Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
31 (2015), 249–257
www.emis.de/journals
ISSN 1786-0091
REST BOUNDED SECOND VARIATION SEQUENCES AND
p-TH POWER INTEGRABILITY OF SOME FUNCTIONS
RELATED TO SUMS OF FORMAL TRIGONOMETRIC
SERIES
XHEVAT Z. KRASNIQI
Abstract. In this paper we have studied p-th power integrability of functions sin xg(x) and sin xf (x) with a weight, where g(x) and f (x) denote the
formal sum functions of sine and cosine trigonometric series respectively.
This study may be taken as a continuation for some recent foregoings results proven by L. Leindler [3] and S. Tikhonov [7] employing the so-called
rest bounded second variation sequences.
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
imposing certain conditions on the coefficients λn .
Some classical results of this type are obtained by Young-Boas-Haywood
(see [1], [2], [8]) which deal with above mentioned trigonometric series whose
coefficients are monotone decreasing.
Theorem 1.1. Let λn ↓ 0. If 0 ≤ α ≤ 2, then
∞
X
−α
x g(x) ∈ L(0, π) ⇐⇒
nα−1 λn < ∞.
n=1
2010 Mathematics Subject Classification. 42A32, 46E30.
Key words and phrases. Integrability of functions with a weight, trigonometric series, rest
bounded second variation sequences.
249
250
XHEVAT Z. KRASNIQI
If 0 < α < 1, then
x
−α
f (x) ∈ L(0, π) ⇐⇒
∞
X
nα−1 λn < ∞.
n=1
The monotonicity condition on the sequence {λn } was replaced by
L. Leindler [3] to a more general ones {λn } ∈ R0+ BV S.
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.
His theorems on integrability of the sum functions of the sine and the cosine
trigonometric series state as follows:
Theorem 1.2. Suppose that {λn } ∈ R0+ BV S, 1 < p < ∞, and 1/p − 1 < θ <
1/p. Then
∞
X
x−pθ |ψ(x)|p ∈ L(0, π) ⇐⇒
npθ+p−2 λpn < ∞,
n=1
where ψ(x) represents either f (x) or g(x).
Later on, J. Németh [4] considered weight functions more general than power
one and obtained some sufficient conditions for the integrability of the sine
series with such weights. Namely, he proved:
Theorem 1.3. Suppose that {λn } ∈ R0+ BV S and the sequence γ := {γn }
satisfies the condition: there exists an > 0 such that the sequence {γn n−2+ }
is almost decreasing. Then
∞
X
γn
λn < ∞ =⇒ γ(x)g(x) ∈ L(0, π).
n
n=1
A sequence γ := {γn } of positive terms will be called almost increasing
(decreasing) if there exists a constant C := C(γ) ≥ 1 such that
Cγn ≥ γm
(γn ≤ Cγm )
holds for any n ≥ m.
Here and in the sequel,
a function γ(x) is defined by the sequence γ in the
following way: γ πn := γn , n ∈ N and there exist positive
constants C1 and
π
C2 such that C1 γn ≤ γ(x) ≤ C2 γn+1 for x ∈ n+1
, πn .
In 2005 S. Tikhonov [7] has proved two theorems providing necessary and
sufficient conditions for the p−th power integrability of the sums of sine and
cosine series with weight γ. His results refine the assertions of Theorems 1.21.3 which show that such conditions depend on the behavior of the sequence
γ.
REST BOUNDED SECOND VARIATION SEQUENCES
251
We present Tikhonov’s results below.
Theorem 1.4. Suppose that {λn } ∈ R0+ BV S and 1 ≤ p < ∞.
(A) If the sequence {γn } satisfies the condition: there exists an ε1 > 0 such
that the sequence {γn n−p−1+ε1 } is almost decreasing, then the condition
∞
X
(1.4)
γn np−2 λpn < ∞
n=1
is sufficient for the validity of the condition
γ(x)|g(x)|p ∈ L(0, π).
