Almost periodic functions in finite-dimensional space with the spectrum in a cone

0.15 MB·English

by  S. Favorov

Almost periodic functions in finite-dimensional space
with the spectrum in a cone
7
0 Favorov S., Udodova O.1
0
2

n
a Kharkiv National University
J Department of Mechanics and Mathematics

8 md. Svobody 4, Kharkiv 61077, Ukraine
2 e-mail: [email protected],

[email protected]
]
V
Abstract
C
We prove that an almost periodic function in finite-dimensional space extends to a holomorphic
.
h bounded function in a tube domain with a cone in the base if and only if the spectrum belongs to
t the conjugate cone. We also prove that an almost periodic function in finite-dimensional space has the
a
boundedspectrum if and only if it extendsto an entirefunction of exponentialtype.
m
2000 Mathematics Subject Classification 42A75, 30B50.
[

1 A continuous function f(z) on a strip
v
0 S ={z =x+iy :x∈R, a≤y ≤b}, −∞≤a≤b≤+∞.
a,b
2
8 is called almost periodic by Bohr on this strip, if for any ε>0 there exists l =l(ε) such that every interval
1
of the real axis of length l contains a number τ (ε-almost period for f(z)) with the property
0
7
sup |f(z+τ)−f(z)|<ε. (1)
0
z∈Sa,b
/
h
In particular, when a=b=0 we obtain the class of almost periodic functions on the real axis.
t
a To each almost periodic function f(z) assign the Fourier series
m
∞
v: an(y)eiλnx, λn ∈R,
Xi nX=0
r where an(y) are continuous functions of the variable y ∈[a,b].
a
In the case a = b = 0 all exponents λ are nonnegative if and only if the function f(x) extends to the
n
upper half-plane asa holomorphic boundedalmostperiodic function; the setof allexponents λ is bounded
n
if and only if f(x) extends to the plane C as an entire function of the exponential type σ =sup|λ |, which
n
n
is almost periodic in every horizontal strip of finite width (see [1], [4]).
A number of works connected with almost periodic functions of many variables on a tube set appeared
recently (see [2], [6]-[9]). Recall that the set T ⊂Cm is a tube set if
K
T ={z =x+iy :x∈Rm,y ∈K},
K
where K ⊂Rm is the base of the tube set.
Definition. (See [6], [9]). A continuous function f(z), z ∈ T is called almost periodic by Bohr on T , if
K K
for any ε > 0 there exists l = l(ε), such that every m-dimensional cube on Rm with the side l contains at
least one point τ ((T ,ε)-almost period from f(z)) with the property
K
sup |f(z+τ)−f(z)|<ε. (2)
z∈TK
1TheworkwassupportedbytheINTAS-grant99-00089
1
Let f, g be locally integrable functions on every real plane
{z =x+iy :x∈Rm}, y ∈K.
0 0
Definition. Stepanoff distance of the order p≥1 between functions f and g is the value
1
p
S (f,g)= sup  |f(z+u)−g(z+u)|pdu .
p,TK z∈TK[0,Z1]m 
 
Using this definition, we can extend the concept of almost periodic functions by Stepanoff on a strip
(see [4] p.197) to almost periodic functions on a tube set:
Definition. A function f(z), z ∈ T , is called almost periodic by Stepanoff on T , if for any ε > 0 there
K K
exists l = l(ε) such that every m-dimensional cube with the side l contains at least one τ ((T ,ε,p)-almost
K
period by Stepanoff of the function f(z)) with the property
S (f(z),f(z+τ))<ε. (3)
p,TK
The Fourier series for an almost periodic (by Bohr or by Stepanoff) function f(z) on a set T is the
K
series
a(λ,y)eihx,λi, (4)
λX∈Rm
where hx,λi is the scalar product on Rm, and
m
a(λ,y)= lim 1 f(x+x′+iy)e−ihx+x′,λidx; (5)
N→∞(cid:18)2N(cid:19) Z
[−N,N]m
this limit exists uniformly in the parameter x′ ∈Rm and does not depend on this parameter (see [6], [8]).
