Property 1. f(t) is an almost periodic function f is weakly almost periodic.
This follows from the estimate
lx *(f(t + T)) - x *(f (t))1 -- Ilx*IIIIN + T) -POI.
Property 2. f(t) is a weakly almost periodic function Rf is bounded and separable.'
Proof For all x* e X* the numerical function x*(f(t)) is almost periodic, and consequently bounded. Therefore, it follows from the
Banach-Steinhaus theorem that sup lif(t)11< co.
teJ
This proves that Rf is bounded. To prove separability we consider
the countable set {f(r)}, where the r are rational points of J. For every to e J there is a sequence rl , r2,. . . , rk, . . . (rk -> to) of rational numbers such that
lim* f(rk)= f(t0)
k-*co
(f(t), being weakly almost periodic, is weakly continuous). There- fore, there is a sequence of linear combinations
Nr,
Yn = E aknxk (xk =Prk), akn E C)
k=1
1 Recall that g21 = {x e X: x = f(t), t E j}.
66 Weakly almost periodic functions
which converges strongly to f (to) as n -> 00. 2 Without loss of general- ity we may assume that all the ak n are rational. Hence the set of all
Yn is countable and so Rf is separable.
Property 3. Suppose that a sequence tfn ( t )1 of weakly almost periodic functions converges weakly to f(t) uniformly in t El. Then f(t) is
weakly almost periodic.
Proof. For all x*e X* we have
lim x*(fn (t)) = x* (f(t)),
uniformly in t e J. Therefore x*(f(t)) is an almost periodic function and so f(t) is weakly almost periodic.
Property 4. Let f be a weakly almost periodic function and Isnl a
sequence of real numbers for which
lim* f(t + sn )= g(t) for all t e j. (1)
Then:
(i) the convergence is uniform in t e f ; (ii) if f2f denotes the convex hull of AS then
12f = (2)
(iii) suPtEJ V(t)II = suPtEJ WWII. (3)
Proof. (i) x*(f(t)) is a numerical almost periodic function for all
X * EX. Therefore, if x*(f(t + sn )) is convergent for all t e J, then the convergence is uniform (see Chapter 1, § 2, the proof of the sufficiency in Bochner's thereom).
(ii) By definition, 12-f is the closure of the set 12f, that is, of the set of points
P
Z = E pif(0, ti, t2, . . • , tp E I,
i=1
E P pi =1, p>0.
1= 1
We consider an arbitrary point of 12g : Y = i p(t).
k=1
2 See, for example, L. A. Lyusternik & V. I. Sobolev, Elements of
functional analysis, `Nauka', Moscow, 1965, p. 216 English translation, Frederick Ungar, New York, 1961, p. 123.
Definition and elementary properties 67 Obviously
y =lim* i Pkf(tk + sn) =lim* zn,
n ->oo k=1 n-*
where
p
Zn = E pkf(tk +sn) G f2f.
k=1
Since the set Ar is closed and convex, by a theorem of Mazur it is also weakly closed. Hence y e fl f and so
il-g g_ flf.
Now we observe that for every fixed x* E X*
lim x*(f(t + s n )) = x*(g(t))
n->00
uniformly in t E J. Therefore, for every E > 0 there is a natural number ne (depending also on x*) such that
sup
tEl
for n > ne ; hence it follows that sup
tej
that is
lim x*(g(t — s n )) = x*(f(t))
n -> co
uniformly in t e J, or
lim* g(t — Sn) = f(t). (4)
n ->00
Therefore,
[if g_ fig,
and so (ii) is proved.
(iii) From (i) and a standard result of functional analysis 3 it follows that
iim l[f(t + sm)I1= sup Itf(t)II,
3 See, fer example, Lyusternik & Sobolev (footnote 2), p. 217; English translation p. 123.
68 Weakly almost periodic functions
and similarly from (4) we have
lif(t)ii = sup Mg(t)119 tEl
which proves (3).