(1.5)
(B) If the sequence {γn } satisfies the condition: there exists an ε2 > 0 such
that the sequence {γn np−1−ε2 } is almost increasing, then the condition
(1.4) is necessary for the validity of condition (1.5).
Theorem 1.5. Suppose that {λn } ∈ R0+ BV S and 1 ≤ p < ∞.
(A) If the sequence {γn } satisfies the condition: there exists an ε3 > 0 such
that the sequence {γn n−1+ε3 } is almost decreasing, then the condition
∞
X
(1.6)
γn np−2 λpn < ∞
n=1
is sufficient for the validity of the inclusion
γ(x)|f (x)|p ∈ L(0, π).
(1.7)
(B) If the sequence {γn } satisfies the condition: there exists an ε4 > 0 such
that the sequence {γn np−1−ε4 } is almost increasing, then the condition
(1.6) is necessary for the validity of condition (1.7).
In 2009 B. Szal [6] introduced a new class of sequences as follows.
Definition 1.1. A sequence α := {ck } of nonnegative numbers tending to
zero is called of Rest Bounded Second Variation sequence, 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 sequence
{ck }, and we assume it to be bounded.
Before we state the purpose of this paper we give the following definition:
Definition 1.2. A sequence α := {ck } of nonnegative numbers tending to
zero is called of Mean Rest Bounded Second Variation sequence, or briefly
{ck } ∈ M RBSV S, if it has the property
∞
X
k=2m
k|ck − ck+2 | ≤
2m−1
K(α) X
k|ck − ck+2 |
m k=m
252
XHEVAT Z. KRASNIQI
for all natural numbers m, where K(α) is positive, depending only on sequence
{ck }, and we assume it to be bounded.
The aim of this paper is to extend Tikhonov’s results ( as well as Leindler’s
result) so that the sequence {λn } belongs the class M RBSV S or RBSV S
which is a wider one than RBV S class. To achieve this goal we need some
helpful statements given in next section.
Closing this section we shall assume, throughout this paper, that λ1 = λ2 =
0.
2. Helpful Lemmas
We shall use the following lemmas for the proof of the main results.
Lemma 2.1 ([5]). Let λn > 0 and an ≥ 0. Then
!p
!p
∞
n
∞
∞
X
X
X
X
p
λn
aν
≤ pp
λ1−p
λν ,
n an
and
n=1
ν=1
∞
X
∞
X
λn
n=1
!p
aν
ν=n
≤ pp
n=1
ν=n
∞
X
n
X
p
λ1−p
n an
n=1
p≥1
!p
λν
,
p ≥ 1.
ν=1
Lemma 2.2. The following representations of g(x) and f (x) hold true:
∞
X
(λk − λk+2 ) cos(k + 1)x
2 sin xg(x) = −
k=1
and
∞
X
(λk − λk+2 ) sin(k + 1)x,
2 sin xf (x) =
k=1
where we have assumed that λ1 = λ2 = 0.
Proof. We start from obvious equality
∞
∞
∞
X
1X
1X
λk cos kx =
(λk + λk+1 ) cos kx +
(λk − λk+1 ) cos kx,
2 k=1
2 k=1
k=1
or
1X
λk cos kx
2 k=1
∞
∞
∞
∞
X
X
1
1
1X
λk cos kx − sin x
λk sin kx.
(λk + λk+1 ) cos kx − cos x
=
2 k=1
2
2
k=2
k=2
Thus we have
∞
1 + cos x X
2
λk cos kx
k=2
∞
∞
X
1
1X
1
λk sin kx − λ1 cos x
=
(λk + λk+1 ) cos kx − sin x
2 k=1
2
2
k=2
REST BOUNDED SECOND VARIATION SEQUENCES
253
or since λ1 = 0 we obtain
X
∞
∞
∞
X
X
1
(2.1)
λk cos kx =
(λk + λk+1 ) cos kx − sin x
λk sin kx .
