ON INTEGRABILITY OF FUNCTIONS DEFINED BY
TRIGONOMETRIC SERIES
Integrability of Functions
L. LEINDLER
Bolyai Institute, University of Szeged
Aradi vértanúk tere 1,
H-6720 Szeged, Hungary
EMail: leindler@math.u-szeged.hu
L. Leindler
vol. 9, iss. 3, art. 69, 2008
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Contents
Received:
15 January, 2008
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Accepted:
19 August, 2008
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Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
26D15, 26A42, 40A05, 42A32.
Key words:
Sine and cosine series, Lp integrability, equivalence of coefficient conditions,
quasi power-monotone sequences.
Abstract:
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 }.
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Contents
1
Introduction
3
2
New Results
5
3
Notions and Notations
7
Integrability of Functions
4
Lemmas
9
L. Leindler
vol. 9, iss. 3, art. 69, 2008
5
Proof of the Theorems
13
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1.
Introduction
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
Integrability of Functions
λn cos nx.
L. Leindler
vol. 9, iss. 3, art. 69, 2008
n=1
As a nice example, we recall Chen’s ([4]) theorem: If λn ↓ 0, thenPx−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.
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;
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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.
Integrability of Functions
L. Leindler
vol. 9, iss. 3, art. 69, 2008
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2.
New Results
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
Integrability of Functions
γ(x)g(x) ∈ Lp (0, π),
(2.1)
L. Leindler
vol. 9, iss. 3, art. 69, 2008
then
(2.2)
∞
X
γn n
n=1
p−2
∞
X
!p
−1
k 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
then (2.1) holds.
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k=n
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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),
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).
Integrability of Functions
L. Leindler
vol. 9, iss. 3, art. 69, 2008
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3.
Notions and Notations
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 )
Integrability of Functions
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
k=n
where [α] denotes the integral part of α.
−1
[λn]
X
k=[λ−1 n]
ck ,
n ∈ N,
L. Leindler
vol. 9, iss. 3, art. 69, 2008
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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 := γn1/p , n ∈ N; and K1 (γ)γn 5 γ(x) 5 K2 (γ)γn
γ
n
π
holds for all x ∈ n+1
, πn .
Integrability of Functions
L. Leindler
vol. 9, iss. 3, art. 69, 2008
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4.
Lemmas
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
p
1−p
5p
βn
βk αnp ,
(4.1)
βn
αk
n=1
n=1
k=1
Integrability of Functions
L. Leindler
k=n
vol. 9, iss. 3, art. 69, 2008
and
(4.2)
∞
X
∞
X
βn
n=1
!p
5 pp
αk
∞
X
βn1−p
n=1
k=n
n
X
!p
βk
αnp .
(4.3)
1
:=
∞
X
βn
n=1
∞
X
!p
bk
<∞
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k=n
implies
(4.4)
Contents
k=1
Lemma 4.2. If bn = 0, p = 1, s > 0, then
X
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X
2
:=
∞
X
n=1
βn n−sp
n
X
!p
k s bk
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<∞
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.
Close
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 )
.
Integrability of Functions
k=1
L. Leindler
By partial summation we obtain
X
2
sp
∞
X
n
X
βn n−sp
n=1
vol. 9, iss. 3, art. 69, 2008
!p
k s−1 Tk
=:
k=1
X
3
.
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Since nδ βn ↓ with δ > 1 − sp, Lemma 4.1 with (4.1) shows that
X
3
∞
X
n=1
∞
X
(ns−1 Tn )p (βn n−sp )1−p
k=n
βn Tnp =
X
1
n=1
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!p
βk k −sp
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this proves that (4.3) ⇒ (4.4).
Now suppose that (4.4) holds. First we show that
(4.5)
∞
X
∞
X
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bn < ∞.
n=1
Denote
Hn :=
n
X
k=1
k s bk .
Then
(4.6)
N
X
bk =
k=n
N
X
N
−1
X
−s
k (Hk − Hk−1 ) 5 s
k=n
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
−s−1
1
− p1
p
Hk βk
N
−1
X
5
k=n
! p1
Hkp βk k −sp
N
−1
X
k=n
! p−1
p
−1/p
(k −1 βk )p/(p−1)
Integrability of Functions
.
k=n
Since, nδ βn ↑ with δ < 1, thus
∞
X
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(k −p βk−1 k −δ+δ )1/(p−1) ∞
X
δ−p
k p−1 < ∞.
k=1
k=1
This, (4.4) and (4.7) imply that
(4.8)
∞
X
k −s−1 Hk < ∞,
k=1
∞
X
k=n
bk Contents
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thus HN N −s tends to zero, herewith, by (4.6), (4.5) is verified, furthermore,
(4.9)
L. Leindler
vol. 9, iss. 3, art. 69, 2008
∞
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).
