusion and apply Zorn's lemma, thus proving the existence of a minimal compact invariant set

The trajectories that belong to a compact minimal set are tionally called recurrent (in the sense of Birkhoff); an example of a recurrent trajectory is an almost periodic trajectory

conven-2 Suppose that we are given two semigroups defined on X and Y, respectively Then there is an obvious semigroup on the cartesian

product X x Y (the `semigroup product')

Two trajectories x(t), y(t) are called compatibly recurrent if the trajectory {x (t), y(t)} is recurrent in X x Y Clearly, compatible recur- rence implies the recurrence of each component, but the converse does not hold

10 Almost periodic functions in metric spaces

We say that a trajectory is absolutely recurrent if it is compatibly recurrent with any recurrent trajectory In Chapter 7 we prove that

an almost periodic trajectory is absolutely recurrent

5 A theorem of A A Markov

We consider a semigroup S(t) (t 0) on a complete metric space X, and call S(t) Lyapunov stable if the transformations S(t) (t ? - 0) are equicontinuous on every compact set from X

Markov's theorem The restriction of a Lyapunov stable semigroup

to a compact invariant subset is a two -sidedly stable group In particular, every continuous compact trajectory is almost periodic

Proof Let X be a compact invariant subset We introduce on X the equivalent metric

d(x i , x2) = sup p(xi t, x2 t ),

to

which has the property d(x i t, x2t )= d(xi, x2) for t_ - 0 Let Z = X x X

We define a metric on Z by the relation

d(z i , z2) , d(xi, x2) + d(Yi, y2),

where z1= {xi, Yi} and z2 = Ix2, Y21 Since X is invariant, through every point z = z(0) e Z at least one trajectory z(t) passes Let A c Z

be the set of elements z = {x, y} such that there is at least one trajectory

z(t) = {x(t), y(t)} with

d(x(t), y(t)) -= d(x, y) (t e j)

The set A is closed and invariant in Z We are going to prove that

A =Z Suppose this is not the case, that is, there is a zo o A

Let zo(t) = zo t be some trajectory and z 1 be a limit point of the form

funda-t E j Therefore we have convergence to some trajectory z i (t):

zo(t + tim)-> zi(t) (t €J)

Markov's theorem 11

Since the function d(x o(t), yo(t)) is non-increasing, we have

d(x i (t), y i (t))= lim d(x 0(t + t' in ), yo(t + t' in )) const.,

m-»CO

that is, z 1 e A The contradiction proves that A = Z

From our conclusion that A = Z it follows easily that S(to)xi 0

S(t o)x 2 for x 1 0 x2(t .- 0), that is, through any point from X a unique

trajectory passes It is also easy to conclude that the mapping

5 (to) : X -+ X is 'onto', that is, the inverse mappings S -1 (t0) are

continuous This proves the theorem

The next proposition is proved by a similar argument

Proposition 1 Suppose that on a compact metric space K there is defined a non-contractive operator T :K -+ K, that is,

P(Txi, Tx2) .- P(xi, x2)

Then TK = K

6 Some simple properties of trajectories

1 The results of this section are purely subsidiary We consider some general properties of the so-called continuous semigroups

A semigroup S(t) (t 0) is called continuous if every tory of it is a continuous function r > X, where r denotes the semiaxis [0, co)

semitrajec-Proposition 2 Suppose that S(t) is a continuous semigroup on a compact metric space X Then when t ranges over a finite interval

on the open semiaxis (0, 00), the transformations S(t) are tinuous

equicon-Proof We set Z = X x X, and consider the space B of all continuous scalar functions 0(z) on Z, and an obvious semigroup of linear operators on B:

O t (z)= 0(S (t)z)

(here z = (x i , x2)) Since the trajectories are continuous, we easily see that the function 0 t :I + > B is measurable But then, as is well known from the theory of semigroups of linear operators (see Dun-ford & Schwartz [40], p.616), the function 0 t is continuous on (0, oo) Hence, by putting 0(z) = p (x i , x 2) we obtain the required result

It follows from Proposition 2 that a compact semitrajectory of a continuous semigroup is uniformly continuous on the semiaxis J,

12 Almost periodic functions in metric spaces

and that trajectories belonging to a compact invariant set are formly continuous on the whole axis

uni-2 To the concept of a recurrent trajectory (see § 4) there corresponds the obvious concept of a recurrent function

Let K be a complete metric space, and let 0(K) denote the set of

all continuous functions J -> K with the topology of uniform gence on each finite segment For f(s) e 0(K) we set f t = f(s +t)

conver-A function f(s)e 0(K) is called recurrent if the trajectory f t is

recurrent in 4P(K)

There is a natural connection between recurrent functions and recurrent trajectories Let x t be the recurrent trajectory of a con-tinuous semigroup defined on a complete metric space X, and let