2 x
2
cos
2
k=2
k=1
k=2
Similarly as above we obtain
∞
∞
∞
X
1X
1X
λk sin kx =
(λk + λk+1 ) sin kx +
(λk − λk+1 ) sin kx,
2
2
k=1
k=1
k=1
or
1X
1X
λk sin kx =
(λk + λk+1 ) sin kx
2 k=1
2 k=1
∞
∞
∞
∞
X
X
1
1
− cos x
λk sin kx + sin x
λk cos kx.
2
2
k=2
k=2
(2.2)
Inserting (2.1) into (2.2) we have (λ1 = 0)
∞
∞
∞
X
1X
1
1X
λk sin kx =
(λk + λk+1 ) sin kx − cos x
λk sin kx
2 k=1
2 k=1
2
k=2
∞
∞
sin x2 sin x X
sin x2 X
(λk + λk+1 ) cos kx −
λk sin kx
+
2 cos x2 k=1
2 cos x2 k=2
∞
∞
sin x2 X
1X
(λk + λk+1 ) cos kx
=
(λk + λk+1 ) sin kx +
2 k=1
2 cos x2 k=1
∞
cos x sin x2 sin x X
−
+
λk sin kx
2
2 cos x2
k=2
or
∞
1 X
1
λk sin kx =
(λk + λk+1 ) sin k +
x
x
2
cos
2
2 k=1
k=1
∞
X
Applying the summation by parts to 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 k=1
2
i=0
k=1
or finally, noting that
k
X
i=0
we get
∞
X
k=1
1
2 sin i +
2
x sin
1 X
(λk − λk+2 ) cos(k + 1)x,
2 sin x k=1
∞
λk sin kx = −
x
= 1 − cos(k + 1)x
2
254
XHEVAT Z. KRASNIQI
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 to
the equality (3.10), see page 167 of [6].
3. Main Results
Our first result deals with p−th power integrability of the function sin xf (x)
with weight γ.
Theorem 3.1. Suppose that 1 ≤ p < ∞. Let {λn } ∈ M RBSV S. If the
sequence {γn } satisfies the condition: there exists an ε1 > 0 such that the
sequence {γn n−p−1+ε1 } is almost decreasing, then the condition
∞
X
(3.1)
γn np−2 |λn − λn+2 |p < ∞
n=1
is sufficient for the validity of the condition
γ(x)| sin xf (x)|p ∈ L(0, π).
(3.2)
Proof. For the proof we shall
i idea of Tikhonov which he used for his
use the
π
π
results. For this, let x ∈ n+1 , n . Based on Lemma 2.2 and applying the
summation by parts we obtain
X
n
X
∞
(k + 1)|λk − λk+2 | + (λk − λk+2 ) sin(k + 1)x
2| sin xf (x)| ≤ x
k=1
1
n
n
X
k=n+1
k|λk − λk+2 | +
∞
X
∗ ∗ e (x) + |λn+1 − λn+3 |D
e (x)
|42 λk + 42 λk+1 |D
k
n
k=n
k=1
e ∗ (x) are defined by
where D
k
e ∗ (x) :=
D
k
k
X
i=0
cos x2 − cos k +
sin(i + 1)x =
2 sin x2
3
2
x
,
k ∈ N,
and 4 λk = λk − 2λk+1 + λk+2 .
e ∗ (x)| = O 1 and {λn } ∈ M RBSV S we have
Taking into account that |D
k
x
that
n
∞
X
1X
2| sin xf (x)| k|λk − λk+2 | +
k|λk − λk+2 | + n|λn+1 − λn+3 |
n k=1
k=n
2
n
n−1
1X
1X
k|λk − λk+2 | +
k|λk − λk+2 | + n|λn+1 − λn+3 |
n k=1
n n
k= 2
1X
k|λk − λk+2 |,
n k=1
n
REST BOUNDED SECOND VARIATION SEQUENCES
255
where we have used the fact that from {λn } ∈ M RBSV S it follows
n−1
1X
n|λn+1 − λn+3 | k|λk − λk+2 | k|λk − λk+2 |.
