DOI: 10.2478/auom-2013-0044
An. Şt. Univ. Ovidius Constanţa
Vol. 21(3),2013, 73–80
Almost Periodic Solutions for Some
Convolution Equations
Silvia-Otilia Corduneanu
Abstract
We use the theory of Fourier series for almost periodic functions to
looking for complex-valued functions f which are almost periodic on R
and satisfy the following equation
Z t Z +∞
1
f (x) = lim
f (x − y − z)ϕ(z)dz g(y)dy + h(x), x ∈ R.
t→∞ 2t −t
−∞
In this context g and h are almost periodic functions on R and ϕ belongs
to L1 (R).
1
Introduction
b be the dual group of the group R and AP (R) the space of all almost periLet R
odic complex-valued functions defined on R. We denote by M (f ) or Mx [f (x)]
the mean of an almost periodic function f and by cγ (f ) the Fourier coefficient
b If the functions f and g belong to
of f corresponding to a character γ ∈ R.
1
AP (R) and ϕ ∈ L (R), then the function
Φ(x) = My [(ϕ ∗ f )(x − y)g(y)],
x ∈ R,
(1.1)
b f ∈ AP (R), g ∈ AP (R) and
is an almost periodic function. Consider γ ∈ R,
1
ϕ ∈ L (R). We calculate the Fourier coefficient of Φ corresponding to the
character γ, more precisely we obtain:
cγ (Φ) = cγ (f )cγ (g)ϕ(γ),
b
Key Words: Almost periodic function, convolution equation, Fourier coefficients.
2010 Mathematics Subject Classification: Primary 42A05,42A10; Secondary 39B32.
Received: February, 2013.
Revised: February, 2013.
Accepted: April, 2013.
73
74
Silvia-Otilia Corduneanu
where ϕ
b is the the Fourier transform of ϕ. In this paper we consider the
following equation
f (x) = My [(ϕ ∗ f )(x − y)g(y)] + h(x),
x ∈ R,
(1.2)
where g ∈ AP (R), h ∈ AP (R) and ϕ ∈ L1 (R). If we use the formula of the
mean we can rewrite equation (1.2) as
Z Z +∞
1 t
(1.3)
f (x − y − z)ϕ(z)dz g(y)dy + h(x),
f (x) = lim
t→∞ 2t −t
−∞
where x ∈ R. For solving equation (1.2) we use the theory of Fourier series
for almost periodic functions. Taking into account the property that two
almost periodic functions coincide if they have the same Fourier coefficients
we establish an existence and uniqueness result in the case of equation (1.2).
Finally we establish an existence and uniqueness result in the case of equation
Z
1 t
(1.4)
f (x) = My [(ϕ ∗ f )(x − y)g(y)] + lim
k(x − y)l(y)dy,
t→∞ 2t −t
where x ∈ R,
1
t→∞ 2t
Z
t
Z
∞
My [(ϕ ∗ f )(x − y)g(y)] = lim
−t
f (x − y − z)ϕ(z)dz g(y)dy,
−∞
for every x ∈ R and the functions k and l belong to AP (R).
2
Preliminaries
Consider C(R) the set of all bounded continuous complex-valued functions on
R and denote by k · k the supremum norm defined on C(R). Let λ be the
Lebesgue
Z measure on R. The space of Lebesgue measurable functions f on R,
with
|f (x)|dλ(x) < ∞ will be denoted by L1 (R), and the norm of L1 (R),
R
b to denote the dual group of the group R and [R]
b the space
by k · k1 . We use R
1
of all trigonometric polynomials on R. If ϕ ∈ L (R), the Fourier transform of
ϕ, denoted by ϕ,
b is given by
Z
b
ϕ(γ)
b
=
ϕ(x)γ(x)dx, γ ∈ R.
R
For f ∈ C(R) and a ∈ R, the translate of f by a is fa (x) = f (x + a) for all
x ∈ R. We remind the definition of an almost periodic function which can be
found in [4], [5], [11], [13], [14].
ALMOST PERIODIC SOLUTIONS FOR SOME CONVOLUTION EQUATIONS
75
Definition 2.1. A function f ∈ C(R) is called an almost periodic function on
R, if the family of translates of f , {fa | a ∈ R} is relatively compact in the
sense of uniform convergence on R.
