Journal of Inequalities in Pure and
Applied Mathematics
ON BELONGING OF TRIGONOMETRIC SERIES TO ORLICZ SPACE
S. TIKHONOV
Department of Mechanics and Mathematics
Moscow State University
Vorob’yovy gory, Moscow 119899 RUSSIA.
EMail: tikhonov@mccme.ru
volume 5, issue 2, article 22,
2004.
Received 18 July, 2003;
accepted 27 December, 2003.
Communicated by: A. Babenko
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
002-04
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Abstract
In this paper we consider trigonometric series with the coefficients from R0+ BV S
class. We prove the theorems on belonging to these series to Orlicz space.
2000 Mathematics Subject Classification: 42A32, 46E30.
Key words: Trigonometric series, Integrability, Orlicz space.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3
Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4
Proofs of Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
On Belonging Of Trigonometric
Series To Orlicz Space
S. Tikhonov
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J. Ineq. Pure and Appl. Math. 5(2) Art. 22, 2004
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1.
Introduction
We will study the problems of integrability of formal sine and cosine series
(1.1)
g(x) =
∞
X
λn sin nx,
n=1
(1.2)
f (x) =
∞
X
λn cos nx.
On Belonging Of Trigonometric
Series To Orlicz Space
n=1
S. Tikhonov
First, we will rewrite the classical result of Young, Boas and Heywood for series
(1.1) and (1.2) with monotone coefficients.
Theorem 1.1 ([1], [2], [11]). Let λn ↓ 0.
If 0 ≤ α < 2, then
g(x)
∈ L(0, π) ⇐⇒
xα
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∞
X
nα−1 λn < ∞.
n=1
If 0 < α < 1, then
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∞
X
f (x)
∈
L(0,
π)
⇐⇒
nα−1 λn < ∞.
xα
n=1
Several generalizations of this theorem have been obtained in the following
directions: more general weighted functions γ(x) have been considered; also,
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integrability of g(x)γ(x) and f (x)γ(x) of order p have been examined for different values of p; finally, more general conditions on coefficients {λn } have
been considered.
Igari ([3]) obtained the generalization of Boas-Heywood’s results. The author used the notation of a slowly oscillating function.
A positive measurable function S(t) defined on [D; +∞), D > 0 is said to
be slowly oscillating if lim S(κt)
= 1 holds for all x > 0.
S(t)
t→∞
Theorem 1.2 ([3]). Let λn ↓ 0, p ≥ 1, and let S(t) be a slowly oscillating
function.
If −1 < θ < 1, then
∞
X
g p (x)S x1
∈ L(0, π) ⇐⇒
npθ+p−1 S(n)λpn < ∞.
pθ+1
x
n=1
S. Tikhonov
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If −1 < θ < 0, then
f p (x)S
xpθ+1
On Belonging Of Trigonometric
Series To Orlicz Space
1
x
∈ L(0, π) ⇐⇒
∞
X
npθ+p−1 S(n)λpn < ∞.
Theorem 1.3 ([10]). Let λn ↓ 0, and p > 0. Then
n=1
II
I
n=1
Vukolova and Dyachenko in [10], considering the Hardy-Littlewood type
theorem found the sufficient conditions of belonging of series (1.1) and (1.2) to
the classes Lp for p > 0.
∞
X
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np−2 λpn < ∞ =⇒ ψ(x) ∈ Lp (0, π),
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where a function ψ(x) is either a f (x) or a g(x).
In the same work it is shown that the converse result does not hold for cosine
series.
Leindler ([5]) 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
|cn − cn+1 | ≤ K(c) cm
On Belonging Of Trigonometric
Series To Orlicz Space
n=m
for all natural numbers m, where K(c) is a constant depending only on c. In
[5] it was shown that the class R0+ BV S was not comparable to the class of
quasi-monotone sequences, that is, to the class of sequences c = {cn } such that
n−α cn ↓ 0 for some α ≥ 0. Also, in [5] it was proved that the series (1.1) and
(1.2) are uniformly convergent over δ ≤ x ≤ π − δ for any 0 < δ < π. In the
same paper the following was proved.
Theorem 1.4 ([5]). Let {λn } ∈ R0+ BV S, p ≥ 1, and
1
p
− 1 < θ < p1 . Then
S. Tikhonov
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p
ψ (x)
∈ L(0, π) ⇐⇒
xpθ
∞
X
npθ+p−2 λpn < ∞,
n=1
where a function ψ(x) is either a f (x) or a g(x).
