Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 2, Article 22, 2004
ON BELONGING OF TRIGONOMETRIC SERIES TO ORLICZ SPACE
S. TIKHONOV
D EPARTMENT OF M ECHANICS AND M ATHEMATICS
M OSCOW S TATE U NIVERSITY
VOROB ’ YOVY GORY, M OSCOW 119899 RUSSIA.
tikhonov@mccme.ru
Received 18 July, 2003; accepted 27 December, 2003
Communicated by A. Babenko
A BSTRACT. 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.
Key words and phrases: Trigonometric series, Integrability, Orlicz space.
2000 Mathematics Subject Classification. 42A32, 46E30.
1. I NTRODUCTION
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.
n=1
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
∞
X
g(x)
∈
L(0,
π)
⇐⇒
nα−1 λn < ∞.
α
x
n=1
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
This work was supported by the Russian Foundation for Fundamental Research (grant no. 03-01-00080) and the Leading Scientific Schools
(grant no. NSH-1657.2003.1).
002-04
2
S. T IKHONOV
If 0 < α < 1, then
∞
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, 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
If −1 < θ < 0, then
f p (x)S
xpθ+1
1
x
∈ L(0, π) ⇐⇒
∞
X
npθ+p−1 S(n)λpn < ∞.
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.
Theorem 1.3 ([10]). Let λn ↓ 0, and p > 0. Then
∞
X
np−2 λpn < ∞ =⇒ ψ(x) ∈ Lp (0, π),
n=1
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
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
∞
X
ψ p (x)
∈
L(0,
π)
⇐⇒
npθ+p−2 λpn < ∞,
xpθ
n=1
where a function ψ(x) is either a f (x) or a g(x).
J. Inequal. Pure and Appl. Math., 5(2) Art. 22, 2004
http://jipam.vu.edu.au/
O N B ELONGING T O O RLICZ S PACE
3
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 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
λn < ∞ =⇒ γ(x)g(x) ∈ L(0, π).
n
n=1
Here
and in the sequel, a function γ(x) is defined by the sequence γ in the following way:
π
γ n := γn , n ∈ N and there exist positive constants A and B such that Aγn+1 ≤ γ(x) ≤ Bγn
π
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 continuous function defined on
[0, ∞) such that Φ(0) = 0 and lim Φ(t) = +∞. For a weight γ(x) the weighted Orlicz space
t→∞
L(Φ, γ) is defined by (see [9], [12])
Z π
(1.3)
L(Φ, γ) = h :
γ(x)Φ(ε |h(x)|)dx < ∞
for some ε > 0 .
0
p
If Φ(x) = x 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).
2. R ESULTS
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
Φ(nλn ) < ∞ =⇒ ψ(x) ∈ L(Φ, γ),
2
n
n=1
where a function ψ(x) is either a sine or cosine series.
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).
J. Inequal. Pure and Appl. Math., 5(2) Art. 22, 2004
http://jipam.vu.edu.au/
4
S. T IKHONOV
Theorem 2.2. Let Φ(x) ∈ 4(p, q) (0 ≤ q ≤ p). If {λn } ∈ R0+ BV S, and sequence {γn } is
such that {γn n−(1+q)+ε } is almost decreasing for some ε > 0, then
∞
X
γn
Φ(n2 λn ) < ∞ =⇒ g(x) ∈ L(Φ, γ).
2+q
n
n=1
Remark 2.3. 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.
3. AUXILIARY R ESULTS
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
λν .
λ1−p
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,
p
n ∈ N. Then
q
(1): θ Φ (t) ≤ Φ (θt) ≤ θ Φ (t) , 0 ≤ θ ≤ 1, t ≥ 0,
!p∗
!
n
n
P
P
1
(2): Φ
, p∗ = max(1, p).
tj ≤
Φ p∗ (tj )
j=1
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).
Proof. Let ξ be an integer such that 2ξ ≤ k < 2ξ+1 . Then
k
X
ν=1
m+1
aν ≤
ξ−1 2
X
X−1
aν +
m=0 ν=2m
k
X
ξ−1
X
aν ≤ C1
!
m
ξ
2 a2 m + 2 a2 ξ
m=0
ν=2ξ
≤ C1
ξ
X
2m a2m .
m=0
Lemma 3.2 implies
Φ
k
X
!
aν
≤ Φ C1
ν=1
ξ
X
!
m
2 a2 m
m=0
ξ
X
≤ C1p Φ
!
