Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 2, Article 35, 2003
ADDITIONS TO THE TELYAKOVSKIǏ’S CLASS S
L. LEINDLER
B OLYAI I NSTITUTE ,
U NIVERSITY OF S ZEGED , A RADI VÉRTANÚK TERE 1,
H-6720 S ZEGED , H UNGARY
leindler@math.u-szeged.hu
Received 17 March, 2003; accepted 07 April, 2003
Communicated by H. Bor
A BSTRACT. 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.
Key words and phrases: Cosine series, Fourier series, Inequalities, Classes of number sequences.
2000 Mathematics Subject Classification. 26D15, 42A20.
1. I NTRODUCTION
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
P∞ S, or briefly 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
∞
a0 X
+
an cos nx
(1.1)
2
n=1
belong to the class S. Then the series (1.1) is a Fourier series and
Z π ∞
∞
X
X
a
0
an cos nx dx ≤ C
an ,
+
0 2
n=1
n=0
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant No. T04262.
035-03
2
L. L EINDLER
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:
{an } belongs to Sr , if there exists a monotonically decreasing sequence
n A null-sequence
o
P∞ r (r)
(r)
(r)
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 integral of the absolute value its sum
P
r (r)
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 null-sequence
{an }
n
o
(α)
belongs to the class S(α), if there exists a monotonically decreasing 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
∞
X
(1.2)
n=1
An ≤ (r + 1)
∞
X
nr 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.
Theorem 1.2. Let γ ≥ β > 0 and Sα := S(α) if αn = nα . If {an } ∈ Sγ then {nβ an } ∈ Sγ−β
and
∞
∞
X
X
γ−β (γ−β)
nγ A(γ)
n
An
≤ (β + 1)
(1.3)
n
n=1
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
Pin the definition of 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 maxk≥n |∆ ak |.
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.
2. R ESULTS
Before formulating the first theorem we recall a definition.
J. Inequal. Pure and Appl. Math., 4(2) Art. 35, 2003
http://jipam.vu.edu.au/
A DDITIONS TO THE T ELYAKOVSKI Ǐ ’ S C LASS S
3
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.
Theorem 2.2. Let γ ≥ β > 0. If {nβ an } ∈ Sγ−β , and
∞
X
(2.1)
nγ |∆ an | < ∞,
n=1
then {an } ∈ Sγ .
Remark 2.3. 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 2.4. If c := {cn } ∈ LAM S and αn := supk≥n ck , then for any δ > −1
∞
X
(2.2)
δ
n αn ≤ K(K(c), δ)
∞
X
nδ cn .
n=1
n=1
Proof. Since c ∈ LAM S thus with K := K(c)
(2.3)
α2n = sup ck ≤ sup K c2m ≤ K sup c2m .
k≥2n
If
P
m≥n
m≥n
nδ 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(1+δ)
2
k=n
α2k ≤ K c2n+p
n+p
X
2k(1+δ)
k=n
(1+δ) (n+p)(1+δ)
≤K2
2
c2n+p
2n+p
≤ K 2 2(1+δ)2
X
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, P
and make these blocks repeatedly, furthermore
k(1+δ)
if we add P
all of these sums, we see that the sum ∞
α2k will be majorized by the sum
k=3 2
∞
2 (1+δ)
δ
K 4
n
c
,
and
this
proves
(2.2).
n
n=1
Remark 2.5. 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.
J. Inequal. Pure and Appl. Math., 4(2) Art. 35, 2003
http://jipam.vu.edu.au/
4
L. L EINDLER
3. P ROOFS
Proof of Theorem 2.1. Using Lemma 2.4 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
|∆(nβ an )| = |nβ (an − an+1 ) − an+1 ((n + 1)β − nβ )|
it is clear that
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
∞
X
|∆ an | ≤ n−β A(γ−β)
+ K n−1
n
(3.2)
|∆ ak |,
k=n+1
thus if we define
A(γ)
n
:= n
−β
A(γ−β)
n
+Kn
∞
X
−1
|∆ ak |,
k=n+1
(γ)
An
(γ)
then this sequence
is clearly monotonically decreasing, and An ≥ |∆ an |, furthermore by
the assumptions of Theorem 1.2 and (3.2)
∞
X
nγ A(γ)
n < ∞,
n=1
since
∞
X
n=1
n
γ−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 2.3. Let an = n−β , then |∆ nβ an | = 0, therefore {nβ an } ∈ Sγ−β holds e.g.
(γ−β)
with An
= nβ−γ−2 . On the other hand |∆ an | ≥ (n + 1)−β−1 , thus, by γ ≥ β,
∞
X
(3.3)
nγ |∆ an | = ∞,
n=1
consequently, if
(γ)
An
≥ |∆ an |, then
∞
X
nγ A(γ)
n = ∞
n=1
also holds, therefore {an } 6∈ Sγ .
In this case, by (3.3), the additional condition (2.1) does not maintain.
Herewith, Remark 2.3 is verified, namely we can also see that the condition (2.1) cannot be
weakened in general.
J. Inequal. Pure and Appl. Math., 4(2) Art. 35, 2003
http://jipam.vu.edu.au/
A DDITIONS TO THE T ELYAKOVSKI Ǐ ’ S C LASS S
5
R EFERENCES
[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.
[4] Ž. TOMOVSKI, An extension of the Sidon-Fomin inequality and applications, Math. Ineq. and
Appl., 4(2) (2001), 231–238.
[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]
J. Inequal. Pure and Appl. Math., 4(2) Art. 35, 2003
http://jipam.vu.edu.au/