ON INTEGRABILITY CONDITIONS OF FUNCTIONS RELATED TO THE
FORMAL TRIGONOMETRIC SERIES BELONGING
TO ORLICZ SPACE
XH. Z. KRASNIQI
Abstract. In this paper we have introduced a new class of numerical sequences named as Mean Rest
Bounded Variation Sequence of second order. This class is used to show some integrability conditions
of the functions sin xg(x) and sin xf (x) such that these functions belong to the Orlicz space, where g(x)
and f (x) denote formal sine and cosine trigonometric series, respectively. This study may be taken as
an continuation of some recent foregoing results proved by L. Leindler [5] and S. Tikhonov [14].
1. Introduction
Many authors have studied the integrability of the formal series
(1.1)
g(x) :=
∞
X
λn sin nx
n=1
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and
(1.2)
f (x) :=
∞
X
λn cos nx
n=1
Received September 24, 2012.
2010 Mathematics Subject Classification. Primary 42A32, 46E30.
Key words and phrases. Trigonometric series; integrability; Orlicz space.
requiring certain conditions on the coefficients λn (see [6]–[7] and [2]–[15]).
As initial example, R. P. Boas in [1] proved the following result for (1.1).
Theorem 1.1. If λn ↓ 0, then for 0 ≤ γ ≤ 1, x−γ g(x) ∈ L[0, π] if and only if
converges.
P∞
n=1
nγ−1 λn
This result had previously been proved for γ = 0 by W.H. Young [15] and was later extended
by P. Heywood [4] for 1 < γ < 2.
Later the monotonicity condition on the coefficients λn was replaced to more general ones by
S. M. Shah [12] and L. Leindler [6].
In 2004 S. Tikhonov [14] proved two theorems providing sufficient conditions of g(x) and f (x)
belonging to Orlicz space. Before we state his theorems, we will recall some notions and notations.
Leindler ([6]) introduced the following definition. A sequence c := {cn } of positive numbers
tending to zero is of rest bounded variation, or briefly R0+ BV S, if it possesses the property
(1.3)
∞
X
|cn − cn+1 | ≤ K(c)cm
n=m
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for all natural numbers m, where K(c) is a constant depending only on c.
A sequence γ := {γn } of positive terms will be called almost increasing (decreasing) if there
exists constant C := C(γ) ≥ 1 such that
Cγn ≥ γm
(γn ≤ Cγm )
holds for any n ≥ m.
Here and further C, Ci denote positive constants that are not necessarily the same at each
occurrence, and also we use the notion u w (u w) at inequalities if there exists a positive
constant C such that u ≤ Cw (u ≥ Cw) holds.
We will denote (see [9]) by 4(p, q), (0 ≤ q ≤ p) the set of all nonnegative functions Φ(x) defined
on [0, 1) such that Φ(0) = 0 and Φ(x)/xp is nonincreasing and Φ(x)/xq is nondecreasing. It is clear
that 4(p, q) ⊂ 4(p, 0), 0 < q ≤ p. As an example, 4(p, 0) contains the function Φ(x) = log(1 + x).
Here
and in the sequel, a function γ(x) is defined by the sequence γ in the following way:
γ nπ := γn , n ∈ N and there exist positive constants C1 and C2 such that C1 γn+1 ≤ γ(x) ≤ C2 γn
for x ∈
π
π
n+1 , n
.
A locally integrable almost everywhere positive function γ(x) : [0, π] → [0, ∞) is said to be a
weight function. Let Φ(t) be a nondecreasing continuous function defined on [0, ∞) such that
Φ(0) = 0 and limt→∞ Φ(t) = +∞. For a weight γ(x) the weighted Orlicz space L(Φ, γ) is defined
by
Z π
(1.4)
L(Φ, γ) = h :
γ(x)Φ(ε|h(x)|)dx < ∞ for some ε > 0 .
0
Tikhonov’s results now can be read as follows.
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Theorem 1.2. Let Φ(x) ∈ 4(p, 0), 0 ≤ p. If λn ∈ R0+ BV S and the sequence {γn } is such that
{γn n−1+ε } is almost decreasing for some ε > 0, then
∞
X
γn
(1.5)
Φ(nλn ) < ∞ ⇒ ψ(x) ∈ L(Φ, γ),
n2
n=1
where a function ψ(x) is either a sine or cosine series.
Theorem 1.3. Let Φ(x) ∈ 4(p, q), 0 ≤ q ≤ p. If λn ∈ R0+ BV S and the sequence {γn } is such
that {γn n−(1+q)+ε } is almost decreasing for some ε > 0, then
∞
X
γn
(1.6)
Φ(n2 λn ) < ∞ ⇒ g(x) ∈ L(Φ, γ).
