Volume 9 (2008), Issue 3, Article 69, 10 pp.
ON INTEGRABILITY OF FUNCTIONS DEFINED BY TRIGONOMETRIC SERIES
L. LEINDLER
B OLYAI I NSTITUTE , U NIVERSITY OF S ZEGED
A RADI VÉRTANÚK TERE 1,
H-6720 S ZEGED , H UNGARY
leindler@math.u-szeged.hu
Received 15 January, 2008; accepted 19 August, 2008
Communicated by S.S. Dragomir
A BSTRACT. The goal of the present paper is to generalize two theorems of R.P. Boas Jr. pertaining to Lp (p > 1) integrability of Fourier series with nonnegative coefficients and weight
xγ . In our improvement the weight xγ is replaced by a more general one, and the case p = 1
is also yielded. We also generalize an equivalence statement of Boas utilizing power-monotone
sequences instead of {nγ }.
Key words and phrases: Sine and cosine series, Lp integrability, equivalence of coefficient conditions, quasi power-monotone
sequences.
2000 Mathematics Subject Classification. 26D15, 26A42, 40A05, 42A32.
1. I NTRODUCTION
There are many classical and newer theorems pertaining to the integrability of formal sine
and cosine series
∞
X
g(x) :=
λn sin nx,
n=1
and
f (x) :=
∞
X
λn cos nx.
n=1
As a nice example, we recall Chen’s ([4]) theorem: If P
λn ↓ 0, then x−γ ϕ(x) ∈ Lp (ϕ means
either f or g), p > 1, 1/p − 1 < γ < 1/p, if and only if npγ+p−2 λpn < ∞.
For notions and notations, please, consult the third section.
We do not recall more theorems because a nice short survey of recent results with references
can be found in a recent paper of S. Tikhonov [7], and classical results can be found in the
outstanding monograph of R.P. Boas, Jr. [2].
The generalizations of the classical theorems have been obtained in two main directions:
to weaken the classical monotonicity condition on the coefficients λn ; to replace the classical
power weight xγ by a more general one in the integrals. Lately, some authors have used both
generalizations simultaneously.
017-08
2
L. L EINDLER
J. Németh [6] studied the class of RBV S sequences and weight functions more general than
the power one in the L(0, π) space.
S. Tikhonov [8] also proved two general theorems of this type, but in the Lp -space for p = 1;
he also used general weights.
Recently D.S. Yu, P. Zhou and S.P. Zhou [9] answered an old problem of Boas ([2], Question
6.12.) in connection with Lp integrability considering weight xγ , but only under the condition
that the sequence {λn } belongs to the class M V BV S; their result is the best one among the answers given earlier for special classes of sequences. The original problem concerns nonnegative
coefficients.
In the present paper we refer back to an old paper of Boas [3], which was one of the first to
study the Lp -integrability with nonnegative coefficients and weight xγ .
We also intend to prove theorems with nonnegative coefficients, but with more general weights
than xγ .
It can be said that our theorems are the generalizations of Theorems 8 and 9 presented in
Boas’ paper mentioned above. Boas names these theorems as slight improvements of results of
Askey and Wainger [1]. Our theorems jointly generalize these by using more general weights
than xγ , and broaden those to the case p = 1, as well.
Comparing our results with those of Tikhonov, as our generalization concerns the coefficients,
we omit the condition {λn } ∈ RBV S and prove the equivalence of (2.2) and (2.3).
In proving our theorems we need to generalize an equivalence statement of Boas [3]. At this
step we utilize the quasi β-power-monotone sequences.
2. N EW R ESULTS
We shall prove the following theorems.
Theorem 2.1. Let 1 5 p < ∞ and λ := {λn } be a nonnegative null-sequence.
If the sequence γ := {γn } is quasi β-power-monotone increasing with a certain β < p − 1,
and
γ(x)g(x) ∈ Lp (0, π),
(2.1)
then
(2.2)
∞
X
γn np−2
n=1
∞
X
!p
k −1 λk
< ∞.
k=n
If γ is also quasi β-power-monotone decreasing with a certain β > −1, then condition (2.2) is
equivalent to
!p
∞
n
X
X
< ∞.
