Journal of Inequalities in Pure and
Applied Mathematics
GENERALIZATION OF A RESULT FOR COSINE SERIES
ON THE L1 NORM
volume 4, issue 5, article 99,
2003.
JÓZSEF NÉMETH
Bolyai Institute
University of Szeged
Aradi vértanúk tere 1.
H-6720 Hungary.
EMail: nemethj@math.u-szeged.hu
Received 13 November, 2002;
accepted 03 July, 2003.
Communicated by: A. Babenko
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
A L1 -estimate will be established for cosine series, considering the generalized
Fomin-class Fϕ , where ϕ is a function more general than the power function
2000 Mathematics Subject Classification: 26D15; 42A20.
Key words: Trigonometric series, Fomin-class, L1 -estimate.
This research was supported by the Hungarian National Foundation for Scientific
Research under Grant No. 19/75/1K703.
Generalization of a Result for
Cosine Series on the L1 Norm
József Németh
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4
Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
References
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1.
Introduction
Let
∞
a0 X
f (x) =
+
an cos nx.
2
n=1
(1.1)
Many authors ([1], [2], [9] – [15]) have investigated coefficient conditions
guaranteeing that (1.1) isR a Fourier series of some function f ∈ L1 and they
π
have given estimates for 0 |f (x)|dx via the sequence {an }.
Recently Z. Tomovski [15] proved a theorem of this type by using the class
of coefficients defined by Fomin [1] as follows: a sequence {an } belongs to
Fp (p > 1) if ak → 0 and
Generalization of a Result for
Cosine Series on the L1 Norm
József Németh
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∞
X
(1.2)
k=1
(
1
k
∞
X
) p1
|∆ai |p
Contents
< ∞.
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J
i=k
Now we can formulate Z. Tomovski’s result [15]:
Theorem 1.1. Let {an } ∈ Fp , 1 < p ≤ 2, then the series (1.1) is a Fourier
series and the following inequality holds:
Z
π
|f (x)|dx ≤ Cp
(1.3)
0
where Cp depends only on p.
∞
X
n=1
(
∞
1X
|∆ak |p
n k=n
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) p1
,
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Recently we ([7]) investigated the properties of classes of numerical sequences obtained by using functions more general than the power functions.
Such functions were used first of all in the works of H.P. Mulholland [5] and M.
Mateljevič and M. Pavlovič [8]. The following definition is due to Mateljevič
and Pavlovič.
∆(q, p) (q ≥ p > 0) denotes the family of the nonnegative real functions
ϕ(x) defined on [0; ∞) with the following properties: ϕ(0) = 0, ϕ(t)
is nontq
ϕ(t)
increasing and tp is nondecreasing on (0; ∞). ∆ will denote the set of the
functions ϕ(x) ∈ ∆(q, p) for some q ≥ p > 0.
Using this notion in [7] we defined the following class: a nullsequence {an }
belongs to the class Fϕ for some ϕ ∈ ∆ if
!
∞
∞
X
1X
ϕ
ϕ(|∆ak |) < ∞,
(1.4)
n k=n
n=1
where ϕ is the inverse of the function ϕ.
The aim of the present paper is to generalize Theorem 1.1 using Fϕ instead
of Fp , where ϕ ∈ ∆. Since our goal is to get a result concerning such functions
as ϕ(x) = x logα (1 + x) and not only for functions which are generalizations
of xp (p > 1), we therefore need to define two subclasses of ∆. Namely we use
the following definitions: ∆(1) denotes the family of functions ϕ(x) belonging
to ∆(q, p) for some q ≥ p > 1 and ∆(2) is the collection of functions ϕ(x) from
∆(q, 1) for some q > 1 such that for all A > 0 there exists p := p(A) > 1
satisfying the condition that ϕ(x)
is nondecreasing on (0; A). It is obvious that
xp
(1)
(2)
∆ ⊂∆ .
Generalization of a Result for
Cosine Series on the L1 Norm
József Németh
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After giving these definitions we can formulate our result which generalizes
Theorem 1.1 (if ϕ ∈ ∆(1) ). Furthermore, it contains the case like ϕ(x) =
x logα (1 + x) (α > 0).
Generalization of a Result for
Cosine Series on the L1 Norm
József Németh
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2.
Result
Theorem 2.1. Let {an } ∈ Fϕ for ϕ ∈ ∆(2) . Then the series (1.1) is a Fourier
series and the following estimate holds
!
Z π
∞
∞
X
1X
ϕ(|∆ak |) ,
(2.1)
|f (x)|dx ≤ Cϕ
ϕ
n k=n
0
n=1
where Cϕ is a constant depending only on ϕ.