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Thus we can apply (4.9) and Lemma 4.1 with (4.2) for any p = 1, whence, by
n βn ↑ with δ < 1, we obtain that
!p
!p
∞
∞
∞
∞
X
X
X
X
βn
k −s−1 Hk
bk
βn
δ
n=1
n=1
k=n
∞
X
k=n
(n−s−1 Hn )p βn1−p
n=1
∞
X
n
X
!p
βk
Integrability of Functions
L. Leindler
k=1
βn n−sp Hnp ;
vol. 9, iss. 3, art. 69, 2008
n=1
herewith (4.4) ⇒ (4.3) is also proved.
The proof of Lemma 4.2 is complete.
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5.
Proof of the Theorems
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)
Z π
p1 Z π
10
Z π
p
p
−p0
|g(x)|dx 5
|g(x)γ(x)| dx
.
γ(x) dx
0
Denote xn :=
0
0
π
,
n
Integrability of Functions
β
n ∈ N. Since γn n ↑ (β < p − 1)
Z π
Z xn
∞
X
−p0
1/(1−p)
γ(x) dx γn
dx
0
=
xn+1
n=1
∞
X
L. Leindler
vol. 9, iss. 3, art. 69, 2008
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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
γ
n
1
0
x
0
n+1
n=1
Integrating g(x), we obtain
Z x
∞
∞
X
X
λn
nx
λn
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
n
.
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Denote
Z
xn
|g(x)|dx,
gn :=
n ∈ N.
xn+1
Then
∞
X
ν+1
−1
k λk =
k=n
∞ 2Xn
X
k −1 λk
ν=0 k=2ν n
∞
X
ν+1
G(2
Integrability of Functions
L. Leindler
n)
vol. 9, iss. 3, art. 69, 2008
gk
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ν=0
∞
∞
X
X
ν=0 k=2ν+1 n
∞
X
ν=0
(5.1)
2ν+1 n i=2ν n
ν+1 n
∞ 2X
X
1
i
ν=0 i=2ν n
∞
∞
X
1X
i=n
i
Contents
ν+1 n
2X
1
∞
X
∞
X
gk
k=2ν+1 n
gk
1
:=
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k=i
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gk .
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k=i
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Now we have
X
JJ
∞
X
n=1
np−2 γn
∞
X
k=n
!p
k −1 λk
∞
X
n=1
np−2 γn
∞
X
k=n
k −1
∞
X
i=k
!p
gi
.
Applying Lemma 4.1 with (4.2) we obtain that
!p
!p
∞
n
∞
X
X
X
X
k p−2 γk .
n−1
gi (np−2 γn )1−p
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
Integrability of Functions
L. Leindler
k=1
vol. 9, iss. 3, art. 69, 2008
and thus
(np−2 γn )1−p
n
X
!p
k p−2 γk
γn n2p−2 ,
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k=1
whence we get
X
1
∞
X
γn n2p−2
n−1
∞
X
n=1
Contents
!p
gi
.
i=n
Using again Lemma 4.1 with (4.2) we have
X
1
∞
X
n−p gnp (n2p−2 γn )1−p
n=1
n
X
!p
k
2p−2
γk
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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
(n2p−2 γn )1−p
n
X
k=1
!p
k 2p−2 γk
γn n3p−2 ,
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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
γn n2p−2
=
Integrability of Functions
|g(x)|dx
Z
L. Leindler
xn
|g(x)|p dx
Z
xn+1
∞ Z
X
Zn=1
π
p
xn
xn+1
n=1
Z
p−1
xn
dx
xn+1
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xn
p
|γ(x)g(x)| dx
Contents
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 n (x) := D
sin nx x−1 ,
n=1
we obtain
(5.4)
|g(x)| x
n
X
k=1
vol. 9, iss. 3, art. 69, 2008
∞
n
∞
X
X
X
kλk + λk sin kx x
kλk + n
|∆λk |.
k=n+1
k=1
k=n
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Denote
∆n :=
∞
X
|∆λk |.
k=n
It is easy to see that
n∆n n
−1
n
X
k∆k
k=1
Integrability of Functions
and, by λn → 0,
L. Leindler
λn 5 ∆n .
vol. 9, iss. 3, art. 69, 2008
Thus, by (5.4), we have
|g(x)| n
−1
n
X
Title Page
k∆k .
Contents
k=1
Hence
Z π
|γ(x)g(x)|p dx =
0
∞ Z xn
X
n=1
|γ(x)g(x)|p dx xn+1
∞
X
γn n−2−p
n
X
n=1
!p
k∆k
k=1
π
p
|γ(x)g(x)| dx 0
∞
X
p
−2−p 1−p
)
∞
X
!p
γk k −2−p
k=n
Since γn nβ ↓ with β > −1 − p, we have
∞
X
k=n
β −2−p−β
γk k k
γn n
β
∞
X
k=n
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n=1
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Page 17 of 22
Applying Lemma 4.1 with (4.1), we obtain
Z
.
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k −2−p−β γn n−1−p ,
.