(2.3)
γn n−2
λk
n=1
k=1
If the sequence γ is quasi β-power-monotone decreasing with a certain β > −1 − p, and
!p
∞
∞
X
X
(2.4)
γn np−2
|∆λk | < ∞,
n=1
k=n
then (2.1) holds.
Theorem 2.2. Let p and λ be defined as in Theorem 2.1.
If the sequence γ is quasi β-power-monotone increasing with a certain β < p − 1, and
(2.5)
γ(x)f (x) ∈ Lp (0, π),
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 69, 10 pp.
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I NTEGRABILITY OF F UNCTIONS
3
then (2.2) holds.
If the sequence γ is quasi β-power-monotone decreasing with a certain β > −1, then (2.4)
implies (2.5).
3. N OTIONS AND N OTATIONS
We shall say that a sequence γ := {γn } of positive terms is quasi β-power-monotone increasing (decreasing) if there exist a natural number N := N (β, γ) and a constant K := K(β, γ) = 1
such that
Knβ γn = mβ γm
(3.1)
(nβ γn 5 Kmβ γm )
holds for any n = m = N.
If (3.1) holds with β = 0, then we omit the attribute "β-power" and use the symbols ↑ (↓).
We shall also use the notations L R at inequalities if there exists a positive constant K
such that L 5 KR.
A null-sequence c := {cn } (cn → 0) of positive numbers satisfying the inequalities
∞
X
|∆cn | 5 K(c)cm ,
(∆cn := cn − cn+1 ),
m ∈ N,
n=m
with a constant K(c) > 0 is said to be a sequence of rest bounded variation, in symbols,
c ∈ RBV S.
A nonnegative sequence c is said to be a mean value bounded variation sequence, in symbols,
c ∈ M V BV S, if there exist a constant K(c) > 0 and a λ = 2 such that
2n
X
|∆ck | 5 K(c)n
−1
[λn]
X
ck ,
n ∈ N,
k=[λ−1 n]
k=n
where [α] denotes the integral part of α.
In this paper a sequence γ := {γn } and a real number p = 1 are associated to a function
γ(x) (= γp (x)), being defined in the following way:
π γ
:= γn1/p , n ∈ N; and K1 (γ)γn 5 γ(x) 5 K2 (γ)γn
n
π
holds for all x ∈ n+1
, πn .
4. L EMMAS
To prove our theorems we recall one known result and generalize one of Boas’ lemmas ([2,
Lemma 6.18]).
Lemma 4.1 ([5]). Let p = 1, αn = 0 and βn > 0. Then
!p
!p
∞
n
∞
∞
X
X
X
X
5 pp
βn1−p
βk αnp ,
(4.1)
βn
αk
n=1
k=1
∞
X
∞
X
n=1
k=n
∞
X
n
X
and
(4.2)
n=1
βn
!p
αk
k=n
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 69, 10 pp.
5 pp
n=1
βn1−p
!p
βk
αnp .
k=1
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4
L. L EINDLER
Lemma 4.2. If bn = 0, p = 1, s > 0, then
X
(4.3)
1
:=
∞
X
∞
X
βn
n=1
!p
<∞
bk
k=n
implies
X
(4.4)
2
:=
∞
X
n
X
βn n−sp
n=1
!p
k s bk
<∞
k=1
if nδ βn ↓ with a certain δ > 1 − sp; and if nδ βn ↑ with a certain δ < 1, then (4.4) implies (4.3).
Thus, if both monotonicity conditions for {βn } hold, then the conditions (4.3) and (4.4) are
equivalent.
Proof of Lemma 4.2. First, suppose (4.3) holds. Write
∞
X
Tn :=
bk ;
k=n
then
X
2
=
∞
X
n
X
βn n−sp
n=1
!p
k s (Tk − Tk+1 )
.
k=1
By partial summation we obtain
X
2
∞
X
sp
n
X
βn n−sp
n=1
!p
k s−1 Tk
=:
X
3
k=1
.
Since nδ βn ↓ with δ > 1 − sp, Lemma 4.1 with (4.1) shows that
X
3
∞
X
(ns−1 Tn )p (βn n−sp )1−p
n=1
∞
X
∞
X
!p
βk k −sp
k=n
βn Tnp =
X
n=1
1
,
this proves that (4.3) ⇒ (4.4).