2 Harmonic analysis of weakly almost periodic functions In this section we shall assume that the Banach space X is weakly complete (for instance, a reflexive space). We are going to prove certain additional properties of weakly almost periodic func- tions that are connected with their harmonic analysis.
Property 5. f(t) is a weakly almost periodic function the mean value
a(A) = Ar{f(t) exp (-iAt)}
== Tlifoo * 2-1T T . T f(t) exp (-iAt) dt j (5) exists for all A E J. Moreover, the mean value
1 T+s
liM * —e, I Fr, f(t) exp (-iAt) dt (6)
T-0,C70 Z 1 f-T±S
exists uniformly with respect to s E J.
Proof. For all A e J the function f(t) exp (-iAI), being weakly con- tinuous, is Riemann integrable on every finite interval. Then the
following mean value exists for all x* e X*:
lim j .T - j T x*(f(t)) exp Hilt) dt T-..ô, 2T -
fT Tf(t)exp (-iAt) dt). (7)
Since x* E X* is arbitrary and the space X is weakly complete, it follows from (7) that the weak limit
1 f T
a(A.) = lim* —n ,T, f(t) exp (-Ott) dt T-,,,, 2T .1 —T
= M*1f(t) exp (-iAt)}
exists. Then for all x* E X* the limit
1 f T+s
rn --
li e.,,,, x*(f(t)) exp (-iAt) dt = x*(a(A)) T _,,,,, z i -T +8
Harmonic analysis 69 exists uniformly with respect to s. Therefore, the limit (6) exists uniformly with respect to s. By analogy with the earlier terminology a(A) is called the Bohr transform of f(t).
Property 6. f(t) is a weakly almost periodic function a(A)= 0 except for at most some sequence {A}.
Proof. By property 2 of § 1 Rf g XCI g X, where X0 is a separable subspace of X. It is obvious that a(A ) e Xo . Since X0 is separable, there exists a determining sequence (with respect to X0) of func- tionals {x* r} c X*. 4 From a property of determining sequences of
functionals it follows that
iia(A)ii= sup ker, a(A))i.
r (8)
For every fixed r, (x*,., f(t)) is a numerical almost periodic function,
and so (for every fixed r) (e r, a(A )) = 0 except for some sequence {AO. It follows from here and (8) that a(A) = 0 except for at most a
sequence
{An } = U Akr , kr
as we required to prove.
We set
an =a(An), (9)
and associate with f(t) the Fourier expansion
f (t) —E an exp (iAn t). (10)
n
Let B = { gn } be a rational basis for the sequence {An } (see Chapter 2, § 4). We are going to extend the Bochner—Fejer summation pro- cedure to weakly almost periodic functions. Let
1V7n
Pin(t)= E tkinkak exp (iAkt), 0 --. kt in k ---1, k=1
be a Bochner—Fejer polynomial for f(t) (see Chapter 2, § 4).
Property 7. f(t) is a weakly almost periodic function lim* Pm (t)=
f(t) uniformly on J.
Proof. First observe that B is also a basis for the Fourier exponents
of the functions (x*, f(t)) for all x* E X * . In fact, it follows from (7)
4 See, for example, Dunford & Schwartz [40].
70 Weakly almost periodic functions that
a (A ; (x*, f(t)))= M{x*, f(t)) exp ( — iAt)} = (x*, a(A)).
The last expression is zero for A 0 {A n }. Furthermore,
Islm
(x*, Pm(t)) = E il in k ( x * , a(Ak)) exP (iAkt),
k=1
that is, (x*, Pm(t)) is a Bochner-Fejer polynomial constructed in
terms of the basis B and the function (x*, f(t)). Therefore, 1imn_ằ.0 (x*, Pm(t)) = (x*, fit)) uniformly on J as we required to
prove.
Property 8. f(t) is a weakly almost periodic function and a (A ) z----. 0 f(t) =0 (the uniqueness theorem for weakly almost periodic func-
tions).