0 :X -+ IC be a given continuous function Then it follows easily from Proposition 2 that f(t)= 0(x t ) is recurrent In particular, if 0

is a scalar function, then since every semitrajectory is everywhere dense in a minimal set, it follows that

sup f(t)= sup f(t)= sup f(t)

teJ

Comments and references to the literature

§ 1 The definition of an almost periodic function and its simplest properties for numerical functions is due to Bohr [17] and [22]

Long before the publication of Bohr's work, Bohl [15] and Esclangon [120], [121] had discussed a special case of almost periodic functions which are now known as conditionally periodic (or sometimes,

quasiperiodic) functions In contrast to Bohr's definition in which the only condition on almost periods was relative denseness, the definition of Bohl and Esclangon imposed further conditions The latter definition is as follows: A continuous function f is called

conditionally periodic with periods 27r/A 1 , 27r/A 2, , 27r/A„, if for every E >0 there is a 3 = 3(E ) > 0 such that each number r satisfying

the system of inequalities

Comments and references 13

tions is discussed in Chapter 3, § 3, and the role of the system of inequalities (8) in the theory of almost periodic functions is con-sidered in Chapter 3, § 2, and Chapter 4, § 1 (Bogolyubov's theorem) The extension of the theory of almost periodic functions to vector-valued (abstract) functions is due to Bochner [27] Bochner's work was preceded by an important article by Muckenhoupt [93] who considered essentially a special class of abstract almost periodic functions with values in a special Hilbert space It is interesting to note that the concept of a Bochner measurable and summable func-tion, which is being widely extended at the present time, had its origins in Bochner's investigations on abstract almost periodic functions

§ 2 The compactness property of an almost periodic function was discovered by Bochner [25] Lyusternik's theorem was published in [83]

§§ 3 and 4 The connection between almost periodicity and stable dynamical systems is well known (see the monograph of Nemytskii

& Stepanov [95], Ch 5); we have presented only the most elementary facts

§ 5 The following result is due to Markov: a compact Lyapunov stable trajectory of a dynamical system is almost periodic In fact, the result which we have called Markov's theorem says slightly more Proposition 1 was first stated in a paper by Brodskii & Mil'man [30], and then in a more general form in a book by Dunford & Schwartz ([40], p 459)

2 Harmonic analysis of almost periodic

functions

1 Prerequisites about Fourier-Stieltjes integrals

1 Let o-(A), A GI, be a numerical (complex-valued) function of bounded variation on the real line

The Fourier - Stieltjes transform of o-(A ) is the function f(t) defined

by

f(t)= f co exp (iAt) do-(A)

Let A 1 , A2, denote the points of discontinuity of cr(A) in any order, and d 1 , d2, the corresponding jumps, that is,

for all ii, E J, and that for kt, 0 A, '= 1, 2, • ,

lim E d, sin (A,, —,(A.)T

T -)00 y T(A„—,u)

The series in (5) is majorised by E, 141/ 71/kw —IL I, and for ii, 0 A„

(v =1, 2, ) each term in the series (5) tends to zero as T -4 00; henQe

we obtain the equality (5)

For the proof of (4) we set for any 8>0

Joe

clAs (A + tk) Loo (A — tk)T f_„, AT

sin AT

fiAl ,s A 1

sin AT + clA s (A + ,u) 1 ki>8 AT

=Ai+A2

Then by using standard estimates for Riemann—Stieltjes integrals

we have

1.8 lAilj - _s idAs(A + ) 1= vae m i,1 Is(A )1

= Var r 8 Is (A )1 — Var r s fs (A )1;

1 1A21= —

ST V ar cfoo Is (A)}

16 Harmonic analysis

The relation (4) now follows from these estimates and from the

continuity of the variation of a continuous function of bounded

variation 1

2 Now suppose that g(t) is the Fourier-Stieltjes transform of a

function 7-(A ) (also of bounded variation on the whole line):

g(t)= f exp (iAt) dr(A)

The integral in (6) is called the convolution of the functions cr and

7 From (6) it follows that if at least one of o(À) or r(A) is continuous, then so is their convolution

Now we consider a special case Let

is representable as a Fourier-Stieltjes integral with a continuous

distribution function This conclusion and Lemma 1 (with p, =0) lead to

Lemma 2 Let h(t) be represented as a Fourier-Stieltjes integral (7)

with a continuous distribution function s(A) Then

1 T lim —, f ih (Or dt =0

1 G E Shilov, Mathematical analysis A special course, Moscow, 1961,

p 280

2 A N Kolmogorov 8r S V Fomin, Elements of the theory of functions

and functional analysis, `Nauka', Moscow, 1972, p 423

00

(7)