Now suppose that (4.4) holds. First we show that
∞
X
(4.5)
bn < ∞.
n=1
Denote
Hn :=
n
X
k s bk .
k=1
Then
N
X
(4.6)
k=n
bk =
N
X
−s
k (Hk − Hk−1 ) 5 s
k=n
N
−1
X
k −s−1 Hk + HN N −s .
k=n
If p > 1 then by Hölder’s inequality, we obtain
(4.7)
N
−1
X
k=n
1
− p1
p
k −s−1 Hk βk
5
N
−1
X
! p1
Hkp βk k −sp
k=n
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 69, 10 pp.
N
−1
X
! p−1
p
−1/p p/(p−1)
(k −1 βk
)
.
k=n
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I NTEGRABILITY OF F UNCTIONS
5
Since, nδ βn ↑ with δ < 1, thus
∞
X
(k −p βk−1 k −δ+δ )1/(p−1)
k=1
∞
X
δ−p
k p−1 < ∞.
k=1
This, (4.4) and (4.7) imply that
∞
X
(4.8)
k −s−1 Hk < ∞,
k=1
thus HN N
−s
tends to zero, herewith, by (4.6), (4.5) is verified, furthermore,
∞
X
(4.9)
bk k=n
∞
X
k −s−1 Hk .
k=n
If p = 1, then without Hölder’s inequality, the assumption nδ βn ↑ with a certain δ < 1 and
(4.4) clearly imply (4.8).
Thus we can apply (4.9) and Lemma 4.1 with (4.2) for any p = 1, whence, by nδ βn ↑ with
δ < 1, we obtain that
!p
!p
∞
∞
∞
∞
X
X
X
X
βn
bk
βn
k −s−1 Hk
n=1
n=1
k=n
∞
X
k=n
(n−s−1 Hn )p βn1−p
n=1
∞
X
n
X
!p
βk
k=1
βn n−sp Hnp ;
n=1
herewith (4.4) ⇒ (4.3) is also proved.
The proof of Lemma 4.2 is complete.
5. P ROOF OF THE T HEOREMS
Proof of Theorem 2.1. First we prove that (2.1) implies g(x) ∈ L(0, π) and (2.2). If p > 1,
then, by Hölder’s inequality, we get with p0 := p/(p − 1)
10
Z π
p1 Z π
Z π
p
p
−p0
γ(x) dx
.
|g(x)|dx 5
|g(x)γ(x)| dx
0
Denote xn :=
π
,
n
0
0
β
n ∈ N. Since γn n ↑ (β < p − 1)
Z π
Z
∞
X
−p0
1/(1−p)
γ(x) dx γn
0
n=1
=
∞
X
xn
dx
xn+1
n−2 (γn nβ )1/(1−p) nβ/(p−1) 1,
n=1
that is, g(x) ∈ L.
If p = 1, then γn nβ ↑ with some β < 0, thus γn ↑, whence
Z π
Z xn
Z
∞
X
1
1 π
|g(x)|dx |g(x)|γ(x)dx |g(x)|γ(x)dx 1.
γ
γ1 0
0
n=1 n xn+1
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 69, 10 pp.
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6
L. L EINDLER
Integrating g(x), we obtain
Z x
∞
∞
X
X
λn
λn
nx
G(x) :=
g(t)dt =
(1 − cos nx) = 2
sin2
.
n
n
2
0
n=1
n=1
Hence
G(x2k ) 2k
X
λn
n=k
Denote
.
n
xn
Z
|g(x)|dx,
gn :=
n ∈ N.
xn+1
Then
∞
X
ν+1
−1
k λk =
∞ 2Xn
X
k=n
k −1 λk
ν=0 k=2ν n
∞
X
ν+1
G(2
n)
ν=0
∞
∞
X
X
gk
ν=0 k=2ν+1 n
∞
X
ν=0
ν+1 n
2X
1
2ν+1 n i=2ν n
∞
X
gk
k=2ν+1 n
ν+1
(5.1)
∞ 2Xn
∞
X
1X
ν=0 i=2ν n
∞
∞
X
X
i=n
Now we have
X
1
:=
∞
X
n
p−2
∞
X
γn
n=1
1
i
k λk
gk
k=i
gk .
k=i
!p
−1
i
∞
X
n
p−2
γn
∞
X
n=1
k=n
k
−1
k=n
∞
X
!p
gi
.
i=k
Applying Lemma 4.1 with (4.2) we obtain that
!p
!p
∞
∞
n
X
X
X
X
n−1
gi (np−2 γn )1−p
k p−2 γk .