Very recently Nemeth [8] has found the sufficient condition of integrability
of series (1.1) with the sequence of coefficients {λn } ∈ R0+ BV S and with quite
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general conditions on a weight function. The author has used the notation of
almost monotonic sequences.
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.
Theorem 1.5 ([8]). If {λn } ∈ R0+ BV S, and the sequence γ := {γn } such that
{γn n−2+ε } is almost decreasing for some ε > 0, then
∞
X
γn
n=1
n
On Belonging Of Trigonometric
Series To Orlicz Space
S. Tikhonov
Title Page
λn < ∞ =⇒ γ(x)g(x) ∈ L(0, π).
Here and in the sequel,
a function γ(x) is defined by the sequence γ in the
π
following way: γ n := γn , n ∈ N and there exist positive constants A and B
π
such that Aγn+1 ≤ γ(x) ≤ Bγn for x ∈ ( n+1
, πn ).
We will solve the problem of finding of sufficient conditions, for which series
(1.1) and (1.2) belong to the weighted Orlicz space L(Φ, γ). In particular, we
will obtain sufficient conditions for series (1.1) and (1.2) to belong to weighted
space Lpγ .
Definition 1.1. 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
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continuous function defined on [0, ∞) such that Φ(0) = 0 and lim Φ(t) = +∞.
t→∞
For a weight γ(x) the weighted Orlicz space L(Φ, γ) is defined by (see [9], [12])
(1.3)
Z
L(Φ, γ) = h :
π
γ(x)Φ(ε |h(x)|)dx < ∞
for some
ε>0 .
0
If Φ(x) = xp for 1 ≤ p < ∞, when the weighted Orlicz space L(Φ, γ)
defined by (1.3) is the usual weighted space Lpγ (0, π).
We will denote (see [6]) by 4(p, q) (0 ≤ q ≤ p) the set of all nonnegative
functions Φ(x) defined on [0, ∞) 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).
On Belonging Of Trigonometric
Series To Orlicz Space
S. Tikhonov
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2.
Results
The following theorems provide the sufficient conditions of belonging of f (x)
and g(x) to Orlicz spaces.
Theorem 2.1. Let Φ(x) ∈ 4(p, 0) (0 ≤ p). If {λn } ∈ R0+ BV S, and sequence
{γn } is such that {γn n−1+ε } is almost decreasing for some ε > 0, then
∞
X
γn
n=1
n2
Φ(nλn ) < ∞ =⇒ ψ(x) ∈ L(Φ, γ),
On Belonging Of Trigonometric
Series To Orlicz Space
where a function ψ(x) is either a sine or cosine series.
S. Tikhonov
For the sine series it is possible to obtain the sufficient condition of its belonging to Orlicz space with more general conditions on the sequence {γn } but
with stronger restrictions on the function Φ(x).
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R0+ BV
Theorem 2.2. Let Φ(x) ∈ 4(p, q) (0 ≤ q ≤ p). If {λn } ∈
S, and
−(1+q)+ε
sequence {γn } is such that {γn n
} is almost decreasing for some ε > 0,
then
∞
X
γn
Φ(n2 λn ) < ∞ =⇒ g(x) ∈ L(Φ, γ).
2+q
n
n=1
Remark 2.1. If Φ(t) = t, then Theorem 2.2 implies Theorem 1.5, and if Φ(t) =
tp with 0 < p and {γn = 1, n ∈ N}, then Theorem 2.1 is a generalization of
Theorem 1.3. Also, if Φ(t) = tp with 1 ≤ p and {γn = nα S(n), n ∈ N} with
corresponding conditions on α and S(n), then Theorems 2.1 and 2.2 imply the
sufficiency parts (⇐=) of Theorems 1.2 and 1.4.
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3.
Auxiliary Results
Lemma 3.1 ([4]). If an ≥ 0, λn > 0, and if p ≥ 1, then
!p
!p
∞
n
∞
∞
X
X
X
X
p
λn
aν
≤C
λ1−p
λν .
n an
n=1
ν=1
n=1
ν=n
Lemma 3.2 ([6]). 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∗
n
n
P
P
1
(2) Φ
tj ≤
Φ p∗ (tj )
, p∗ = max(1, p).
j=1
On Belonging Of Trigonometric
Series To Orlicz Space
S. Tikhonov
j=1
Lemma 3.3. Let Φ ∈ 4(p, q) (0 ≤ q ≤ p). If λn > 0, an ≥ 0, and if there
exists a constant K such that aν+j ≤ Kaν holds for all j, ν ∈ N, j ≤ ν, then
!
