Journal of Inequalities in Pure and
Applied Mathematics
ADDITIONS TO THE TELYAKOVSKIı̌’S CLASS S
L. LEINDLER
Bolyai Institute,
University of Szeged, Aradi vértanúk tere 1,
H-6720 Szeged, Hungary
E-Mail: leindler@math.u-szeged.hu
volume 4, issue 2, article 35,
2003.
Received 17 March, 2003;
accepted 07 April, 2003.
Communicated by: H. Bor
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
035-03
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Abstract
A sufficient condition of new type is given which implies that certain sequences
belong to the Telyakovskiı̌’s class S. Furthermore the relations of two subclasses of the class S are analyzed.
2000 Mathematics Subject Classification: 26D15, 42A20.
Key words: Cosine series, Fourier series, Inequalities, Classes of number sequences.
Additions to the Telyakovskiı̌’s
class S
This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant No. T04262.
L. Leindler
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Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3
6
8
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1.
Introduction
In 1973, S.A. Telyakovskiı̌ [3] defined the class S of number sequences which
has become a very flourishing definition. Several mathematicians have wanted
to extend this definition, but it has turned out that most of them are equivalent to
the class S. For some historical remarks, we refer to [2]. These intentions show
that the class S plays a very important role in many problems.
The definition of the class S is the following: A null-sequence a := {an } belongs to the class S, or briefly
P∞ a ∈ S, if there exists a monotonically decreasing
sequence {An } such that n=1 An < ∞ and |∆ an | ≤ An hold for all n.
We recall only one result of Telyakovskiı̌ [3] to illustrate the usability of the
class S.
Theorem 1.1. Let the coefficients of the series
∞
(1.1)
a0 X
+
an cos nx
2
n=1
belong to the class S. Then the series (1.1) is a Fourier series and
Z π ∞
∞
X
X
a0
an cos nx dx ≤ C
an ,
+
0 2
n=1
n=0
where C is an absolute constant.
Recently Ž. Tomovski [4] defined certain subclasses of S, and denoted them
by Sr , r = 1, 2, . . . as follows:
Additions to the Telyakovskiı̌’s
class S
L. Leindler
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A null-sequence
to Sr , if there exists a monotonically decreasn
o{an } belongs
P∞ r (r)
(r)
(r)
ing sequence An such that n=1 n An < ∞ and |∆ an | ≤ An .
In [5] Tomovski established, among others, a theorem which states that if
{an } ∈ Sr then the r-th derivative of the series (1.1) is a Fourier series and the
P
r (r)
integral of the absolute value its sum function less than equal to C(r) ∞
n=1 n An ,
where C(r) is a constant.
His proof is a constructive one and follows along similar lines to that of
Theorem 1.1.
In [1] we also defined a certain subclass of S as follows:
Let α := {αn } be a positive monotone sequence tending to infinity. A nullsequence {an } belongs
to
n
o the class S(α), if there exists a monotonically de(α)
creasing sequence An
such that
∞
X
αn A(α)
n < ∞
and |∆ an | ≤ A(α)
n .
n=1
Clearly S(α) with αn = nr includes Sr .
In [2] we verified that if {an } ∈ Sr , then {nr an } ∈ S, with a sequence {An }
that satisfies the inequality
(1.2)
∞
X
n=1
An ≤ (r + 1)
∞
X
Additions to the Telyakovskiı̌’s
class S
L. Leindler
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n
r
A(r)
n .
n=1
Thus, this result and Theorem 1.1 immediately imply the theorem of Tomovski mentioned above.
Our theorem which yields (1.2) reads as follows.
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Theorem 1.2. Let γ ≥ β > 0 and Sα := S(α) if αn = nα . If {an } ∈ Sγ then
{nβ an } ∈ Sγ−β and
(1.3)
∞
X
n=1
n
γ−β
An(γ−β)
≤ (β + 1)
∞
X
nγ A(γ)
n
n=1
holds.
(0)
It is clear that if γ = β = r then (1.3) gives (1.2) An = An .
In [2] we also verified that the statement of Theorem 1.2 is not reversible in
general.
In [3] Telyakovskiı̌ realized that in the definition of P
the class S we can take
An := maxk≥n |∆ ak |, that is, {an } ∈ S if an → 0 and ∞
n=1 maxk≥n |∆ ak | <
∞.
This definition of S has not been used often, as I know.
The reason, perhaps, is the appearing of the inconvenient addends max |∆ ak |.
k≥n
In the present note first we give a sufficient condition being of similar character as this definition of S but without maxk≥n |∆ ak |, which implies that
{an } ∈ S.
Second we show that with a certain additional assumption, the assertion of
Theorem 1.2 is reversible and the additional condition to be given is necessary
in general.
Additions to the Telyakovskiı̌’s
class S
L. Leindler
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2.
Results
Before formulating the first theorem we recall a definition.
A non-negative sequence c := {cn } is called locally almost monotone if
there exists a constant K(c) depending only on the sequence c, such that
cn ≤ K(c)cm
holds for any m and m ≤ n ≤ 2m. These sequences will be denoted by c ∈
LAM S.
P
Theorem 2.1. If a := {an } is a null-sequence, a ∈ LAM S and ∞
n=1 |∆ an | <
∞, then a ∈ S.
Additions to the Telyakovskiı̌’s
class S
L. Leindler
β
Theorem 2.2. Let γ ≥ β > 0. If {n an } ∈ Sγ−β , and
∞
X
(2.1)
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nγ |∆ an | < ∞,
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n=1
then {an } ∈ Sγ .