Remark 2.1. In [3] and [4] L. Leindler investigated a relation among classes
of numerical sequences other than Fp (see the classes denoted by Sp , Fp∗ , Sp (δ),
Sp (A)) and he proved that all these classes coincide. Later in [7] we defined the
classes of sequences Sϕ , Fϕ∗ , Sϕ (δ), Sϕ (A) exchanging the functions xp to ϕ(x)
and showed that all these classes also coincide. Therefore in Theorem 2.1 the
class Fϕ can be replaced by any of the above mentioned classes, if ϕ ∈ ∆(2) .
Generalization of a Result for
Cosine Series on the L1 Norm
József Németh
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3.
Lemmas
Lemma 3.1. ([11]). Let the nullsequence {an } be of bounded variation and
[i/2] ∞ X
X
∆ai−k − ∆ai+k <∞
k
i=2 k=1
then (1.1) is a Fourier series and the following estimate holds:


Z π
∞
∞ X
X
X
[i/2] ∆ai−k − ∆ai+k  ,

(3.1)
|f (x)|dx ≤ C
|∆ak | +
k
0
i=2 k=1
k=0
Generalization of a Result for
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József Németh
where C is some absolute constant.
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Lemma 3.2. ([1]). Let {an } be a nullsequence. Then the following estimate
holds:
Contents
(3.2)
[i/2] ∞ X
∞
X
X
∆ai−k − ∆ai+k ≤ Cp
∆(p)
s ,
k
s=1
i=2 k=1
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where
(
∆(p)
s =
1
2s−1
) p1
2s
X
|∆ak |p
k=2s−1 +1
and Cp is a constant depending only on p.
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,
1 < p ≤ 2,
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Lemma 3.3. ([5]). If Ψ(x)
is increasing then for all sequences {an } and {bn }
x
of nonnegative numbers
Pn
Pn
ai b i
a Ψ(2bi )
i=1
Pn i
Ψ Pn
≤ i=1
i=1 ai
i=1 ai
holds.
Lemma 3.4. ([6]). Let ρ(x) denote a nonnegative function increasing to infinity
such that ρ(x)
is decreasing to zero when x is increasing from zero to infinity.
x
Furthermore, if an ≥ 0, λn > 0 for all n, then
!
!
∞
n
∞
∞
X
X
X
an X
λn ρ
ak ,
(3.3)
λn ρ
λk ≤ Cρ
λn k=1
n=1
n=1
k=n
where Cρ depends only on the function ρ(x).
Lemma 3.5. Let bn ≥ 0 for all n and let ϕ be the inverse of the function
ϕ(x) ∈ ∆(2) . Then
P∞
∞
∞
X
X
k=n bk
(3.4)
ϕ(bn ) ≤ Kϕ ·
ϕ
,
n
n=1
n=1
where Kϕ is a constant depending only on ϕ.
Proof. Since ϕ(x)
is decreasing to zero if x → ∞, therefore using Lemma 3.4
x
and taking ρ(x) = ϕ(x), bn = n · an , λn = 1, we get
!
∞
∞
∞
X
X
X
bk
,
ϕ(bn ) ≤ Kϕ ·
ϕ
k
n=1
n=1
k=n
Generalization of a Result for
Cosine Series on the L1 Norm
József Németh
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whence the statement of Lemma 3.5 is obtained.
Generalization of a Result for
Cosine Series on the L1 Norm
József Németh
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4.
Proof
Proof of Theorem 2.1. Let ϕ ∈ ∆(2) and bn = ϕ(|∆ an |). Using Lemma 3.5 and
that {an } ∈ Fϕ we have that
!
∞
∞
∞
X
X
1X
ϕ(|∆an |) < ∞,
(4.1)
|∆an | ≤ K ·
ϕ
n k=n
n=1
n=1
where K depends only on ϕ. It now follows that the sequence {an } is of
bounded variation.
Further on, we use the following notations:
)
(
2s
X
1
∆(ϕ)
:= ϕ
ϕ(|∆ ak |) , p0 > 1
s
2s−1
s−1
k=2
denotes a number for which
ϕ(x)
xp 0
+1
↑ on (0; A), where A = sup |∆ak |, and by q
k
ϕ(x)
xq
we denote a number satisfying
↓ (see the definition of ϕ ∈ ∆(2) ).
Now we will prove that for any 1 < p < p0 ,
∞
∞
X
X
s (p)
q/p
(4.2)
2 ∆s ≤ 2
2s ∆(ϕ)
s
s=1
s=1
holds.
is increasing thus using Lemma
Since for ψ(t) = ϕ(t1/p ) the function ψ(t)
t
q
1/p
p
3.3 and that ϕ(2 x) ≤ 2 ϕ(x) we get
!
P2s
P2s
p
q
k=2s−1 +1 ϕ(|∆ak |)
k=2s−1 +1 |∆ak |
p
.