Fourier—Stieltjes integrals 17

3 Now we assume that the distribution function o(A) is decreasing and bounded for —co < A < oo This case often occurs in various applications Let

non-.0

th t2, • • • , tn be arbitrary real numbers, and el, e2, • • • , 6, be arbitrary complex numbers From (8) it follows that

num-We need the following classical theorem of Bochner and Khinchin,

which we state without proof 3

Every continuous positive-definite function f can be resented (in a unique way) as an integral (8) with a non- decreasing bounded function a-(A)

rep-Remark In an expansion (1) for a positive-definite function we have

dy > 0 and s(A) is a continuous non-decreasing function

2 Proof of the approximation theorem

1 The approximation theorem For every continuous almost odic function f: J > X and for every E > 0 there is a trigonometric polynomial

Remark 2 The function a exp (iÀt) is periodic of period 27r/IA I for all a EX and for all A € J Property 6 (Chapter 1, § 2) implies that every trigonometric polynomial is an almost periodic function, and

it follows from Property 3 (Chapter 1, § 4) that every uniform limit

of trigonometric polynomials is an almost periodic function The approximation theorem asserts that all almost periodic functions can

be obtained in this way

Proof of the approximation theorem For any E >0 we choose numbers 1 = 4E18) and 5 = 5(E18) in accordance with Definition 3 (Chapter 1, § 1) and Property 2 (Chapter 1, § 1), respectively Then

in any interval of length 1 there is a number T such that

0 for sit An

It has the following obvious properties, which we shall need later on:

(1) (1/2n/) f_nni/ Ks (s)ds = 1 (n = 1, 2, )

(2) For any s E J and any natural number m

1 mi+s

2m1 K8 (r) dr =1+ 77(s),

Proof of the approximation theorem 19

exists uniformly in every finite interval

(4) The limit function 08 (u) is positive definite For

From the Bochner—Khinchin theorem we obtain the representation

08(u) = E a exP (iAvt)+ exp (in) ds(A), (12)

v =1

where av >0, and s(A) is continuous, non-decreasing and bounded

2 Next we choose natural numbers m and n arbitrarily and set

1 fs,m,n(t)

ml+s

= 4mn/ 2 f-m/+s _ni

From (11) we see that if K8(S)K8(r) 0, then —s + r is an (6/2)-almost

period of f Therefore, it follows from properties (1) and (2) of K8 (s)

that

r

(13)

2 T 2R -n+s

TT

{ 2-1 R f R R Ks(S)Ks(U + s)f(t + u) du} ds

Thus by setting T = Tk (see property (4) in the preceding subsection)

and taking the limit, after using properties (3) and (4) of Ks (s) and the equality (12), we obtain

d f8,„,(t)Lf lki310 Arnmik (0

Hence, from (13) with n = mk, in the limit as k -* 00 we obtain

To complete the proof of the approximation theorem we need the

following simple lemma

Then the function

F (T) = — 1 T J f(t) dt (T > 1)

2 T _T

also has a compact trajectory

Proof of the approximation theorem 21

Proof It is easy to see from the definition of a Riemann integral that F(T) belongs to the closed convex hull of gif: But, as is well known, the latter is compact together with

Now we can complete the proof of the approximation theorem From Lemma 2 it follows that

R

R ii_.00 1121R f -Rf(t + u)h(u) dull

\ 1/2 liM ( -2-1 R f R R I l f(t + U)II 2 du) R->c0

X lirn —,17., f R th (u )1 2 du =0 (15) R .00 zn -R

Thus, if a l , a2, , aN are chosen so that

It follows from (14), (15), and (16) that

Mt) -E Avav exP ( – iAut)ii < 6,

and this completes the proof of the approximation theorem

3 The mean-value theorem; the Bohr transformation;

Fourier series; the uniqueness theorem

1 We make the important point that in proving the approximation

theorem we have used only the definition of an almost periodic function and the elementary Property 2 in Chapter 1 On the other

hand, as we are going to show in this section, basic properties of an

almost periodic function can be deduced comparatively simply from

the approximation theorem Clearly, there is no need to derive once

again the properties we have already mentioned, but as an example

22 Harmonic analysis

we consider the theorem about a sum of almost periodic functions Let f(t), g(t) be two almost periodic functions For every E > 0 there are trigonometric polynomials PE12 (t) and Qe12(t) such that

and so f(t)+g(t) is an almost periodic function (see Chapter 1, § 1,

Property 3) We show that other basic theorems in the theory of almost periodic functions can also be deduced from the approxima- tion theorem

2 Property 1 The mean-value theorem For every almost periodic function f(t), the mean value

1

T

def liM — ,,,, j f(t) dt = M{f}

exists uniformly with respect to a

Proof For all a e J and for all T > 0

1 if A = 0,

sin AT

exp (iAa)