n n
k=n+1
∞
X
k= 2
Hence, we get
Z π
γ(x)| sin xf (x)|p dx
0
∞ Z
X
n=1
∞
X
γn
p
γ(x)| sin xf (x)| dx np+2
π/(n+1)
n=1
π/n
n
X
!p
k|λk − λk+2 |
.
k=1
> 0 and an = n|λn − λn+2 | we obtain
!p
Z π
∞
∞
γ 1−p X
X
γ
n
ν
γ(x)| sin xf (x)|p dx (n|λn − λn+2 |)p p+2
.
p+2
n
ν
0
n=1
ν=n
Applying Lemma 2.1 with λn =
γn
np+2
Moreover, by the assumption on {γn }, we get
∞
∞
X
γν
γn X 1
γn
,
p+2
1+p−ε1
1+ε1
1+p
ν
n
ν
n
ν=n
ν=n
which along with above inequality we have
Z π
∞
X
p
γ(x)| sin xf (x)| dx γn np−2 |λn − λn+2 |p .
0
n=1
Theorem 3.2. Suppose that {λn } ∈ M RBSV S and 1 ≤ p < ∞. If the
sequence {γn } satisfies the condition: there exists an ε3 > 0 such that the
sequence {γn n−1+ε3 } is almost decreasing, then the condition
∞
X
(3.3)
γn np−2 |λn − λn+2 |p < ∞
n=1
is sufficient for the validity of the inclusion
γ(x)| sin xg(x)|p ∈ L(0, π).
(3.4)
Proof. Based on Lemma 2.2 and applying the summation by parts we obtain
∞
n
X
X
(λk − λk+2 ) cos(k + 1)x
2| sin xg(x)| ≤
|λk − λk+2 | + k=n+1
k=1
n
X
k=1
|λk − λk+2 | +
∞
X
k=n
|42 λk + 42 λk+1 |Dk∗ (x) + |λn+1 − λn+3 |Dn∗ (x)
256
XHEVAT Z. KRASNIQI
where Dk∗ (x) are defined by
sin k + 32 x − sin x2
:=
cos(i + 1)x =
,
x
2
sin
2
i=0
∗
Since |Dk (x)| = O x1 and {λn } ∈ M RBSV S then
k
X
Dk∗ (x)
2| sin xg(x)| n
X
|λk − λk+2 | + n
k=1
∞
X
k ∈ N.
|λk − λk+2 | + n|λn+1 − λn+3 |
k=n
n−1
1X
k|λk − λk+2 | + n|λn+1 − λn+3 |
|λk − λk+2 | +
n
n
k=1
n
X
k= 2
for x ∈
follows
π
,π
n+1 n
i
n
X
|λk − λk+2 |,
k=1
, where we have used the fact that from {λn } ∈ M RBSV S it
∞
X
n−1
n
X
1X
|λk − λk+2 | |λk − λk+2 |.
n|λn+1 − λn+3 | ≤ n
k|λk − λk+2 | n
n
k=n+1
k=1
k= 2
Therefore, applying Lemma 2.1 and based on conditions imposed on γn we
have
Z π
∞ Z π/n
X
p
γ(x)| sin xg(x)| dx γ(x)| sin xf (x)|p dx
0
n=1
∞
X
n=1
∞
X
k=1
∞
X
π/(n+1)
γn
n2
n
X
!p
|λk − λk+2 |
k=1
|λk − λk+2 |p
γ 1−p
k
k2
∞
X
γj
j=n
!p
j2
γk k p−2 |λk − λk+2 |p < +∞,
k=1
which implies γ(x)| sin xg(x)| ∈ L(0, π).
p
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Received April 13, 2013.
University of Prishtina,
Faculty of Education,
Department of Mathematics and Informatics,
Avenue Mother Theresa, 10000 Prishtina, Kosova
E-mail address: xhevat.krasniqi@uni-pr.edu