(4.3)
ψ
≤
2
2s−1
2s−1
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Cosine Series on the L1 Norm
József Németh
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From (4.3), taking into account that ϕ(cx) ≤ cϕ(x) if c > 1, we obtain
"
!#
!
P2s
P2s
p
q
|∆a
|
ϕ(|∆a
|)
s−1
s−1
k
k
k=2
+1
k=2
+1
(4.4)
ϕ ψ
≤ 2p ϕ
.
2s−1
2s−1
Since ϕ(ψ(t)) = t1/p so from (4.4) we have
( P2s
k=2s−1 +1
∆(p)
s =
(4.5)
|∆ak |p
Generalization of a Result for
Cosine Series on the L1 Norm
2s−1
P2s
k=2s−1 +1
≤ 2q/p ϕ
) p1
ϕ(|∆ak |)
2s−1
!
= 2q/p ∆(ϕ)
s .
From (4.5), (4.2) immediately follows.
Now we show that
∞
X
(4.6)
s
2
∆s(ϕ)
s=1
s=1
(4.7)
2s ∆s(ϕ)
P∞
ϕ(|∆ak |) is monotone decreasing, we obtain
!
2s
n
X
X
1
ϕ(|∆ak |)
=2·
2s−1 ϕ
s−1
2
s=1
k=2s−1 +1
!
n−1
n−1
2X
2X
∞
1X
ϕ(Us ) = 4 ·
ϕ
ϕ(|∆ak |) .
≤4·
s
s=1
s=1
k=s
Since the sequence Us =
n
X
1
s
P∞
∞
X
k=s ϕ(|∆ak |)
.
≤4·
ϕ
s
s=1
József Németh
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Setting n → ∞ we have from (4.7) the inequality (4.6).
Collecting (4.1), (4.2), (4.6), using Lemma 3.1 and Lemma 3.2, we get that
(1.1) is a Fourier series and (2.1) is true. Thus Theorem 2.1 is proved.
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Cosine Series on the L1 Norm
József Németh
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References
[1] G.A. FOMIN, A class of trigonometric series, Math. Zametki, 23 (1978),
117–124 (Russian).
[2] A.N. KOLMOGOROV, Sur l’ordre de grandeur des coefficients de la serie
de Fourier-Lebesgue, Bull. Acad. Polon. Ser. A, Sci. Math. (1923), 83–86.
[3] L. LEINDLER, On the equivalence of classes of Fourier coefficients,
Math. Inequal. Appl., 3 (2000), 45–50.
[4] L. LEINDLER, A note on some classes of real sequences, Math. Inequal.
Appl., 4 (2001), 53–58.
[5] H.P. MULHOLLAND, The generalization of certain inequality theorems
involving powers, Proc. London Math. Soc., 33 (1932), 481–516.
[6] J. NÉMETH, Generalization of the Hardy-Littlewood inequality. II, Acta
Sci. Math., 35 (1973), 127–134.
[7] J. NÉMETH, A note on two theorems of Leindler, Math. Inequal. Appl., 5
(2002), 225–233.
p
[8] M. MATELJEVIČ AND M. PAVLOVIČ, L -behavior of power series with
positive coefficients and Hardy spaces, Proc. Amer. Math., Soc., 87 (1983),
309–316.
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József Németh
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[9] S. SIDON, Hinreichende Bedingungen für den Fourier-charakter einer
trigonometrischen Reihe, J. London Math. Soc., 14 (1939), 158–160.
J. Ineq. Pure and Appl. Math. 4(5) Art. 99, 2003
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[10] S. SIDON, Reihen theoretische Sätze und ihre Anwendungen in der Theorie der Fourierschen Reihen, Math. Z., 10 (1921), 121–127.
[11] S.A. TELYAKOVSKǏI, Integrability conditions for trigonometric series
and their application to the study of linear summation methods of Fourier
series, Izv. Akad. Nauk SSSR Ser. Mat., 28 (1964), 1209–1238.
[12] S.A. TELYAKOVSKǏI, Some estimates for trigonometric series with
quasi-convex coefficients, Mat. Sb., 63(105) (1964), 426–444. (Russian)
[13] S.A. TELYAKOVSKǏI, An asymptotic estimate of the integral of the absolute value of a function given by sine series, Sibirsk. Mat. Ž., 8 (1967),
1416–1422. (Russian)
[14] S.A. TELYAKOVSKǏI, An estimate, useful in problems of approximation
theory of the norm of a function by means of its Fourier coefficients, Trudy
Mat. Inst. Steklov, 109 (1971), 73–109. (Russian)
[15] Ž. TOMOVSKI, Two new L1 -estimates for trigonometric series with
Fomin’s coefficient condition, RGMIA Research Report Collection, 2(6)
(1999), http://rgmia.vu.edu.au/.
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