Proof. If a(A) — 0, then Pm(t)0 for all m. Therefore, (x*, f(t)):::---. 0
for all x* € X* and hence it follows that f(t) ---= O.
9. Bochner's criterion. Let f(t) be weakly continuous. For f(t) to be weakly almost periodic it is necessary and sufficient that from each sequence {s n } we can extract a subsequence {s'} such that {fit + s'n)}
is weakly convergent uniformly on J.
Proof. The sufficiency of the condition is obvious. We shall prove
the necessity. In the proof of Property 7 we remarked that for all x* E X* the Fourier exponents of the numerical almost periodic function x*f(t)) are contained in a fixed countable set {A n }. It follows from this and Theorem 4 of Chapter 3 that it is enough to distinguish
a sequence Is n' } satisfying the condition: the following limits exist
for any k(= 1, 2, ..
lim exp (is'nikk) = Ok;
n -* c0
this is clearly possible.
3 Criteria for almost periodicity
The following theorem gives a general criterion for almost
periodicity.
Theorem 1. For a bounded function f :J -+ X to be almost periodic it is necessary and sufficient that:
(1) For each x* from a set D everywhere dense in X* the scalar function (x*, fit)) is almost periodic;
Criteria for almost periodicity 71 (2) f(t) is compact in the sense that the closure of the set of its values is compact.
In particular, for a weakly almost periodic function to be almost periodic it is necessary and sufficient that it is compact.
Proof. The necessity of both conditions is obvious. We shall prove the sufficiency.
From the boundedness of f(t) and condition (1) it follows that f (t) is weakly almost periodic. In fact, for any x* E X * we can find a sequence of elements x* n e D such that Ile- enli -> O.
Then
RX * ) fitằ — (x*, f (t))1 .-ỗ-IIX * — X * nil sup lif Mil,
t EJ
from which it follows that (x*, fit)) is almost periodic. 6
We proved earlier that the set R.!. is separable. Therefore, without loss of generality we may assume that X itself is separable, and so is isomorphic to a subspace Y of the space of all functions continuous on the interval [0, 11. 6
We take a sequence of finite-dimensional linear operators En : Y ->
Y with the property
Emy---> m,. y (y e Y) (11)
(for this we can use any basis in the space of continuous functions).
The function fm(t) --- E rj(t) (m =1, 2, . . .) has values in a finite- dimensional space, and so for it the concepts of almost periodicity and weak almost periodicity are equivalent; this is obtained from the equality
(Y * , fm(t)) = (y, Etnf(t)) = (Em * Y, f(t)), where Ern * denotes the adjoint operator of En.
By (11), fm(t) -> f(t) for every t e J, and in view of the next lemma this convergence is uniform with respect to t e J.
Lemma 1. The strong convergence of bounded linear operators is uniform on every compact set K c Y.
Proof. Let Am: Y -* Y be a sequence of bounded operators for which Amy -> Ay for every y e Y. Then by the Banach-Steinhaus theorem the norms IlAm Il are bounded by some number 1.
5 Thus, the boundedness of fj -0 X and condition (1) are necessary and sufficient for the weak almost periodicity of the function.
6 See L. A. Lyusternik 8c V. I. Sobolev (footnote 2), p. 256; English translation, p. 126.
72 Weakly almost periodic functions
Let lyi l (i = 1, 2, ... , p) be a finite (s/41)-net for K, that is, for all y K we can find a y; (j = 1, ... p) such that
IlY el4l. (12)
Furthermore, it is obvious that there is an N = N (s) such that
IIAyj — AyII =- e/ 2 (13)
for n >N and for all j = 1, 2, ... , p. From (12) and (13) we obtain
= IRA —An)(Y yi)+Ayi AnyjIl
An )(yyj)IH llAyjAnyjll ---21(614l)+812=e
for all y E K This proves the lemma, and also Theorem 1.