Remark 2.1. The condition (2.1) is not dispensable, moreover it cannot be
weakened in general.
The following lemma will be required in the proof of Theorem 2.1.
Lemma 2.3. If c := {cn } ∈ LAM S and αn := supk≥n ck , then for any δ > −1
(2.2)
∞
X
n=1
nδ αn ≤ K(K(c), δ)
∞
X
n=1
nδ cn .
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Proof. Since c ∈ LAM S thus with K := K(c)
(2.3)
α2n = sup ck ≤ sup K c2m ≤ K sup c2m .
k≥2n
m≥n
m≥n
P δ
If
n cn < ∞, then cn → 0, thus by (2.3) there exists an integer p = p(n) ≥
0 such that
α2n ≤ K c2n+p .
Then, by the monotonicity of the sequence {αn },
n+p
X
k=n
k(1+δ)
2
α2k ≤ K c2n+p
n+p
X
2k(1+δ)
k=n
(1+δ) (n+p)(1+δ)
≤K2
2
≤ K 2 2(1+δ)2
Additions to the Telyakovskiı̌’s
class S
L. Leindler
c2n+p
n+p
2X
ν δ cν
ν=2n+p−1 +1
clearly follows. If we start this arguing with n = 0, and repeat it with n + p in
place of n, if p ≥ 1; and if p = 0 then with n + 1 in place of n, and make these
blocks
furthermore if we add all of these sums,P
we see that the sum
P∞ repeatedly,
∞
k(1+δ)
2 (1+δ)
δ
2
α
will
be
majorized
by
the
sum
K
4
k
2
k=3
n=1 n cn , and this
proves (2.2).
Remark 2.2. Following the steps of the proof it is easy to see that with ϕn in
place of nδ , (2.2) also holds if {ϕn } ∈ LAM S and 2n ϕ2n is quasi geometrically
increasing.
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3.
Proofs
Proof of Theorem 2.1. Using Lemma 2.3 with cn = an and δ = 0, we immediately get that
∞
X
(3.1)
n=1
max |∆ ak | < ∞,
k≥n
namely the assumption an → 0 yields that sup |∆ ak | = max |∆ ak |, and thus
(3.1) implies that {an } ∈ S.
Proof of Theorem 2.2. With respect to the equality
L. Leindler
|∆(nβ an )| = |nβ (an − an+1 ) − an+1 ((n + 1)β − nβ )|
it is clear that
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nβ |∆ an | ≤ A(γ−β)
+ K nβ−1 |an+1 |,
n
where K is a constant K = K(β) > 0 independent of n.
Hence, multiplying with n−β , we get that
(3.2)
Additions to the Telyakovskiı̌’s
class S
|∆ an | ≤ n−β A(γ−β)
+ K n−1
n
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∞
X
|∆ ak |,
k=n+1
thus if we define
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−β (γ−β)
A(γ)
An
+ K n−1
n := n
∞
X
k=n+1
|∆ ak |,
J. Ineq. Pure and Appl. Math. 4(2) Art. 35, 2003
http://jipam.vu.edu.au
(γ)
(γ)
then this sequence An is clearly monotonically decreasing, and An ≥ |∆ an |,
furthermore by the assumptions of Theorem 1.2 and (3.2)
∞
X
nγ An(γ) < ∞,
n=1
since
∞
X
n
γ−1
n=1
∞
X
|∆ ak | ≤ K(γ)
k=n+1
∞
X
k γ |∆ ak | < ∞.
k=1
Thus {an } ∈ Sγ is proved. The proof is complete.
Proof of Remark 2.1. Let an = n−β , then |∆ nβ an | = 0, therefore {nβ an } ∈
(γ−β)
Sγ−β holds e.g. with An
= nβ−γ−2 . On the other hand |∆ an | ≥ (n+1)−β−1 ,
thus, by γ ≥ β,
∞
X
(3.3)
Additions to the Telyakovskiı̌’s
class S
L. Leindler
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nγ |∆ an | = ∞,
n=1
(γ)
consequently, if An ≥ |∆ an |, then
∞
X
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n
γ
A(γ)
n
=∞
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n=1
also holds, therefore {an } 6∈ Sγ .
In this case, by (3.3), the additional condition (2.1) does not maintain.
Herewith, Remark 2.1 is verified, namely we can also see that the condition
(2.1) cannot be weakened in general.
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References
[1] L. LEINDLER, Classes of numerical sequences, Math. Ineq. and Appl., 4(4)
(2001), 515–526.
[2] L. LEINDLER, On the utility of the Telyakovskiı̌’s class S, J. Inequal. Pure
and Appl. Math., 2(3) (2001), Article 32. [ONLINE http://jipam.
vu.edu.au/v2n3/008_01.html]
[3] S.A. TELYAKOVSKIǏ, On a sufficient condition of Sidon for integrability
of trigonometric series, Math. Zametki, (Russian) 14 (1973), 317–328.
Additions to the Telyakovskiı̌’s
class S
[4] Ž. TOMOVSKI, An extension of the Sidon-Fomin inequality and applications, Math. Ineq. and Appl., 4(2) (2001), 231–238.
L. Leindler
[5] Ž. TOMOVSKI, Some results on L1 -approximation of the r-th derivative
of Fourier series, J. Inequal. Pure and Appl. Math., 3(1) (2002), Article 10.
[ONLINE http://jipam.vu.edu.au/v3n1/005_99.html]
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