A set of all vectors λ∈Rm such that a(λ,y)6≡0 is called the spectrum of f(z) and is denoted by spf;
this set is at most countable, therefore the series (4) can be written in the form
∞
a (y)eihx,λni.
n
nX=0
Notethatpartialsumsoftheseries(4),generallyspeaking,donotconvergetothefunctionf(z). However
the Bochner-Feyer sums 1
q−1
σq(z)= knqan(y)eihx,λni, 0≤knq <1, knq →1 as q →∞
nX=0
convergetothefunctionf(z)uniformlyforalmostperiodicfunctionsbyBohrandwithrespecttothemetric
S foralmostperiodic functionsbyStepanoff; inparticular,iftwofunctions havethe sameFourierseries,
p,TK
then these functions coincide identically. For holomorphic almost periodic functions the series (4) can be
written in the form
∞ ∞
ane−hy,λnieihx,λni = aneihz,λni, an ∈, (6)
nX=0 nX=0
(see [8]). Any series of the form (6) is called Dirihlet series .
By Γ we always denote a convex closed cone in Rm; by Γ we denote the conjugate cone to Γ:
Γ={t∈Rm :ht,yi≥b0 ∀y ∈Γ},
◦ b
note that Γ=Γ. Also, Γ is the interior of a cone Γ.
1Forn=bb1see[4],forn>1see[8].
2
Theorem 1. Let f(x) be an almost periodic function by Bohr on Rm with Fourier series
∞
aneihx,λni, (7)
nX=0
where all the exponents λ belong to a cone Γ⊂Rm. Then f(x) continuously extends to the tube set T as
n
Γ
an almost periodic by Bohr function F(z) with Fourier series (6). The function F(z) is holomorphic on the
interior TΓ, and for any Γ′ ⊂⊂Γ uniformly w.r.t. z ∈TΓ′ b
b
b lim F(z)=a0, (8)
kyk→∞
where a is the Fourier coefficient corresponding to the exponent λ = 0 (if 0 ∈/ sp f, put a = 0.) If
0 0 0
◦
spf ⊂Γ, then (8) is true uniformly w.r.t. z ∈T .
Γ
Here the inclusion Γ′ ⊂⊂ Γ means that thebintersection of Γ′ with the unit sphere is contained in the
interior of the intersection of Γ with this sphere.
b
To prove this theorem, we use the following lemmas.
b
Lemma 1. Supposethat aplurisubharmonic functionϕ(z)on Cm is boundedfrom aboveon asetT , where
K
K ⊂Rm is a convex set. Then the function
ψ(y)= sup ϕ(x+iy)
x∈Rm
is convex on K.
Proof of Lemma 1. Fix y , y ∈K. The plurisubharmonic on Cm function
1 2
ψ(y )−ψ(y )
2 1
ϕ (z)=ϕ(z)− hImz,y −y i
1 ky −y k2 2 1
2 1
is boundedfromaboveonthe setT . Thereforethe subharmonicfunctionϕ (w)=ϕ ((y −y )w+iy )
[y1,y2] 2 1 2 1 1
is bounded on the strip {w =u+iv :u∈ R, 0 ≤ v ≤1}. Hence, the value of ϕ at any point of this strip
1
does not exceed
max{supϕ (u),supϕ (u+i)}≤max{ sup ϕ (x+iy ), sup ϕ (x+iy )}.
2 2 1 1 1 2
u∈R u∈R x∈Rm x∈Rm
Therefore, for any z =x+iy ∈T ,
[y1,y2]
ψ(y )−ψ(y ) ψ(y )−ψ(y )
2 1 2 1
ϕ (z)≤max{ψ(y )− hy ,y −y i,ψ(y )− hy ,y −y i}
1 1 ky −y k2 1 2 1 2 ky −y k2 2 2 1
2 1 2 1
ky k2−hy ,y i ky k2−hy ,y i
2 1 2 1 1 2
= ψ(y )+ ψ(y ).
ky −y k2 1 ky −y k2 2
2 1 2 1
Hence for any y ∈[y ,y ] we have
1 2
ky k2−hy ,y i−hy,y −y i ky k2−hy ,y i+hy,y −y i
2 1 2 2 1 1 1 2 2 1
ψ(y)≤ ψ(y )+ ψ(y ).
ky −y k2 1 ky −y k2 2
2 1 2 1
If y =λy +(1−λ)y , λ∈(0,1), then we obtain the inequality
1 2
ψ(λy +(1−λ)y )≤λψ(y )+(1−λ)ψ(y ).