1
n=1
i=n
k=1
β
Since γn n ↑ with β < p − 1, we have
n
n
X
X
β p−2−β
β
(5.2)
γk k k
γn n
k p−2−β γn np−1 ,
k=1
k=1
and thus
(np−2 γn )1−p
n
X
!p
k p−2 γk
γn n2p−2 ,
k=1
whence we get
X
1
∞
X
n=1
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 69, 10 pp.
γn n2p−2
n−1
∞
X
!p
gi
.
i=n
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I NTEGRABILITY OF F UNCTIONS
7
Using again Lemma 4.1 with (4.2) we have
X
1
∞
X
n
X
n−p gnp (n2p−2 γn )1−p
n=1
!p
k 2p−2 γk
.
k=1
A similar calculation and consideration as in (5.2) give that
n
X
k 2p−2 γk γn n2p−1 ,
k=1
and
n
X
(n2p−2 γn )1−p
!p
k 2p−2 γk
γn n3p−2 ,
k=1
thus
X
(5.3)
1
∞
X
γn n2p−2 gnp .
n=1
Since
∞
X
γn n2p−2 gnp
=
n=1
∞
X
γn n
n=1
∞
X
2p−2
Z
|g(x)|dx
xn+1
γn n
Z
2p−2
∞ Z
X
Zn=1
π
=
xn
p
Z
p−1
xn
|g(x)| dx
dx
xn+1
xn+1
n=1
p
xn
xn
|γ(x)g(x)|p dx
xn+1
|γ(x)g(x)|p dx.
0
This and (5.3) prove the implication (2.1) ⇒ (2.2).
Next we verify that (2.4) implies (2.1). Let x ∈ (xn+1 , xn ]. Then, using the Abel transformation and the well-known estimation
k
X
e
Dn (x) := sin nx x−1 ,
n=1
we obtain
(5.4)
|g(x)| x
n
X
k=1
∞
n
∞
X
X
X
kλk + λk sin kx x
kλk + n
|∆λk |.
k=n+1
k=1
Denote
∆n :=
∞
X
k=n
|∆λk |.
k=n
It is easy to see that
n∆n n
−1
n
X
k∆k
k=1
and, by λn → 0,
λn 5 ∆n .
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 69, 10 pp.
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8
L. L EINDLER
Thus, by (5.4), we have
|g(x)| n
−1
n
X
k∆k .
k=1
Hence
Z
π
|γ(x)g(x)|p dx =
0
∞ Z
X
xn
|γ(x)g(x)|p dx xn+1
n=1
∞
X
γn n−2−p
!p
n
X
n=1
k∆k
.
k=1
Applying Lemma 4.1 with (4.1), we obtain
!p
Z π
∞
∞
X
X
|γ(x)g(x)|p dx (n∆n )p (γn n−2−p )1−p
γk k −2−p .
0
n=1
k=n
β
Since γn n ↓ with β > −1 − p, we have
∞
∞
X
X
γk k β k −2−p−β γn nβ
k −2−p−β γn n−1−p ,
k=n
k=n
and thus
∞
X
(γn n−2−p )1−p
!p
γn n−2 .
γk k −2−p
k=n
Collecting these estimations we obtain
Z π
∞
∞
X
X
p−2 p
p
γn np−2
γn n ∆n =
|γ(x)g(x)| dx 0
∞
X
n=1
n=1
!p
|∆λk |
,
k=n
herewith the implication (2.4) ⇒ (2.1) is also proved.
In order to prove the equivalence of the conditions (2.2) and (2.3), we apply Lemma 4.2 with
s = 1,
βn = γn np−2
and bk = k −1 bk .