The space AP (R) is a Banach algebra with respect to the supremum norm
b is dense in AP (R) in the sense of uniform convergence. There exists a
and [R]
unique positive linear functional M : AP (R) → C such that M (fa ) = M (f ),
for all a ∈ R, f ∈ AP (R) and M (1) = 1. (In this context we denote by 1 the
constant function which is 1 for all x ∈ R). If f ∈ AP (R) we define the mean
of f as being the above complex number M (f ), we put cγ (f ) = M (γf ) for all
b and we call cγ (f ), the Fourier coefficient of f corresponding to γ ∈ R.
b
γ∈R
b
If f ∈ AP (R) the set {γ ∈ R | M (γf ) 6= 0} is at most a countable set (see
b | n ∈ N}. The Fourier
Theorem 1.15 from [5]) and we denote it by {γn ∈ R
series of f is
∞
X
cγn (f )γn .
n=1
1
If ϕ ∈ L (R) and f ∈ AP (R), their convolution ϕ ∗ f belongs to AP (R). We
recall that
Z
ϕ ∗ f (x) =
f (x − y)ϕ(y)dy, x ∈ R.
R
3
Convolution equations with almost periodic solutions
Theorem 3.1. If g and f belong to AP (R) and ϕ belongs to L1 (R) then the
function Φ : R → C given by
Φ(x) = My [(ϕ ∗ f )(x − y)g(y)],
x ∈ R,
(3.1)
b the following equality is true
is almost periodic and for every γ ∈ R,
cγ (Φ) = cγ (f )cγ (g)ϕ(γ).
b
(3.2)
b
Proof. We first check the conclusion of the theorem for characters of R,
b
after that for trigonometric polynomials of [R] and finally for arbitrary alb suppose that f = γ0 and denote
most periodic functions. Consider γ0 ∈ R,
Φ0 (x) = My [(ϕ ∗ γ0 )(x − y)g(y)], x ∈ R. With this choice we calculate ϕ ∗ γ0
as
Z ∞
ϕ ∗ γ0 (x) =
γ0 (x − y)ϕ(y) = ϕ(γ
b 0 )γ0 (x), x ∈ R.
−∞
We observe that
Φ0 (x) = My [(ϕ ∗ γ0 )(x − y)g(y)] = ϕ(γ
b 0 )cγ0 (g)γ0 (x),
x ∈ R,
76
Silvia-Otilia Corduneanu
b then Φ0 ∈ AP (R). On the other hand we get, for
therefore, if f = γ0 ∈ R,
b
γ ∈ R,

 1, γ = γ0
cγ (f ) = M (γγ0 ) =

0, γ 6= γ0
hence
cγ (Φ0 ) = cγ (f )cγ (g)ϕ(γ).
b
b the function Φ which is given in
It immediately follows, that for f ∈ [R],
relation (3.1) is an almost periodic function on R and relation (3.2) is valid.
Suppose that f ∈ AP (R) and is arbitrary. There exists a sequence (fn )n such
that
b ∧ (kfn − f k → 0).
((∀n ∈ N)(fn ∈ [R]))
If we denote
Φn (x) = My [(ϕ ∗ fn )(x − y)g(y)],
x ∈ R, n ∈ N,
from the previous part it results that (∀n ∈ N)(Φn ∈ AP (R)). Taking into
account that for every n ∈ N and x ∈ R
|Φn (x) − Φ(x)|
= |My [(ϕ ∗ fn )(x − y)g(y)] − My [(ϕ ∗ f )(x − y)g(y)]|
≤ λ(|ϕ|)M (|g|)kfn − f k,
it follows that Φn → Φ uniformly, hence Φ ∈ AP (R). We see also from the
b
previous part that for γ ∈ R,
(∀n ∈ N)(cγ (Φn ) = cγ (fn )cγ (g)ϕ(γ)).
b
Using the fact that Φn → Φ uniformly it results that cγ (Φn ) → cγ (Φ) and
b
further, for γ ∈ R,
cγ (Φ) = cγ (f )cγ (g)ϕ(γ).
b
Consider g ∈ AP (R), f ∈ AP (R) and ϕ ∈ L1 (R). Suppose that the function
h ∈ AP (R) has the Fourier series
∞
X
n=1
cγn (h)γn .
