243
Acta Math. Univ. Comenianae
Vol. LXXXII, 2 (2013), pp. 243–252
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
∞
X
(1.1)
g(x) :=
λn sin nx
n=1
and
(1.2)
f (x) :=
∞
X
λn cos nx
n=1
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
P∞
γ−1
λn converges.
n=1 n
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.
Received September 24, 2012.
2010 Mathematics Subject Classification. Primary 42A32, 46E30.
Key words and phrases. Trigonometric series; integrability; Orlicz space.
244
XH. Z. KRASNIQI
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
∞
X
(1.3)
|cn − cn+1 | ≤ K(c)cm
n=m
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
exist
following way: γ nπ := γn , n ∈ N and there
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.
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
(1.5)
∞
X
γn
Φ(nλn ) < ∞
n2
n=1
⇒
ψ(x) ∈ L(Φ, γ),
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
ON INTEGRABILITY CONDITIONS FOR TRIGONOMETRIC SERIES
245
A null-sequence c of nonnegative numbers possessing the property
∞
X
(1.7)
|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 (1.5) and (1.6) may be modified as follows:
!
2n−1
∞
X
X
γn
Φ
λν < ∞ ⇒ ψ(x) ∈ L(Φ, γ),
(1.8)
n2
ν=n
n=1
and
(1.9)
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
∞
X
|ck − ck+2 | ≤ K(α)cm
k=m
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
n=2m
|42 cn + 42 cn+1 | ≤
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.
246
XH. Z. KRASNIQI
2. Auxiliary Lemmas
We shall use the following lemmas for the proof of the main results.
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
2m+1
X−1
(2.1)
aν ν=2m
m
2X
−1
aν
ν=1
holds for all m ∈ N, then
∞
X
ρk Φ
k
X
!
aν
ν=1
k=1
∞
X
Φ
k=1
2k−1
X
!
aν
ρk
∞
1 X
ρν
kρk
!p∗
,
ν=k
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
∞
∞
∞
X
1X
1X
λk cos kx =
(λk + λk+1 ) cos kx +
(λk − λk+1 ) cos kx,
2
2
k=1
or
k=1
k=1
∞
∞
k=1
k=1
∞
X
1X
1X
1
λk cos kx =
(λk + λk+1 ) cos kx − cos x
λk cos kx
2
2
2
−
1
sin x
2
k=2
∞
X
λk sin kx.
k=2
Thus we have
∞
1 + cos x X
λk cos kx
2
k=2
=
∞
1X
2
k=1
∞
(λk + λk+1 ) cos kx −
X
1
1
sin x
λk sin kx − λ1 cos x
2
2
k=2
ON INTEGRABILITY CONDITIONS FOR TRIGONOMETRIC SERIES
247
or since λ1 = 0, we obtain
∞
X
λk cos kx
k=2
(2.2)
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
λk sin kx =
k=1
∞
∞
k=1
k=1
1X
1X
(λk + λk+1 ) sin kx +
(λk − λk+1 ) sin kx,
2
2
or
∞
∞
k=1
k=1
1X
1X
λk sin kx =
(λk + λk+1 ) sin kx
2
2
(2.3)
−
∞
∞
k=2
k=2
X
X
1
1
cos x
λk sin kx + sin x
λk cos kx.
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
sin x2 sin x X
+
(λ
+
λ
)
cos
kx
−
λk sin kx
k
k+1
2 cos x2
2 cos x2
k=1
=
k=2
∞
1X
∞
sin x2 X
(λk + λk+1 ) sin kx +
(λk
2
2 cos x2
k=1
k=1
∞
cos x sin x2 sin x X
−
+
λk sin kx
2
2 cos x2
k=2
+ λk+1 ) cos kx
or
∞
X
k=1
∞
1 X
1
λk sin kx =
(λk + λk+1 ) sin k +
x
2 cos x2
2
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 =
(λk − λk+2 )
sin i +
x,
2 cos x2
2
i=0
k=1
k=1
or finally, noting that
k
X
1
x
2 sin i +
x sin = 1 − cos(k + 1)x,
2
2
i=0
248
XH. Z. KRASNIQI
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].
Lemma 2.4. If λ := {λn } ∈ M RBSV S and Dn :=
1
n
P2n−1
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
|42 λk + 42 λk+1 |
|λk − λk+2 | `
k=2`
k=`
∞
X
≥
≥
k=m
∞
X
|42 λk + 42 λk+1 |
kλk − λk+2 | − |λk+1 − λk+3 k ≥ |λm − λm+2 |.
k=m
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
|λk − λk+2 | |λm − λm+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 we use the idea
i which Tikhonov and Leindler used for
π
π
their results. For this, let x ∈ n+1 , n . Based on Lemma 2.3 and applying the
ON INTEGRABILITY CONDITIONS FOR TRIGONOMETRIC SERIES
summation by parts, we obtain
n
X
2| sin xf (x)| ≤
k=1
n
X
≤
X
∞
(λk − λk+2 ) sin(k + 1)x
|λk − λk+2 | + k=n
∞
X
|λk − λk+2 | +
k=1
∗ e k (x)
|42 λk + 42 λk+1 |D
k=n
∗ e (x)
+ |λn − λn+2 |D
n
e ∗ (x) are defined by
where D
k
k
X
cos x2 − cos k + 32 x
sin(i + 1)x =
, k ∈ N.