1 2 1 2
Therefore, the function ψ(y) is convex on K.
Lemma 2. Let ψ(y) be a convex bounded function on a cone Γ. Then ψ(y)≤ψ(0) for all y ∈Γ.
3
Proof of Lemma 2. Since ψ(y) is convex, we have
1 1
ψ(y)≤ 1− ψ(0)+ ψ(ty), t>1.
(cid:18) t(cid:19) t
Taking t→∞, we obtain ψ(y)≤ψ(0).
Proof of Theorem 1. Let σ (x), q = 0,1,2,... be the Bochner-Feyer sums for the series (7). Obviously,
q
these functions are also defined for z ∈Cm. Assume that
ϕ (z)=log(|σ (z)−σ (z)|).
q,l q l
For any fixed q and l, q >l, the function ϕ (z) is plurisubharmonic on Cm. Moreover,for z ∈T we have
q,l
Γ
hy,λ i≥0 and
n
|σq(z)−σl(z)|≤|σq(z)|+|σl(z)|≤ b
q−1 l−1 q−1
|an|e−hy,λni+ |an|e−hy,λni ≤2 |an|.
nX=0 nX=0 nX=0
Consider the function
ψ (y)= sup log(|σ (z)−σ (z)|).
q,l q l
x∈Rm
Using lemma 1, we obtain that ψ (y) is convex in Γ. Therefore, by lemma 2, we have
q,l
b
sup(|σ (z)−σ (z)|)≤ sup (|σ (x)−σ (x)|). (9)
q l q l
z∈T x∈Rm
Γ
Further, the function f(x) is almobst periodic, therefore the Bochner-Feyersums convergeuniformly on Rm,
and for q,l≥N(ε)
sup (|σ (x)−σ (x)|)≤ε.
q l
x∈Rm
Hence, sup (|σ (z)−σ (z)|)≤ε for all z ∈T , q,l≥N(ε).
q l
x∈Rm Γ
Thus the Bochner-Feyer sums uniformly converge on T , their limit is an almost periodic function by
b Γ
Bohr and holomorphic on the interior of T with the Dirihlet series (6).
Γ
Further, passing to the limit in (9) as q →∞, we get b
b
sup(|F(z)−σ (z)|)≤ sup (|f(x)−σ (x)|).
l l
z∈T x∈Rm
Γ
Choose l such that the right handbside of this inequality is less than ε. We have for Γ′ ⊂⊂Γ
b
sup |F(z)−a |≤ sup |F(z)−σ (z)|+ sup |σ (z)−a |≤ε+ sup |σ (z)−a |.
0 l l 0 l 0
z∈TΓ′ z∈TΓ′ z∈TΓ′ z∈TΓ′
Note that for any fixed λ ∈Γ\{0} the value hy,λ i tends to +∞ as kyk→∞, y ∈Γ′, therefore, we have
n n
l−1 l−1
|σl(z)−a0|=| kjlajeihx,λjie−hy,λji|≤ |aj|e−hy,λji →0
Xj=1 Xj=1
as kyk→∞ on Γ′. Hence, uniformly w.r.t. z ∈TΓ′
lim |F(z)−a |≤ε. (10)
0
kyk→∞
This is true for arbitraryε>0, then (8) follows.
◦
If spf ⊂ Γ, then for any λn ∈spf, hy,λni→+∞ as kyk→∞ uniformly w.r.t. y ∈Γ, therefore (10) is
true uniformly w.r.t. z ∈T , and (8) is also true. The theorem has been proved.