ALMOST PERIODIC SOLUTIONS FOR SOME CONVOLUTION EQUATIONS
77
In what follows we treat equation
f (x) = My [(ϕ ∗ f )(x − y)g(y)] + h(x),
x ∈ R.
(3.3)
Using the formula of the mean we can rewrite equation (3.3) as
1
t→∞ 2t
Z
t
+∞
Z
f (x) = lim
−t
f (x − y − z)ϕ(z)dz g(y)dy + h(x),
(3.4)
−∞
where x ∈ R. Taking into account the property that two almost periodic
functions coincide if they have the same Fourier coefficients, we establish an
existence and uniqueness result in the case of equation (3.3).
Theorem 3.2. Consider g ∈ AP (R), f ∈ AP (R) and ϕ ∈ L1 (R). If
∞
X
|cγn (h)| < ∞ and inf 1 − cγ (g)ϕ(γ)
b
∈ (0, ∞), then equation (3.3) has a
n=1
b
γ∈R
unique solution f ∈ AP (R).
Proof. Let us denote
δ = inf 1 − cγ (g)ϕ(γ)
b
b
γ∈R
b Taking into account the conclusion of Theorem 3.1 we
and consider γ0 ∈ R.
deduce that if f ∈ AP (R) would be solution of the equation (3.3) then
cγ0 (f ) = cγ0 (f )cγ0 (g)ϕ(γ
b 0 ) + cγ0 (h).
Therefore we consider the series
∞
X
cγn (h)
γn
1 − ϕ(γ
b n )cγn (g)
n=1
(3.5)
which is uniformly convergent, because
cγn (h)
1
γn (x) ≤
cγn (h) ,
1 − ϕ(γ
b n )cγn (g)
δ
x ∈ R.
We denote by f the sum of the series (3.5). It is obvious that f is an almost
periodic function. Based on the property that two almost periodic functions
coincide if they have the same Fourier series we conclude that f is the unique
solution of the equation (3.3). 78
Silvia-Otilia Corduneanu
Consider ϕ ∈ L1 (R), g ∈ AP (R), k ∈ AP (R) and l ∈ AP (R). Next we treat
equation
Z
1 t
(3.6)
k(x − y)l(y)dy,
f (x) = My [(ϕ ∗ f )(x − y)g(y)] + lim
t→∞ 2t −t
where x ∈ R. Equation (3.6) is a a particular case of equation (3.3).
Theorem 3.3. If in the previous context we consider that kϕk1 kgk ∈ (0, 1)
then equation (3.6) has a unique solution f ∈ AP (R).
Proof. Let us denote
1
h(x) = lim
t→∞ 2t
Z
t
k(x − y)l(y)dy,
x ∈ R.
−t
Taking into account that h(x) = My [k(x − y)l(y)], x ∈ R, we get the equality
cγ (h) = cγ (k) · cγ (l),
b
γ ∈ R.
(3.7)
We observe that equation (3.6) is a a particular case of equation (3.3) so we
check the first hypothesis of Theorem 3.2. Using a Cauchy inequality we obtain
∞
X
|cγn (h)| =
n=1
∞
X
"
|cγn (k)| |cγ (l)| ≤
n=1
∞
X
# 21 "
|cγn (k)|
2
∞
X
# 21
2
|cγ (l)|
.
n=1
n=1
The functions k and l are almost periodic on R, hence their Fourier series
satisfy the Parseval equality, i.e.,
∞
X
|cγn (k)|2 = M (|k|2 )
n=1
and
∞
X
|cγn (l)|2 = M (|l|2 ).
n=1
On the other hand taking into account that
cγ (g)ϕ(γ)
b
≤ kgk · kϕk1 ,
b
γ ∈ R,
we obtain
inf 1 − cγ (g)ϕ(γ)
b
≥ δ,
b
γ∈R
where δ = 1 − kgk · kϕk1 > 0. We can apply Theorem 3.2, and we conclude
that equation (3.6) has a unique solution f ∈ AP (R). ALMOST PERIODIC SOLUTIONS FOR SOME CONVOLUTION EQUATIONS
79
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Silvia-Otilia CORDUNEANU,
Department of Mathematics,
Gh. Asachi Technical University of Iaşi,
Bdul Carol I 11, 700506 Iaşi, Romania.
Email: scorduneanu@yahoo.com