2m a2m
m=0
!p∗
ξ
≤C
X
Φ
1
p∗
(2m a2m )
m=0
≤C
J. Inequal. Pure and Appl. Math., 5(2) Art. 22, 2004
1
k
X
Φ p∗ (mam )
m
m=1
!p∗
.
http://jipam.vu.edu.au/
O N B ELONGING T O O RLICZ S PACE
By Lemma 3.1, we have
!
k
∞
∞
X
X
X
λk
λk Φ
aν ≤ C
k=1
ν=1
1
k
X
Φ p∗ (mam )
m
m=1
k=1
!p∗
≤C
5
∞
X
P∞
ν=k
Φ (kak ) λk
kλk
k=1
λν
p∗
.
Note that this Lemma was proved in [7] for the case 0 < p ≤ 1.
4. P ROOFS OF T HEOREMS
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
Dk (x) = +
cos nx, k ∈ N.
2 n=1
and λn ∈ R0+ BV S, we see that
!
n
∞
X
X
|f (x)| ≤ C
λk + n
|λk − λk+1 | ≤ C
Since |Dk (x)| = O
1
x
k=1
k=n
n
X
!
λk + nλn .
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
≤C
k=n
n
X
k=1
n
X
λk + n
∞
X
!
|λk − λk+1 |
k=n
!
λk + nλn ,
k=1
e k (x) are the conjugate Dirichlet kernels, i.e. D
e k (x) := Pk sin nx, k ∈ N.
where D
n=1
Therefore,
!
n
X
|ψ(x)| ≤ C
λk + nλn ,
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 + λn
1 ≤C
λk .
k=1
J. Inequal. Pure and Appl. Math., 5(2) Art. 22, 2004
k=1
k=1
http://jipam.vu.edu.au/
6
S. T IKHONOV
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
n=1
≤ C1p πB
π/(n+1)
k=1
∞
X
n=1
n
X
γn
Φ
n2
!
λk
k=1
∞
X
γk
≤C
Φ (kλk ) 2
k
k=1
∞
k X γν
γk ν=k ν 2
!p∗
,
where p∗ = max(1, p). Since there exists a constant ε > 0 such that {γn n−1+ε } is almost
decreasing, then
∞
∞
X
γν
γk X −ε−1
γk
≤ C 1−ε
ν
≤C .
2
ν
k
k
ν=k
ν=k
Then
Z
π
γ(x)Φ (|ψ(x)|) dx ≤ C
0
∞
X
γk
k=1
k2
Φ (kλk ) .
The proof of Theorem 2.1 is complete.
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
(4.2)
|g(x)| ≤
kxλk + λk sin kx
k=1
≤
n
X
k=n+1
∞ X
e k (x)
kxλk +
(λk − λk+1 ) D
k=1
≤C
≤C
k=n
1
n
1
n
n
X
k=1
n
X
!
kλk + nλn
n
X
1
kλk + λn
k
n k=1
k=1
!
n
1X
≤ C1
kλk .
n k=1
Using Lemma 3.2, Lemma 3.3 and the estimate (4.2), we can write
!Z
Z π
∞
n
π/n
X
X
1
γ(x)Φ (|g(x)|) dx ≤
Φ C1
kλk
γ(x)dx
n k=1
0
π/(n+1)
n=1
!
∞
n
X
X
γn
p
≤ C1 πB
Φ
kλk
n2+q
n=1
k=1
!p∗
∞
∞
1+q X
X
γ
k
γ
k
ν
≤ C2
Φ k 2 λk 2+q
,
2+q
k
γk ν=k ν
k=1
where p∗ = max(1, p).
J. Inequal. Pure and Appl. Math., 5(2) Art. 22, 2004
http://jipam.vu.edu.au/
O N B ELONGING T O O RLICZ S PACE
7
By the assumption on {γn },
Z π
∞
X
γk
γ(x)Φ (|g(x)|) dx ≤ C
Φ k 2 λk ,
2+q
k
0
k=1
and the proof of Theorem 2.2 is complete.
R EFERENCES
[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.
[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.
J. Inequal. Pure and Appl. Math., 5(2) Art. 22, 2004
http://jipam.vu.edu.au/