Γ b
b
4
Theorem 2. Let f(x) be an almost periodic function by Stepanoff on Rm with the Fourier series (7). Let
all the exponents λ belong to a cone Γ ⊂ Rm. Then there exists an almost periodic by Stepanoff function
n
F(z) in the tubeset T with the Fourier series (6) such that F(x)=f(x). The function F(z) is holomorphic
Γ
◦
almost periodic by Bohr on any domain T , b∈Γ. Besides, for any cone Γ′ ⊂⊂Γ we have uniformly w.r.t.
b Γ+b
z ∈TΓ′ b b
b lim F(z)=a0, (11)
kyk→∞
◦
where a0 is the Fourier coefficient for the exponent λ = 0. If spf ⊂Γ, then (11) is true uniformly w.r.t.
◦
z ∈T for any b∈Γ.
Γ+b
b
Prboof. To prove the first part of the theorem, we need to replace ϕq,l(z) by
1
p
ϕ (z)=log |σ (z+u)−σ (z+u)|pdu .
q,l q l
Z
[0,1]m 
b  
Arguingasintheproofoftheorem1,weobtainthattheBochner-Feyersumsσ (z)convergeintheStepanoff
q
metric uniformly w.r.t. z ∈T to an almost periodic function by Stepanoff F(z) with Fourier series (6).
Γ
◦
Let b ∈Γ. The module of the function σ (z)−σ (z) is estimated from above by the mean value on the
b q l
corresponding ball contained in T . Using the Ho¨lder inequality, we have
b Γ
1
b p
sup |σ (x+bi)−σ (x+bi)|≤C sup  |σ (z+u)−σ (z+u)|pdu ,
q l q l
x∈Rm z∈T Z
Γ[0,1]m 
 
b
where the constant C depends only on b and Γ.
Applying lemmas 1 and 2 to the functions
b
ψ˜ (y)= sup log|σ (z+bi)−σ (z+bi)|,
q,l,b q l
x∈Rm
wegetthatthe Bochner-Fouriersums convergeuniformly onT to F(z),thusF(z)isholomorphicalmost
Γ+b
◦
periodic by Bohr in T for any b∈Γ. b
Γ+b
Then the other statements of the theorem follow from theorem 1.
b
b
Now we prove the inverse statements to theorems 1 and 2.
Theorem 3. Suppose that an almost periodic by Bohr function f(x) continuously extends to the interior of
TΓ as a holomorphic function F(z). If F(z) is bounded on any set TΓ′, Γ′ being a the cone in Rm, Γ′ ⊂⊂Γ,
then F(z) is an almost periodic function by Bohr on T and the spectrum of F(z) is contained in Γ.
Γ
◦ b
Proof. Take λ∈/ Γ. Then there exists y0 ∈ Γ such that hy0,λi<0.
◦
ChooseaneighboburhoodU ⊂Γofy0 suchthathy,λi≤ 12hy0,λiforally ∈U.LetAbeanynondegenerate
operator in Rm such that A maps all the vectors e =(1,0,...,0),...,e =(0,0,...,1) into U.
1 m
The function F(Aζ) is holomorphic and bounded on the set
{ζ =ξ+iη ∈Cm :ξ ∈Rm, ηj >0,j =1,...,m}
because A{η :ηj ≥0}⊂Γ.
If for each coordinates ζ1,...,ζm we change the integration over the segments −N ≤ξj ≤N, ηj =0 to
the integration over the half-circles ζj =Neiθj, 0≤θj ≤π, j =1,...,m we obtain the equality
m m m
1 F(Aξ)e−ihAξ,λidξ = i F(ANeiθ) eiθj−iNeiθjhAej,λidθ, (12)
(cid:18)2N(cid:19) Z (cid:18)2(cid:19) Z
jY=1
[−N,N]m [0,π]m
5
where θ =(θ1,...,θm), eiθ =(eiθ1,...,eiθm).