Then the assumptions nδ βn ↑ with δ < 1 and nδ βn ↓ with δ > 1 − p, determine the following
conditions pertaining to γn ;
(5.5)
nβ γn ↑
with
β < p − 1 and nβ γn ↓
with
β > −1.
The equivalence of (2.2) and (2.3) clearly holds if both monotonicity conditions required in
(5.5) hold.
This completes the proof of Theorem 2.1.
Proof of Theorem 2.2. As in the proof of Theorem 2.1, first we prove that (2.5) implies (2.2)
and f (x) ∈ L. The proof of f (x) ∈ L runs as that of g(x) ∈ L in Theorem 2.1.
Integrating f (x), we obtain
Z x
∞
X
λn
F (x) :=
f (t)dt =
sin nx,
n
0
n=1
and integrating F (x) we get
Z
F1 (x) :=
x
F (t)dt = 2
0
∞
X
λn
n=1
n2
sin2
nx
.
2
Thus we obtain
2k
π
X
λn
F1
.
2k
n2
n=k
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 69, 10 pp.
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I NTEGRABILITY OF F UNCTIONS
Denote
Z
xn
|f (x)|dx,
fn :=
9
n ∈ N,
xn+1
π
xn =
.
n
Then
Z
x2n
F1 (x2n ) =
F (t)dt
0
∞
X
xk
Z
xk+1
k=2n
∞
X
1
k2
k=2n
thus
xk
Z
|f (t)|dt du
0
∞ Z x`
X
∞
∞
X
1 X
|f (t)|dt =
f` ,
k 2 `=k
x`+1
k=2n
`=k
∞
∞
X
1 X
n
f` .
k
k 2 `=k
k=2n
2n
X
λk
k=n
Using the estimation obtained above we have
∞
X
ν+1
−1
k λk =
∞ 2Xn
X
k=n
k −1 λk
ν=0 k=2ν n
∞
X
ν
2 n
∞
X
k
ν=0
k=2ν+1 n
∞
X
i+2 n
∞
2X
X
ν
2 n
∞
X
−2
f`
`=k
k
−2
∞
X
f`
ν=0
∞
X
i=ν k=2i+1 n
`=2i+1 n
∞
∞
X
X
2ν n
f`
(2in )−1
i+1
ν=0
i=ν
`=2
n
∞
X
i=0
∞
X
(2i n)−1
∞
X
f`
!
2ν n
ν=0
`=2i+1 n
∞
X
i
X
f` .
i=0 `=2i+1 n
Hereafter, as in (5.1), we get that
∞
X
−1
k λk k=n
∞
∞
X
1X
i=n
i
f` ,
`=i
and following the method used in the proof of Theorem 2.1 with fn in place of gn , the implication (2.5) ⇒ (2.2) can be proved.
The proof of the statement (2.4) ⇒ (2.5) is easier. Namely
n
∞
n
∞
X
X
X
1X
λk + λk cos kx λk +
|f (x)| 5
|∆λk |.
x k=n
k=n+1
k=1
k=1
Using the notations of Theorem 2.1 and assuming x ∈ (xn+1 , xn ], we obtain
!p
!p
Z xn
n
∞
X
X
|γ(x)f (x)|p dx γn n−2
λk + γn n−2 n
|∆λk |
xn+1
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 69, 10 pp.
k=1
k=n
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L. L EINDLER
and thus, by λn → 0,
Z π
∞
X
p
(5.6)
|γ(x)f (x)| dx γn n−2
0
n=1
n X
∞
X
!p
|∆λm |
+
∞
X
γn np−2
n=1
k=1 m=k
∞
X
!p
∆λk
.
k=n
To estimate the first sum, we again use Lemma 4.1 with (4.1), thus, by γn nβ ↓ with some
β > −1,
!p
!p
∞
∞
n
∞
X
X
X
X
∆k
γk k −2
γn n−2
∆pn (γn n−2 )1−p
n=1
n=1
k=1
∞
X
γn np−2 ∆pn ≡
n=1
k=n
∞
X
∞
X
n=1
k=n
γn np−2
!p
|∆λk |
.
This and (5.6) imply the second assertion of Theorem 2.2, that is, (2.4) ⇒ (2.5).
We have completed our proof.
R EFERENCES
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