2+q
n
n=1
A null-sequence c of nonnegative numbers possessing the property
(1.7)
∞
X
|cn − cn+1 | ≤
n=2m
2m−1
K(c) X
cν
m ν=m
is called a sequence of mean rest bounded variation, in symbols, c ∈ M RBV S.
In [5], L. Leindler extended Theorem 1.2 and Theorem 1.3, so that the sequence {λn } belongs
to the class M RBV S instead of the class R0+ BV S. His results are formulated as follows.
Theorem 1.4. Theorems 1.2 and 1.3 can be improved when the condition λn ∈ R0+ BV S is
replaced by the assumption λn ∈ M RBV S. Furthermore the conditions of (1.8) and (1.6) may be
modified as follows:
!
2n−1
∞
X
X
γn
(1.8)
Φ
λν < ∞ ⇒ ψ(x) ∈ L(Φ, γ),
n2
ν=n
n=1
and
(1.9)
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∞
2n−1
X
X
γn
Φ
n
λν
n2+q
ν=n
n=1
!
< ∞ ⇒ g(x) ∈ L(Φ, γ),
respectively.
In 2009, B. Szal [11] introduced a new class of sequences as follows.
Definition 1.1. A sequence α := {ck } of nonnegative numbers tending to zero is called Rest
Bounded Second Variation of second order, or briefly, {ck } ∈ RBSV S, if it has the property
Close
∞
X
Quit
k=m
|ck − ck+2 | ≤ K(α)cm
for all natural numbers m, where K(α) is positive, depending only on the sequence {ck }, and we
assume that the sequence is bounded.
Motivated by the above definition, we introduce a new class of numerical sequences.
Definition 1.2. A null-sequence c of nonnegative numbers possessing the property
(1.10)
∞
X
|42 cn + 42 cn+1 | ≤
n=2m
2m−1
K(c) X
|cν − cν+2 |
m ν=m
is said to be a sequence of Mean Rest Bounded Variation of second order, in symbols, c ∈
M RBSV S, where 42 cn = cn − 2cn+1 + cn+2 .
The aim of this paper is to extend Tikhonov’s results and Leindler’s result, so that the sequence
{λn } belongs to the class M RBSV S instead of the classes R0+ BV S and M RBV S. To achieve
this aim, we need some helpful statements given in next section.
2. Auxiliary Lemmas
We shall use the following lemmas for the proof of the main results.
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Lemma 2.1 ([9]). Let Φ ∈ 4(p, q), 0 ≤ q ≤ p, and tj ≥ 0, j = 1, 2, . . . , n, n ∈ N. Then
(1) θp
Φ(t) ≤ Φ(θt)
θq Φ(t), 0 ≤ θ ≤
1,
t ≥ 0,
≤
P
p∗
Pn
n
1/p∗
(2) Φ
(tj )
,
p∗ := max(1, p).
j=1 tj ≤
j=1 Φ
Lemma 2.2 ([5]). Let Φ ∈ 4(p, q), 0 ≤ q ≤ p. If ρn > 0, an ≥ 0 and if
Close
(2.1)
Quit
2m+1
X−1
ν=2m
aν m
2X
−1
ν=1
aν
holds for all m ∈ N, then
∞
X
k=1
ρk Φ
k
X
!
aν
ν=1
∞
X
Φ
k=1
2k−1
X
∞
1 X
ρν
kρk
!
aν
ρk
ν=k
!p∗
,
ν=k
where p∗ := max(1, p).
Lemma 2.3. The following representations of g(x) and f (x)
2 sin xg(x) = −
∞
X
(λk − λk+2 ) cos(k + 1)x
k=1
and
2 sin xf (x) =
∞
X
(λk − λk+2 ) sin(k + 1)x,
k=1
where we have assumed that λ1 = λ2 = 0, hold.
Proof. We start from obvious equality
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X
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λk cos kx =
k=1
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or
∞
∞
k=1
k=1
1X
1X
(λk + λk+1 ) cos kx +
(λk − λk+1 ) cos kx,
2
2
∞
∞
k=1
k=1
∞
X
1X
1X
1
λk cos kx =
(λk + λk+1 ) cos kx − cos x
λk cos kx
2
2
2
k=2
∞
X
1
λk sin kx.