Since hAe ,λi<0 for j =1,...,m, we see that the integrand in the right-hand side of (12) is uniformly
j
bounded for all N >1. By Lebesgue theorem (12) tends to zero as N →∞.
Thus
1
lim f(x)e−ihx,λidx=0. (13)
N→∞(2N)m Z
A([−N,N]m)
CoverthesetA([−N2,N2]m)bycubesL =x′ +[−N,N]m suchthattheinteriorsofthesecubesarenotin-
j j
tersected. WemayassumethatthenumberofthecubesintersectingtheboundaryofthesetA([−N2,N2]m)
is O(Nm−1) as N → ∞. Taking into account boundedness of the function f(x)e−ihx,λi on Rm and equal-
ity (13), we have
1 1
f(x)e−ihx,λidx=  f(x)e−ihx,λidx+O(N2m−1)=o(1) (14)
(2N)2m Z (2N)2m Z
∪Lj A([−N2,N2]m) 
Since (5) we see that uniformly w.r.t. x′ ∈Rm as N →∞
1
f(x)e−ihx,λidx=a(λ,f)+o(1). (15)
(2N)m Z
x′+[−N,N]m
On the other hand, the number of the cubes L equals O(Nm) as N →∞, then the equality a(λ,f)=0
j
follows from (14) and (15). This yields the inclusion spf ⊂ Γ. Using theorem 1 we complete the proof of
our theorem.
b
Theorem 4. If F(z) is bounded on each set TΓ′, Γ′ ⊂⊂ Γ, and the nontangential limit value of F(z) as
y → 0 is an almost periodic function by Stepanoff on Rm, then F(z) extends to T as an almost periodic
Γ
function by Stepanoff and the spectrum of F(z) is contained in Γ.
Proof. The proof of this theorem is the same as of theoremb3, but we have to use theorem 2 instead of
theorem 1.
To formulate further results we need the concept of P-indicator. ( See, for example, [5], p. 275.)
Definition. P-indicator of an entire function F(z) on Cm is the function
1
h (y)= sup lim log|F(x+iry)|.
F
x∈Rmr→∞r
Theorem 5. (For m = 1 see [3], [4].) Let f(x), x ∈ Rm be an almost periodic function by Stepanoff with
the Fourier series (7), and let kλ k ≤ C < ∞ for all n. Then f(x) extends to Cm as an entire function
n
F(z) of exponential type, which is almost periodic by Bohr on any tube domain in Cm with bounded base;
F(z) has the Dirichlet series (6), and P-indicator h (y) satisfies the equation h (y) = H (−y), where
F F spf
H (µ):= sup hx,µi is the support function of the set spf.
spf
x∈spf
Proof. Take µ∈Rm such that kµk=1. Put
fµ(x)=f(x)e−i[Hspf(µ)+ε]hx,µi.
∞
The Fourier series aneihx,λn−(Hspf(µ)+ε)µi corresponds to the function fµ(x), hence
nP=0
spf ⊂{x∈Rm :hx,µi ≤−ε}.
µ
Since spf is bounded, we obtain for some δ >0
µ
spf (x)⊂Γ ={λ∈Rm :hλ,−µi≥δkλk}.
µ δ,−µ
6
Theorem 2 yields that f (x) extends to the interior of the domain T , where
µ
Γδ,−µ
Γ ={y :hy,−µi≥ 1−δ2kybk}
δ,−µ
p
is the conjugate cone to Γ , as anbalmost periodic function by Bohr F (z). This function is holomorphic
δ,−µ µ
◦
on any domain T , b∈Γ with the Dirichlet series
Γδ,−µ+b δ,−µ
b
b ∞
aneihz,λn−[Hspf(µ)+ε]µi,
nX=0
and Fµ(z)→0 as kyk→∞ uniformly w.r.t. z ∈TΓ′ for any cone Γ′ ⊂⊂ Γδ,−µ. Using (5) we get
b
ane−hy,λn−[Hspf(µ)+ε]µi ≤ sup |Fµ(x+iy)|, y ∈Γ′. (16)
(cid:12) (cid:12) x∈Rm
(cid:12) (cid:12)
(cid:12) (cid:12)
Put
F(z):=Fµ(z)ei[Hspf(µ)+ε]hz,µi.