P∞
p∗
∞
k
∞
X
X
X
ν=k λν
λk Φ
aν ≤ C
Φ (kak ) λk
,
kλ
k
ν=1
k=1
k=1
where p∗ = max(1, p).
ν=1
m+1
aν ≤
ξ−1 2
X
X−1
m=0 ν=2m
aν +
k
X
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aν
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ν=2ξ
ξ−1
≤ C1
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Proof. Let ξ be an integer such that 2ξ ≤ k < 2ξ+1 . Then
k
X
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ξ
!
2m a2m + 2ξ a2ξ
≤ C1
X
m=0
2m a2m .
J. Ineq. Pure and Appl. Math. 5(2) Art. 22, 2004
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Lemma 3.2 implies
Φ
k
X
!
aν
≤ Φ C1
ν=1
ξ
X
!
2m a2m
m=0
ξ
≤
C1p Φ
X
!
2m a2m
m=0
≤C
ξ
X
!p∗
Φ
1
p∗
On Belonging Of Trigonometric
Series To Orlicz Space
(2m a2m )
m=0
≤C
1
k
X
Φ p∗ (mam )
m
m=1
!p∗
S. Tikhonov
.
Contents
By Lemma 3.1, we have
∞
X
k=1
λk Φ
k
X
ν=1
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!
aν
!p∗
1
k
X
Φ p∗ (mam )
≤C
λk
m
m=1
k=1
P∞
p∗
∞
X
ν=k λν
≤C
Φ (kak ) λk
.
kλ
k
k=1
∞
X
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Note that this Lemma was proved in [7] for the case 0 < p ≤ 1.
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4.
Proofs of Theorems
π
Proof of Theorem 2.1. Let x ∈ ( n+1
, πn ]. Applying Abel’s transformation we
obtain
n
∞
∞
n
X
X
X
X
|f (x)| ≤
λk + λk cos kx ≤
λk +
|(λk − λk+1 ) Dk (x)| ,
k=1
k=n+1
k=1
k=n
where Dk (x) are the Dirichlet kernels, i.e.
k
1 X
cos nx, k ∈ N.
Dk (x) = +
2 n=1
Since |Dk (x)| = O x1 and λn ∈ R0+ BV S, we see that
!
!
n
n
∞
X
X
X
|f (x)| ≤ C
λk + n
|λk − λk+1 | ≤ C
λk + nλn .
k=n
k=1
k=1
The following estimates for series (1.2) can be obtained in the same way:
∞
n
X
X
|g(x)| ≤
λk + λk sin kx
k=1
≤
n
X
k=n+1
∞ X
e k (x)
λk +
(λk − λk+1 ) D
k=1
≤C
k=1
λk + n
∞
X
k=n
!
|λk − λk+1 |
≤C
S. Tikhonov
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k=n
n
X
On Belonging Of Trigonometric
Series To Orlicz Space
n
X
k=1
!
λk + nλn ,
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e k (x) are the conjugate Dirichlet kernels, i.e. D
e k (x) :=
where D
N.
Therefore,
!
n
X
|ψ(x)| ≤ C
λk + nλn ,
Pk
n=1
sin nx, k ∈
k=1
where a function ψ(x) is either a f (x) or a g(x).
One can see that if {λn } ∈ R0+ BV S, then {λn } is almost decreasing sequence, i.e. there exists a constant K ≥ 1 such that λn ≤ Kλk holds for any
k ≤ n. Then
!
n
n
n
X
X
X
(4.1)
|ψ(x)| ≤ C
λk + λ n
1 ≤C
λk .
k=1
k=1
n=1
≤ C1p πB
n=1
∞
X
γn
Φ
n2
γk
≤C
Φ (kλk ) 2
k
k=1
n
X
!
