a function takes its values is a Hilbert space.
Let X be a complex Hilbert space, x, y be elements of X, (x, y)
be a scalar product in X, and 11x11= (x, x) l/2 be the norm of x e X.
Theorem. For every almost periodic function f(t) - E an exP (iAnt):J -4X,
n
Parseval' s relation holds:
00
Mt{(f(t), f(t))} =E (an, . 1 an). (32) Proof. We take arbitrary elements c l , c2, ... , cn e X and arbitrary numbers Mi, M2,. . . , An cf and consider the function d = d(c i , c2, . . . , cn)= Mtlilf(t) — Enk=i ck exP (4041 21. We call NI d the deviation of the sum
n
s(t)= E ck exp (itikt)
k=1
from the almost periodic function f(t). We are going to find
minciex d(ci, c2,. . . , cn). In fact, it is easily obtained from an identity which is derived as follows:
n n
d = Mtl(f(t)— E
k1 ck exp (itint), f(t) - E ci exp (440)1
= 1=1
n
= mtl(f(t), fit)ằ - E (ck, )Wat) exP (-iiIkt)l)
k=1 n
- E (Mat) exP ( — ikLit)}, ct)
1=1 n n
+ E E (ck, ct)MtlexP (ip.kt) exp (—ittit)}
k=1 1=1
n
= MICRO, POằ - E (ck, a(t4k; D)
k=1
n n
- E (a(ihk; f), ck)+ k1 E (ck, Ck)
= k=1
n n
+ E (a(ktk; A aCuk; D) — E
k1 (a(uk; f), a(Ak; A
k=1 =
32 Harmonic analysis
n
= Mtl(fit), POằ — kE 1 (a(lu k; f), a(k; D) tk
n
+ E (ck—aCak; D, Ck — a(/uk; f)). (33)
k=1
It is clear from this identity that min d is attained when and only when
ck = aCuk; D = Mtlf(t) exP (—it)}, that is,
0 if Pk does not coincide with any of the Fourier exponents of f,
Ck a(/.Lk; f) 0 0 if ,uk coincides with one of the Fourier exponents of f.
This property is called the minimal property of the Fourier coefficients. When we set uk = Ak and ck = a(Ak ; f)= ak in the identity
(33) we obtain
d = Mtl(fit), f(t))}- (34)
k=1
Since the left-hand side of this last identity is non-negative, we obtain Bessel's inequality, namely:
n
E kIlak112 ---Aftl(f(t), f(t))}. (35)
=1
Here n is arbitrary so that we deduce, in particular, the convergence of Ec:=1 ilak112. Now we are going to prove Parseval's relation (32); for this we take any E > 0 and let
ne
P(t) = E bk e exp (iA kt)
k=1
be a trigonometric polynomial for which6
sup II fit) — P8 (t)11- e. (36)
tei
It follows from (36) and the minimal property of Fourier coefficients that
0 -.5. d(a i , a2, • • • , ane )--- d(bi e, b2E, • • • , bn e )
= Altfil fit) — Pe(t)11 21 -- e2.
6 We recall that the exponents of an approximating polynomial can be selected from the Fourier exponents of the function (see § 2, Property 3).
Functions with values in a Hilbert space 33 By combining this last inequality with the identity (34) we find that
0 --Mt{lif (t)112} - ilaki12 --. s 2,
and therefore since s was chosen arbitrarily, we have proved Parseval's relation.
6 The almost periodic functions of Stepanov
1. Let F:J-4X (X is a Banach space) be measurable .in the sense of
Lebesgue-Bochner. V. V. Stepanov suggested a generalisation of the concept of almost periodicity for this class of functions which is fully justified. Subsequently, Bochner pointed out that by using a
very simple construction, a Stepanov function can be reduced to a Bohr function, which is vector valued even when the original is a
scalar function; we reproduce this construction below. Let Y "(X) be the Banach space of measurable functions 4: i = [0, 1] - X with
the norm
„1 ‘ lip
(J0 110(n)11' dn) (p...-1).
Clearly, for every t EJ the function f(t +n) ( n E A) is a measurable function from A into X. We now state a definition of almost period- icity in the sense of Stepanov which takes account of the observation of Bochner mentioned above.
Definition. We say that a function f(t):J - > X is almost periodic in the sense of Stepanov if f(t)= {fit +n), n E Al is almost periodic as a function J .'(X).
More fully, f(t) is almost periodic in the sense of Stepanov if for
every e >0 there is a relatively dense set of numbers {Te} satisfying
the inequality7
1 lip
sup (i. ilf(t +71 +7)- f(t +n)F dn) --. E.
tEl 0
The simplest properties of Stepanov almost periodic functions can be derived from the corresponding ones of Bohr almost periodic functions. For instance, since a Bohr almost periodic function is
7 The continuity of f(t+ n) as a function from J into .'(X) follows from
the continuity in the mean of a function integrable in the sense of Lebesgue—Bochner.
34 Harmonic analysis
bounded, we have 1
sup flifit +,,o r d,„ <03. (37)
tel 0
Let Ar (X) stand for the class of all measurable functions f:f -*X
satisfying (37), and Ye (x) for the subclass of all Stepanov almost periodic functions. By using Bochner's criterion it can be proved that an element fe MP (X) belongs to 11%/P(X) if and only if the family
of translates ft = {f(t + s)} is compact in MP (X).
We shall frequently find it convenient to use the following norm
for the space Ile (X) which is equivalent to (37):
1 1 1/p
sup {-, j. ilf(t + ,„)r d-r7} (1>0). (38)
tel t 0
Lemma 4. If a Stepanov almost periodic function is uniformly continuous as a function J X, then it is almost periodic.
Proof. For any natural number n we set
fn(t) = n f 1/
n
fit + 71) chi.
o
Since f(t) is uniformly continuous, it follows that fn (t)-4 f(t) as n --> Cla
uniformly in t (see Chapter 1, § 1, Property 5). Therefore it is sufficient to prove that fn (t) is almost periodic for a fixed n. With this aim we use the norm (38) with 1 = 1/n. For any e >0 we can find a relatively dense set of numbers _r such that
1/n
sup n f Ilf(t +r + n) - f(t + 77)r dli
tEJ 0
Then by using Holder's inequality it follows that for any t cf
1/n
ilfn(t + 7) — fn(t)II --- n fo YU' + 7 ± 70—fit+ n)li dii
1/n 1/p
lif(t + T + 71) — f(t + 7011 1' dril -.. E,