F(z) is almost periodic on TΓ′ with Dirichlet series (6). Therefore it follows from (16) that
|an|≤ sup |F(x+iy)|ehy,λni. (17)
x∈Rm
On the other hand, the function Fµ(z) is bounded on TΓ′, hence
|F(z)|≤C(Γ′)e−[Hspf(µ)+ε]hy,µi, z ∈TΓ′ (18)
Coverthe space Rm by the interiors of a finite number of cones Γ′,...,Γ′ . There exist holomorphic on
1 N
the interior of Γ′ almost periodic functions F (z), k = 1,...,N with identical Dirichlet series (6). Using
k k
the uniqueness theorem, we obtain that these functions coincide on the intersections of the cones and thus
defineaholomorphicfunctionF(z)onCm\Rm. TheBochner-FeyersumsforF(z)convergetothisfunction
uniformly on any set
{z =x+iy :x∈Rm, kyk=r >0}.
Hence, these sums converge on the tube domain T . Thus F(z) extends to Cm as the holomorphic
{kyk<r}
function, which is almost periodic on any tube set with a bounded base. Owing to the uniqueness of
expansion into Fourier series, we have F(x)=f(x).
Let us prove that h (y)=H (−y). From inequality (18) with µ=−y it follows that
F spf
1
h (y)≤ lim [H (−y)+ε]hry,yi=H (−y)+ε.
F spf spf
r→∞r
The functions h (y) and H (y) are positively homogenious, hence the inequality
F spf
h (y)≤H (−y)
F spf
is true for all y ∈Rm.
Further, fix x,y ∈ Rm. The holomorphic on C function ϕ(w) = F(x+wy) is bounded on the axis
Imw =0. Then the estimate
|ϕ(w)|≤Cea|Imw|
for some a>0 and all w∈C follows from (18). Using the definition of P-indicator, we get
1
lim log|ϕ(iv)|≤h (y).
F
v→+∞v
Therefore the function ϕ(w)ei(hF(y)+ε)w is bounded on the positive part of the imaginary axis. Applying
the Fragmen-Lindelof principle to the quadrants Rew ≥ 0, Imw ≥ 0 and Rew ≤ 0, Imw ≥ 0, we get
boundedness of this function on the upper half-plane. Applying the Fragmen-Lindelof principle to the
half-plane Imw ≥0, we get the inequality
|ϕ(w)|≤ sup |ϕ(w)| ehF(y)Imw (Imw>0).
(cid:18) (cid:19)
Imw=0
7
Hence, for all z ∈Cm, we have
|F(z)|≤ sup |F(x)|ehF(y).
x∈Rm
Now using formula (17) for coefficients of the Dirichlet series of the function F(z), we get the estimate
|an|≤ sup |f(x)|ehF(y)+hy,λni. (19)
x∈Rm
Suppose hy ,λ i+h (y ) < 0 for some y ∈Rm. Put y =ty in (19) and let t→∞. We obtain a =0.
0 n F 0 0 0 n
This is impossible because λ ∈spf.
n
Thus for all y ∈Rm and λ ∈spf we have h (y)+hy,λ i≥0, hence
n F n
H (−y)= sup h−y,λ i≤h (y).
spf n F
λn∈spf
This completes the proof of the theorem.
The following theorem is inverse to the previous one.
Theorem 6. (For m = 1 see [3], [4].) Let F(z) be an entire function on Cm, |F(z)| ≤ Cebkzk, let F(x),
x ∈ Rm be an almost periodic function by Stepanoff with the Fourier series (7). Then F(z) is an almost
periodic function by Bohr on any tube domain T ⊂ Cm with the bounded base, F(z) has the Dirichlet
D
series (6), and spF ⊂ {λ:kλk≤b}.
Proof. It follows from theorem 5, that it suffices to prove the inclusion
spF ⊂{λ:kλk≤b}.