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k=1
∞
k X γν
γk ν=k ν 2
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π/(n+1)
k=1
∞
X
S. Tikhonov
Title Page
k=1
We will use (4.1) and the fact that {λk } is almost decreasing sequence; also, we
will use Lemmas 3.2 and 3.3:
!Z
Z π
∞
n
π/n
X
X
γ(x)Φ (|ψ(x)|) dx ≤
Φ C1
λk
γ(x)dx
0
On Belonging Of Trigonometric
Series To Orlicz Space
!p∗
Page 12 of 17
,
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where p∗ = max(1, p). Since there exists a constant ε > 0 such that {γn n−1+ε }
is almost decreasing, then
∞
X
γν
ν=k
Then
Z
ν2
≤C
∞
γk X −ε−1
γk
ν
≤
C
.
k 1−ε ν=k
k
π
γ(x)Φ (|ψ(x)|) dx ≤ C
0
∞
X
γk
k=1
k2
Φ (kλk ) .
On Belonging Of Trigonometric
Series To Orlicz Space
The proof of Theorem 2.1 is complete.
S. Tikhonov
Proof of Theorem 2.2. While proving Theorem 2.2 we will follow the idea of
the proof of Theorem 2.1.
π
Let x ∈ ( n+1
, πn ]. Then
∞
n
X
X
kxλk + λk sin kx
(4.2)
|g(x)| ≤
k=1
≤
n
X
k=n+1
∞ X
e k (x)
kxλk +
(λk − λk+1 ) D
k=1
≤C
≤C
1
n
n
X
k=1
kλk +
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n
1X
kλk + nλn
n k=1
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n
X
1
λn
k
n k=1
!
≤ C1
1
n
n
X
k=1
kλk .
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Using Lemma 3.2, Lemma 3.3 and the estimate (4.2), we can write
!Z
Z π
∞
n
π/n
X
1X
kλk
γ(x)Φ (|g(x)|) dx ≤
Φ C1
γ(x)dx
n
0
π/(n+1)
n=1
k=1
!
∞
n
X
X
γ
n
≤ C1p πB
Φ
kλk
2+q
n
n=1
k=1
!p∗
∞
∞
1+q X
X
k
γ
γ
k
ν
Φ k 2 λk 2+q
,
≤ C2
k
γk ν=k ν 2+q
k=1
S. Tikhonov
where p∗ = max(1, p).
By the assumption on {γn },
Z
π
γ(x)Φ (|g(x)|) dx ≤ C
0
and the proof of Theorem 2.2 is complete.
On Belonging Of Trigonometric
Series To Orlicz Space
Title Page
∞
X
k=1
γk
Φ k 2 λk ,
2+q
k
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References
[1] R.P. BOAS JR, Integrability of trigonometric series. III, Quart. J. Math.
Oxford Ser., 3(2) (1952), 217–221.
[2] P. HEYWOOD, On the integrability of functions defined by trigonometric
series, Quart. J. Math. Oxford Ser., 5(2) (1954), 71–76.
[3] S. IGARI, Some integrability theorems of trigonometric series and monotone decreasing functions, Tohoku Math. J., 12(2) (1960), 139–146.
[4] L. LEINDLER, Generalization of inequalities of Hardy and Littlewood,
Acta Sci. Math., 31 (1970), 279–285.
[5] L. LEINDLER, A new class of numerical sequences and its applications
to sine and cosine series, Anal. Math., 28 (2002), 279–286.
[6] M. MATELJEVIC AND M. PAVLOVIC, Lp -behavior of power series with
positive coefficients and Hardy spaces, Proc. Amer. Math. Soc., 87 (1983),
309-316.
[7] J. NEMETH, Note on the converses of inequalities of Hardy and Littlewood, Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 17 (2001), 101–105.
[8] J. NEMETH, Power-Monotone Sequences and integrability of trigonometric series, J. Inequal. Pure and Appl. Math., 4(1) (2003). [ONLINE
http://jipam.vu.edu.au]
[9] M.M. RAO AND Z.D. REN, Theory of Orlicz spaces, M. Dekker, Inc.,
New York, 1991.
On Belonging Of Trigonometric
Series To Orlicz Space
S. Tikhonov
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[10] T.M. VUKOLOVA AND M.I. DYACHENKO, On the properties of sums
of trigonometric series with monotone coefficients, Moscow Univ. Math.
Bull., 50 (3), (1995), 19–27; translation from Vestnik Moskov. Univ. Ser. I.
Mat. Mekh., 3 (1995), 22–32.
[11] W. H. YOUNG, On the Fourier series of bounded variation, Proc. London
Math. Soc., 12(2) (1913), 41–70.
[12] A. ZYGMUND, Trigonometric series, Volumes I and II, Cambridge University Press, 1988.
On Belonging Of Trigonometric
Series To Orlicz Space
S. Tikhonov
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