2 sin x2
i=0
e ∗ (x)| = O 1 and {λn } ∈ M RBSV S, we have
Taking into account that |D
k
x
e k∗ (x) :=
D
2| sin xf (x)| ≤
n
X
|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
2| sin xg(x)|
≤
≤
≤
n
X
k=1
n
X
k=1
n
X
X
∞
(λk − λk+2 ) cos(k + 1)x
|λk − λk+2 | + k=n
|λk − λk+2 | +
|λk − λk+2 | + n
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
|42 λk + 42 λk+1 |Dk∗ (x) + |λn − λn+2 |Dn∗ (x)
k=n
∞
X
k=1
∞
X
|λk − λk+2 | + n|λn − λn+2 |,
k=1
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
k ∈ N.
249
250
XH. Z. KRASNIQI
Thus
| sin xψ(x)| n
X
|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|λn − λn+2 | ≤ n
∞
X
|42 λk + 42 λk+1 | k=n
n
X
|λk − λk+2 |,
k=1
and hence
| sin xψ(x)| (3.2)
n
X
|λk − λk+2 |.
k=1
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
π/n
X
X
γ(x)Φ(| sin xψ(x)|)dx Φ
|λk − λk+2 |
γ(x)dx
0
n=1
∞
X
γn
Φ
n2
n=1
∞
X
n=1
π/(n+1)
k=1
Φ
n
X
!
|λk − λk+2 |
k=1
2n−1
X
!
|λk − λk+2 |
k=n
γn
n2
∞
n X γν
γn ν=n ν 2
!p∗
,
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)
k|λk − λk+2 | < ∞ ⇒ sin xf (x) ∈ L(Φ, γ).
Φ
n2+q
n=1
k=n
ON INTEGRABILITY CONDITIONS FOR TRIGONOMETRIC SERIES
Proof. Let x ∈
2| sin xf (x)| ≤
π
π
n+1 , n
n
X
i
. Then
X
∞
(λk − λk+2 ) sin(k + 1)x
(k + 1)x|λk − λk+2 | + k=1
n
X
x
k=n+1
k|λk − λk+2 |
k=1
∞
X
+
(3.4)
251
∗ ∗ e k (x) + |λn − λn+2 |D
e n (x)
|42 λk + 42 λk+1 |D
k=n
n−1
n
X
k|λk − λk+2 | +
|λk − λk+2 | + n|λn − λn+2 |
k= n
2
k=1
n−1
n−1
X
n
X
k|λk − λk+2 |.
k=1
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
−1
Φ n
k|λk − λk+2 |
γ(x)dx
(3.5)
n=1
∞
X
γn
Φ
n2+q
n=1
∞
X
n=1
Φ
k=1
n
X
2n−1
X
k=n
π/(n+1)
!
k|λk − λk+2 |
k=1
!
k|λk − λk+2 |
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
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.
252
XH. Z. KRASNIQI
References
1. Boas R. P., Jr., Integrability of trigonometrical series III, Quart. J. Math. (Oxford) 3(2)
(1952), 217–221.
2. Yung-Ming Chen, On the integrability of functions defined by trigonometrical series, Math.
Z. 66 (1956), 9–12.
3.
, Some asymptotic properties of Fourier constants and integrability theorems, Math.
Z., 68 (1957), 227–244.
4. Heywood P., On the integrability of functions defined by trigonometric series, Quart. J.
Math. (Oxford), 5(2) (1954), 71–76.
5. Leindler L., Integrability conditions pertaining to Orlicz space, J. Inequal. Pure and. Appl.
Math. 8(2) (2007), Art. 38, 6 pp.
6.
, A new class of numerical sequences and its applications to sine and cosine series,
Analysis Math. 28 (2002), 279–286.
7. Leindler L. and Németh J., On the connection between quasi power-monotone and quasi
geometrical sequences with application to integrability theorems for power series, Acta Math.
Hungar. 68(1-2) (1995), 7–19.
8. Lorentz G. G., Fourier Koeffizienten und Funktionenklassen, Math. Z. 51 (1948), 135–149.
9. Mateljevic M. and Pavlovic M., Lp -behavior of power series with positive coefficients and
Hardy spaces, Proc. Amer. Math. Soc. 87 (1983), 309–316.
10. O’Shea S., Note on an integrability theorem for sine series, Quart. J. Math. (Oxford) 8(2)
(1957), 279–281.
11. Szal B., Generalization of a theorem on Besov-Nikol’skiı̆ classes, Acta Math. Hungar. 125
(1–2) (2009), 161–181.
12. Shah S. M., Trigonometric series with quasi-monotone coefficients, Proc. Amer. Math. Soc.
13 (1962), 266–273.
13. Sunouchi G., Integrability of trigonometric series, J. Math. Tokyo, 1 (1953), 99-103.
14. Tikhonov S., On belonging of trigonometric series of Orlicz space, J. Inequal. Pure and.
Appl. Math. 5(2) (2004), Art. 22, 7 pp. 416–427.
15. Young W. H., Integrability of trigonometric series, Proc. London Math. Soc. 12 (1913),
41–70.
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