Therefore, for all A E j,

The mean-value theorem 23

Next let f be an arbitrary almost periodic function; then for every

e >0 there is a trigonometric polynomial Pe (t) such that

sup lif(t)— Pe (t)il< E

Therefore, for every e> 0 there exists a 1 6 > 0 such that for T', T">

and for any a EJ we have

IIMIf ; r, al—Mlf; T", < 3 s,

as we required to prove Notice that for any almost periodic function

f and for all A e j the function f(t) exp (—iitt) is almost periodic Hence, the function

The set {An } of all those A for which a(A; f)00 is called the

spectrum of f; obviously, {An} g- {An} Let

an = a (An ; f)•

With each almost periodic function f we associate (formally, for the

time being) the Fourier series

f(t)— E an exp (U na

n (^' means that there is a relation between the an and f (t) and conveys

no implication of convergence.) The elements a n EX are called the

Fourier coefficients and the numbers {An } the Fourier exponents of

f The next property follows easily from the proof of the tion theorem and from the mean-value theorem (see (18))

approxima-Property 3 The Fourier exponents of approximating polynomials

As a simple consequence of Property 3 we obtain the following

important result

The uniqueness theorem Let f(t) and g(t) be two almost periodic functions If a (A ; f) a (A ; g), then f g

Proof If a (A ; f) -=- a (A ; g), then a (A ; f — g) O Therefore, we can

assume that the approximating polynomial P8 (t; f — g) 0 for every

e >0 Consequently, f(t)—g(t)O

Property 4 For any almost periodic function f we have lim n _ :, an = O

In fact, let Pe (t) be a trigonometric polynomial for which

sup WO — P.(t)II- E,

tef

The Bohr transformation 25

and ne be such that M {PE W exp (-iAnt)} = 0 for n > ne Then for n > ne

lianii=

Ilmlf(t) exp ( - U.0111

= 11Mtl[fit) -Pe (t)] exp ( - iAnt)}11- E

4 Bochner - Fejer polynomials

1 Let f be a 27r-periodic function with a Fourier series

The sums crn (t) are called Fejer sums For every continuous periodic

function f, the Fejer sums converge uniformly to f(t) as n -> oo On the one hand, the Fejer sums are the arithmetic means of the partial

sums of the Fourier series of f, and on the other hand they can be obtained from the Fourier series by introducing into the series the

Bochner has proved that one can introduce in a Fourier series of

any almost periodic function multipliers depending essentially on the Fourier exponents of the function, so as to obtain finite trigonometric sums that converge uniformly to the almost periodic function These sums are obvious generalisations of Fejer sums, and

so we call them Bochner-Fejer sums The present section is devoted

to the construction of Bochner-Fejer sums, and to the proof of their

convergence

26 Harmonic analysis

2 We assemble some simple concepts that will play a significant role in other questions

Definition 1 A finite or countable set of real numbers 01,

02, , o n, is said to be linearly independent (over the field of

rational numbers) if the equality

rigi+ 1'202 + ' • ' + rnif3n = 0

(1.1 , r2, , rn are rational and n is an arbitrary natural number)

implies that all of r1, r2, , rn are zero

Definition 2 A finite or countable set of linearly independent real numbers 0 1 , 02, , on, is called a rational basis of a countable

set of real numbers A1, A2, • • • , An, if every An is representable as

a finite linear combination of the 0j with rational coefficients, that is,

An = r(in)0 1 + 4°02+ • + r ( n n zkV3mk (n = 1, 2, ), (20)

where the rr are rational numbers

Theorem Every countable set of real numbers has a basis contained

in the set

Proof Let

be a given set of real numbers We denote the first non-zero number

in the set by 0 1 and delete from the set (21) all numbers A satisfying

Remark Clearly, a given set of numbers can have several rational

bases, but in a specific basis the representation (20) is unique

If a basis consists of a finite number of terms, then it is called a

finite basis, otherwise, it is infinite If in the representation (20) all

the r (in ) are integers, then the basis is called an integer basis A basis can be both finite and integer For instance, let(31, 02 be non-coprime

Bochner-Fejer polynomials 27

real numbers and consider the countable set of numbers of the form

n iP i +n202, where n i and n2 are integers Obviously, the numbers

(01, 02) form an integer basis for this set

3 Let f(t) be periodic of period p =27r/0I, and

+00

f(t) — E ak exp (ik0t),

-00

where ak= (1/P) g fit) exp (-ik0t) dt We prove that

In fact, for an arbitrary T >0 we have

and so we obtain (22) from (23) by letting T 00 (N 00) Let us

calculate the Fejer sum of order n of the periodic function f First