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and thus
(γn n−2−p )1−p
∞
X
!p
γk k −2−p
γn n−2 .
k=n
Collecting these estimations we obtain
Z π
∞
∞
X
X
p
p−2 p
|γ(x)g(x)| dx γn n ∆n =
γn np−2
0
n=1
∞
X
n=1
!p
|∆λk |
,
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
Integrability of Functions
k=n
L. Leindler
vol. 9, iss. 3, art. 69, 2008
Title Page
and bk = k −1 bk .
Contents
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.
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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
sin nx,
F (x) :=
f (t)dt =
n
0
n=1
Close
and integrating F (x) we get
x
Z
F1 (x) :=
F (t)dt = 2
0
Thus we obtain
∞
X
λn
n=1
n2
sin2
nx
.
2
2k
π
X
λn
F1
.
2
2k
n
n=k
Denote
Integrability of Functions
L. Leindler
vol. 9, iss. 3, art. 69, 2008
Z
xn
|f (x)|dx,
fn :=
n ∈ N,
xn =
xn+1
π
.
n
Title Page
Then
Z
F1 (x2n ) =
thus
Contents
x2n
0
∞
X
F (t)dt
Z xk Z
k=2n
∞
X
xk+1
1
k2
k=2n
∞
∞
X
1 X
|f (t)|dt =
f` ,
2
k
x`+1
k=2n
`=k
x`
∞
∞
X
1 X
n
f` .
2
k
k
k=2n
`=k
2n
X
λk
k=n
0
∞ Z
X
`=k
xk
|f (t)|dt du
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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
ν=0
k=2ν+1 n
∞
X
i+2 n
∞
2X
X
2ν n
∞
X
k −2
f`
`=k
k −2
Integrability of Functions
∞
X
L. Leindler
f`
ν=0
∞
X
i=ν k=2i+1 n
`=2i+1 n
∞
∞
X
X
f`
2ν n
(2in )−1
i+1
ν=0
i=ν
`=2
n
!
∞
∞
i
X
X
X
(2i n)−1
f`
2ν n
ν=0
i=0
`=2i+1 n
∞
∞
X
X
f` .
i=0
vol. 9, iss. 3, art. 69, 2008
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Page 20 of 22
`=2i+1 n
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Hereafter, as in (5.1), we get that
∞
X
k=n
k −1 λk ∞
∞
X
1X
i=n
i
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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.
Close
The proof of the statement (2.4) ⇒ (2.5) is easier. Namely
∞
n
n
∞
X
X
X
1X
λk + λk cos kx λk +
|∆λk |.
|f (x)| 5
x k=n
k=1
k=n+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
k=1
Integrability of Functions
L. Leindler
k=n
vol. 9, iss. 3, art. 69, 2008
and thus, by λn → 0,
Z π
(5.6)
|γ(x)f (x)|p dx
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0
∞
X
γn n−2
n=1
n X
∞
X
!p
|∆λm |
+
∞
X
γn np−2
n=1
k=1 m=k
∞
X
!p
∆λk
.
k=n
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β
To estimate the first sum, we again use Lemma 4.1 with (4.1), thus, by γn n ↓ with
some β > −1,
!p
!p
∞
∞
n
∞
X
X
X
X
γn n−2
∆k
∆pn (γn n−2 )1−p
γk k −2
n=1
k=1
n=1
∞
X
n=1
Page 21 of 22
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k=n
γn n
p−2
∆pn
≡
∞
X
n=1
γn n
p−2
∞
X
!p
|∆λk |
Close
.
k=n
This and (5.6) imply the second assertion of Theorem 2.2, that is, (2.4) ⇒ (2.5).
We have completed our proof.
References
[1] R. ASKEY AND S. WAINGER, Integrability theorems for Fourier series. Duke
Math. J., 33 (1966), 223–228.
[2] R.P. BOAS, Jr., Integrability Theorems for Trigonometric Transforms, Springer,
Berlin, 1967.
[3] R.P. BOAS, Jr., Fourier series with positive coefficients, J. of Math. Anal. and
Applics., 17 (1967), 463–483.
[4] Y.-M. CHEN, On the integrability of functions defined by trigonometrical series,
Math. Z., 66 (1956), 9–12.
Integrability of Functions
L. Leindler
vol. 9, iss. 3, art. 69, 2008
Title Page
[5] L. LEINDLER, Generalization of inequalities of Hardy and Littlewood, Acta
Sci. Math., 31 (1970), 279–285.
[6] J. NÉMETH, Power-monotone-sequences and integrability of trigonometric series, J. Inequal. Pure and Appl. Math., 4(1) (2003), Art. 3. [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=239].
[7] S. TIKHONOV, On belonging of trigonometric series of Orlicz space, J. Inequal.
Pure and Appl. Math., 5(2) (2004), Art. 22. [ONLINE: http://jipam.vu.
edu.au/article.php?sid=395].
[8] S. TIKHONOV, On the integrability of trigonometric series, Math. Notes, 78(3)
(2005), 437–442.
[9] D.S. YU, P. ZHOU AND S.P. ZHOU, On Lp integrability and convergence of
trigonometric series, Studia Math., 182(3) (2007), 215–226.
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