− sin x
2
k=2
Thus we have
∞
1 + cos x X
λk cos kx
2
k=2
=
∞
1X
2
∞
(λk + λk+1 ) cos kx −
k=1
X
1
1
sin x
λk sin kx − λ1 cos x
2
2
k=2
or since λ1 = 0, we obtain
∞
X
(2.2)
λk cos kx
k=2
1
=
2 cos2
X
∞
x
2
(λk + λk+1 ) cos kx − sin x
k=1
∞
X
λk sin kx .
k=2
Similarly as above, we obtain
∞
X
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k=1
∞
k=1
k=1
∞
∞
k=1
k=1
1X
1X
λk sin kx =
(λk + λk+1 ) sin kx
2
2
(2.3)
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∞
1X
1X
(λk + λk+1 ) sin kx +
(λk − λk+1 ) sin kx,
2
2
or
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λk sin kx =
−
∞
∞
k=2
k=2
X
X
1
1
λk sin kx + sin x
λk cos kx.
cos x
2
2
Inserting (2.2) into (2.3), we have (λ1 = 0)
∞
∞
∞
k=1
k=1
k=2
X
1X
1X
1
λk sin kx =
(λk + λk+1 ) sin kx − cos x
λk sin kx
2
2
2
∞
sin x2 X
(λk
+
2 cos x2
k=1
∞
X
∞
sin x2 sin x X
+ λk+1 ) cos kx −
λk sin kx
2 cos x2
k=2
∞
sin x2 X
(λk + λk+1 ) sin kx +
(λk + λk+1 ) cos kx
2 cos x2
k=1
k=1
∞
cos x sin x2 sin x X
−
+
λk sin kx
2
2 cos x2
1
=
2
k=2
or
∞
X
λk sin kx =
k=1
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k=1
Applying the summation by parts to the above equality and taking into account that λ1 = λ2 = 0,
we obtain
∞
k
∞
X
X
1 X
1
λk sin kx =
sin i +
(λk − λk+2 )
x,
2 cos x2
2
i=0
k=1
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∞
1
1 X
(λ
+
λ
)
sin
k
+
x
k
k+1
2 cos x2
2
k=1
or finally, noting that
k
X
1
x
2 sin i +
x sin = 1 − cos(k + 1)x,
2
2
i=0
we get
∞
X
∞
λk sin kx = −
k=1
1 X
(λk − λk+2 ) cos(k + 1)x,
2 sin x
k=1
which clearly proves the first part of this lemma.
For the proof of the second part of this lemma, it is enough to put n = 1 into the equality
(3.10), see [11, page 167].
P
2n−1
Lemma 2.4. If λ := {λn } ∈ M RBSV S and Dn := n1 k=n |λk − λk+2 |, then
Dk D`
holds for all k ≥ 2`.
Proof. For m ≥ 2`, we note that
2`−1
∞
X
1 X
|λk − λk+2 | |42 λk + 42 λk+1 |
`
k=`
k=2`
∞
X
≥
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≥
k=m
∞
X
|42 λk + 42 λk+1 |
kλk − λk+2 | − |λk+1 − λk+3 k ≥ |λm − λm+2 |.
k=m
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Summing up the both sides of the last inequality, when m goes from k to 2k − 1, we obtain
2`−1
2k−1
X
k X
|λm − λm+2 |,
|λk − λk+2 | `
k=`
m=k
whence the required inequality follows immediately.
3. Main Results
Our first theorem deals with integrability of both functions sin xg(x) and sin xf (x) simultaneously.
Theorem 3.1. Let Φ(x) ∈ 4(p, 0), 0 ≤ p. If λn ∈ M RBSV S and the sequence {γn } is such
that {γn n−1+ε } is almost decreasing for some ε > 0, then
!
∞
2n−1
X
X
γn
(3.1)
Φ
|λν − λν+2 | < ∞ ⇒ sin xψ(x) ∈ L(Φ, γ),
n2
ν=n
n=1
where a function ψ(x) is either a sine or cosine series.
Proof. For the proof
i we use the idea which Tikhonov and Leindler used for their results. For
π
π
this, let x ∈ n+1 , n . Based on Lemma 2.3 and applying the summation by parts, we obtain
X
n
X
∞
2| sin xf (x)| ≤
|λk − λk+2 | + (λk − λk+2 ) sin(k + 1)x
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≤
k=1
n
X
k=1
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k=n
|λk − λk+2 | +
∞
X
∗ e k (x)
|42 λk + 42 λk+1 |D
k=n
∗ e n (x)
+ |λn − λn+2 |D
e ∗ (x) are defined by
where D
k
e k∗ (x)
D
k
X
cos x2 − cos k +
:=
sin(i + 1)x =
2 sin x2
i=0
3
2
x
,
k ∈ N.
e ∗ (x)| = O
Taking into account that |D
k
2| sin xf (x)| ≤
n
X
1
x
and {λn } ∈ M RBSV S, we have
|λk − λk+2 | + n
k=1
n
X
n
X
|42 λk + 42 λk+1 | + n|λn − λn+2 |
k=n
|λk − λk+2 | +
n−1
X
|λk − λk+2 | + n|λn − λn+2 |
k= n
2
k=1
∞
X
|λk − λk+2 | + n|λn − λn+2 |.