Let the function F(x) be bounded on Rm. Arguing as in theorem 5, we see that for all z ∈Cm
|F(z)|≤ sup |F(x)|ehF(y),
x∈Rm
where h (y) is P-indicator for F(z). Further, for all x∈Rm we have
F
1 1
h (y)= sup lim log|F(x+iry)|≤ sup lim (logC+bkx+iryk)≤bkyk,
F
x∈Rmr→∞r x∈Rmr→∞r
therefore for all z ∈Cm
|F(z)|≤Cebkyk.
Take ε>0, µ∈Rm, kµk=1. Consider the function
F (z)=F(z)e−ihz,µi(b+ε).
µ
Since |F (z)| ≤ Cebkyk+(b+ε)hy,µi uniformly w.r.t. x ∈ Rm, then F (z) is uniformly bounded for z ∈ T ,
µ µ Γ−µ
where Γ is the cone {y : hy,−µi ≥ (1− ε )kyk}. Using theorem 4, we obtain that the spectrum F is
−µ b+ε µ
contained in Γ and
−µ
spF =spF +(b+ε)µ⊂Γ +(b+ε)µ.
b µ −µ
Finally, using the inclusion b
(Γ +(b+ε)µ)⊂{λ:kλk≤b+ε}
−µ
µ:k\µk=1
b
andthe arbitrarityofchoiceofε we getthe assertionofthe theoreminthe case ofboundedonRm function
F(z).
Now let the function F(z) be unbounded on Rm. Put for some N >0
1
g(z)= F(z+t)dt.
Nm Z
[0,N]m
The function g(z) satisfies the estimate on Cm
|g(z)|≤CebmNebkzk.
8
Asinthecasem=1(see [4]),wecanprovethatg(x)isanalmostperiodicfunctionbyBohrandisbounded
on Rm. The function g(x) has the Fourier series
∞ a eiλ1nN −1...eiλmnN −1eihx,λni,
n Nλ1 Nλm
nX=0 n n
where λj are coordinatesof the vectorλ (if λj =0, the correspondingmultiplier shouldbe replacedby 1).
n n n
Using countabilityofspF,we canchooseN in sucha waythatnone ofthe numbersλjN coincideswith
n
2πk, k ∈ Z\{0}. In this case spg = spF. Applying the proved above statement to the function g(z), we
obtain the inclusion
spF ⊂{λ:kλk≤b}.
REFERENCES
[1] BohrH. Zur Theorie der fastperiodischen Funktionen. III Teil: Dirichletentwicklung analytischerFunk-
tionen.//Acta math. – 47. – 1926. – P.237-281.
[2] FavorovS.Yu.,RashkovskiiA.Yu.andRonkinL.I.Almostperiodiccurrentsandholomorphicchains.//
C. R. Acad. Sci. Paris.– Serie I. – 327. – 1998. – P.302-307.
[3] Levin B.Ya. Almost periodic functions with the bounded spectrum.//Sb. Actual’nyje voprosy matem-
aticheskogo analisa. – Published by Rostov university. – 1978. – P.112-124.(Russian)
[4] Levitan B.M. Almost periodic functions.– M.: Gostehizdat. – 1953. – 396 p. (Russian)
[5] Ronkin L.I. Introduction to the theory of entire functions of several variables. – M.: ”Nayka”. – 1971.
– 430 p. (Russian)
[6] Ronkin L.I. CR-functions and holomorphic almost periodic functions with an integer base.//at. Fiz.
Anal. Geom. – .4. – 1997.– P.472-490.(Russian)
[7] Ronkin L.I. On a certain class of holomorphic almost periodic functions.//Sb.Mat. – 33. – 1992. –
P.135-141.(Russian)
[8] RonkinL.I.Almostperiodicdistributionsintube domains.–Zap.Nauchn.Sem.POMI.–247.–1997.
– P.210-236.(Russian)
[9] RonkinL.I.Jessentheoremforholomorphicalmost-periodicfunctionsintubedomains.//Sb.Mat–28.
– 1987. – P.199-204.(Russian)
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