k=1
The following estimates can be obtained by the same technique. We get
∞
n
X
X
2| sin xg(x)| ≤
|λk − λk+2 | + (λk − λk+2 ) cos(k + 1)x
≤
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≤
k=1
n
X
k=n
|λk − λk+2 | +
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|42 λk + 42 λk+1 |Dk∗ (x) + |λn − λn+2 |Dn∗ (x)
|λk − λk+2 | + n
n
X
k=1
Close
∞
X
k=n
∞
X
k=1
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k=1
n
X
n
X
k=1
|42 λk + 42 λk+1 | + n|λn − λn+2 |
k=n
|λk − λk+2 | +
n−1
X
|λk − λk+2 | + n|λn − λn+2 |
k= n
2
|λk − λk+2 | + n|λn − λn+2 |,
where Dk∗ (x) are defined by
Dk∗ (x)
k
X
sin k + 32 x − sin x2
cos(i + 1)x =
:=
,
2 sin x2
i=0
Thus
| sin xψ(x)| n
X
k ∈ N.
|λk − λk+2 | + n|λn − λn+2 |,
k=1
where a function ψ(x) is either f (x) or g(x).
Moreover, since {λn } ∈ M RBSV S,
∞
n
X
X
n|λn − λn+2 | ≤ n
|42 λk + 42 λk+1 | |λk − λk+2 |,
k=n
k=1
and hence
(3.2)
| sin xψ(x)| n
X
|λk − λk+2 |.
k=1
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According to Lemma 2.4, the condition (2.1) with |λν − λν+2 | in place of aν is satisfied, and thus
we are ready to apply Lemma 2.2. Therefore, by (3.2), we obtain
!Z
!
Z π
∞
n
∞
n
π/n
X
X
X
X
γn
|λk − λk+2 |
γ(x)dx γ(x)Φ(| sin xψ(x)|)dx Φ
Φ
|λk − λk+2 |
n2
π/(n+1)
0
n=1
n=1
k=1
k=1
!
!p∗
∞
2n−1
∞
X
X
γn n X γν
|λk − λk+2 |
Φ
,
n2 γn ν=n ν 2
n=1
k=n
where p∗ := max(1, p).
Finally, by the assumption on {γn }, we get
∞
n X γν
1
γn ν=n ν 2
which along with the above inequality immediately imply (3.1). The proof is completed.
Theorem 3.2. Let Φ(x) ∈ 4(p, q), 0 ≤ q ≤ p. If λn ∈ M RBSV S and the sequence {γn } is
such that {γn n−(1+q)+ε } is almost decreasing for some ε > 0, then
!
∞
2n−1
X
X
γn
(3.3)
Φ
k|λk − λk+2 | < ∞
⇒
sin xf (x) ∈ L(Φ, γ).
n2+q
n=1
k=n
Proof. Let x ∈
2| sin xf (x)| ≤
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π
π
n+1 , n
n
X
x
(3.4)
i
. Then
X
∞
(k + 1)x|λk − λk+2 | + (λk − λk+2 ) sin(k + 1)x
k=1
n
X
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k=n+1
k|λk − λk+2 | +
k=1
n−1
n
X
∞
X
k=n
k|λk − λk+2 | +
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n−1
n
X
k=1
n−1
X
k= n
2
k=1
Close
∗ ∗ e k (x) + |λn − λn+2 |D
e n (x)
|42 λk + 42 λk+1 |D
k|λk − λk+2 |.
|λk − λk+2 | + n|λn − λn+2 |
According to Lemmas 2.1, 2.2, 2.4, and the estimate (3.4), we have
Z π
γ(x)Φ(| sin xf (x)|)dx
0
!Z
∞
n
π/n
X
X
Φ n−1
k|λk − λk+2 |
γ(x)dx
(3.5)
n=1
∞
X
γn
Φ
2+q
n
n=1
∞
X
n=1
Φ
k=1
n
X
2n−1
X
π/(n+1)
!
k|λk − λk+2 |
k=1
!
k|λk − λk+2 |
k=n
γn
n2+q
∞
n1+q X γν
γn ν=n ν 2+q
!p∗
,
where p∗ := max(1, p).
By the assumption on {γn }, we get
∞
n1+q X γν
1,
γn ν=n ν 2+q
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and hence (3.5) takes this form
Z π
∞
X
γn
γ(x)Φ(| sin xf (x)|)dx Φ
n2+q
0
n=1
2n−1
X
!
k|λk − λk+2 | ,
k=n
which proves (3.3). With this the proof of theorem is finished.
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Xh. Z. Krasniqi, University of Prishtina, Faculty of Education, Department of Mathematics and Informatics, Avenue
“Mother Theresa” 5, 10000 Prishtina, Kosova
e-mail: xhevat.krasniqi@uni-pr.edu