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volume 4, issue 2, article 35, 2003.

Received 17 March, 2003;

accepted 07 April, 2003.

Communicated by:H. Bor

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Journal of Inequalities in Pure and Applied Mathematics

ADDITIONS TO THE TELYAKOVSKIˇı’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

c

2000Victoria University ISSN (electronic): 1443-5756 035-03

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Additions to the Telyakovskiˇı’s classS

L. Leindler

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J. Ineq. Pure and Appl. Math. 4(2) Art. 35, 2003

http://jipam.vu.edu.au

Abstract

A sufficient condition of new type is given which implies that certain sequences belong to the Telyakovskiˇı’s class S. Furthermore the relations of two subclasses of the classSare analyzed.

2000 Mathematics Subject Classification:26D15, 42A20.

Key words: Cosine series, Fourier series, Inequalities, Classes of number se- quences.

This research was partially supported by the Hungarian National Foundation for Sci- entific Research under Grant No. T04262.

1. Introduction

In 1973, S.A. Telyakovskiˇı [3] defined the classSof 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 classS.For some historical remarks, we refer to [2]. These intentions show that the classSplays a very important role in many problems.

The definition of the classSis the following: A null-sequencea:={a_n}belongs to the classS, or brieflya∈S, if there exists a monotonically decreasing sequence{A_n}such thatP∞

n=1A_n <∞and|∆a_n| ≤A_nhold for alln.

We recall only one result of Telyakovskiˇı [3] to illustrate the usability of the classS.

Theorem 1.1. Let the coefficients of the series

(1.1) a₀

2 +

∞

X

n=1

a_n cosnx

belong to the classS.Then the series (1.1) is a Fourier series and

Z π

0

a0

2 +

∞

X

n=1

a_n cosnx

dx≤C

∞

X

n=0

a_n,

whereCis an absolute constant.

Recently Ž. Tomovski [4] defined certain subclasses ofS,and denoted them bySr, r= 1,2, . . .as follows:

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A null-sequence{a_n}belongs toSr,if there exists a monotonically decreasing sequence

n A^(r)n

o

such thatP∞

n=1n^rA^(r)n <∞and|∆a_n| ≤A^(r)n .

In [5] Tomovski established, among others, a theorem which states that if {a_n} ∈ Sr then ther-th derivative of the series (1.1) is a Fourier series and the integral of the absolute value its sum function less than equal toC(r)P∞

n=1n^rA^(r)n , whereC(r)is a constant.

His proof is a constructive one and follows along similar lines to that of Theorem1.1.

In [1] we also defined a certain subclass ofSas follows:

Letα :={α_n}be a positive monotone sequence tending to infinity. A null- sequence {a_n} belongs to the class S(α), if there exists a monotonically decreasing sequencen

A^(α)n

o

such that

∞

X

n=1

α_nA^(α)_n <∞ and |∆a_n| ≤A^(α)_n . ClearlyS(α)withα_n=n^rincludesSr.

In [2] we verified that if{an} ∈S^r,then{n^ran} ∈S,with a sequence{An} that satisfies the inequality

(1.2)

∞

X

n=1

A_n≤(r+ 1)

∞

X

n=1

n^rA^(r)_n .

Thus, this result and Theorem 1.1 immediately imply the theorem of To- movski mentioned above.

Our theorem which yields (1.2) reads as follows.

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Theorem 1.2. Let γ ≥ β > 0andSα := S(α)ifα_n = n^α.If{a_n} ∈ S_γ then {n^βan} ∈S^γ−β and

(1.3)

∞

X

n=1

n^γ−βA^(γ−β)_n ≤(β+ 1)

∞

X

n=1

n^γA^(γ)_n

holds.

It is clear that ifγ =β=rthen (1.3) gives (1.2)

A⁽⁰⁾n =A_n .

In [2] we also verified that the statement of Theorem1.2is not reversible in general.

In [3] Telyakovskiˇı realized that in the definition of the classSwe can take A_n:= maxk≥n|∆a_k|,that is,{a_n} ∈Sifa_n →0andP∞

n=1maxk≥n|∆a_k|<

∞.

This definition ofShas not been used often, as I know.

The reason, perhaps, is the appearing of the inconvenient addendsmax

k≥n |∆a_k|.

In the present note first we give a sufficient condition being of similar char- acter as this definition of S but without maxk≥n|∆a_k|, which implies that {a_n} ∈S.

Second we show that with a certain additional assumption, the assertion of Theorem1.2is reversible and the additional condition to be given is necessary in general.

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2. Results

Before formulating the first theorem we recall a definition.

A non-negative sequence c := {c_n} is called locally almost monotone if there exists a constantK(c)depending only on the sequencec, such that

c_n ≤K(c)c_m

holds for anym and m ≤ n ≤ 2m. These sequences will be denoted by c ∈ LAM S.

Theorem 2.1. Ifa:={a_n}is a null-sequence,a ∈LAM SandP∞

n=1|∆a_n|<

∞,thena∈S.

Theorem 2.2. Letγ ≥β >0.If{n^βa_n} ∈S^γ−β,and (2.1)

∞

X

n=1

n^γ|∆a_n|<∞,

then{a_n} ∈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 Theorem2.1.

Lemma 2.3. Ifc:={c_n} ∈LAM Sandα_n := sup_k≥nc_k,then for anyδ >−1

(2.2)

∞

X

n=1

n^δα_n≤K(K(c), δ)

∞

X

n=1

n^δc_n.

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Proof. Sincec∈LAM S thus withK :=K(c)

(2.3) α2ⁿ = sup

k≥2ⁿ

ck ≤ sup

m≥n

K c2^m ≤Ksup

m≥n

c2^m. IfP

n^δcn<∞,thencn→0,thus by (2.3) there exists an integerp=p(n)≥ 0such that

α₂ⁿ ≤K c₂^n+p. Then, by the monotonicity of the sequence{α_n},

n+p

X

k=n

2^k(1+δ)α₂^k ≤K c₂^n+p

n+p

X

k=n

2^k(1+δ)

≤K2^(1+δ)2^(n+p)(1+δ)c₂^n+p

≤K²2^(1+δ)2

2^n+p

X

ν=2^n+p−1+1

ν^δc_ν

clearly follows. If we start this arguing withn = 0,and repeat it withn+pin place ofn,ifp≥1;and ifp= 0then withn+ 1in place ofn,and make these blocks repeatedly, furthermore if we add all of these sums, we see that the sum P∞

k=32^k(1+δ)α₂^k will be majorized by the sum K²4^(1+δ)P∞

n=1n^δc_n, and this proves (2.2).

Remark 2.2. Following the steps of the proof it is easy to see that with ϕ_nin place ofn^δ,(2.2) also holds if{ϕ_n} ∈LAM Sand2ⁿϕ₂ⁿis quasi geometrically increasing.

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3. Proofs

Proof of Theorem2.1. Using Lemma2.3 withc_n =a_nandδ = 0,we immediately get that

(3.1)

∞

X

n=1

maxk≥n |∆a_k|<∞,

namely the assumption a_n → 0yields that sup|∆a_k| = max|∆a_k|,and thus (3.1) implies that{a_n} ∈S.

Proof of Theorem2.2. With respect to the equality

|∆(n^βan)|=|n^β(an−an+1)−an+1((n+ 1)^β −n^β)|

it is clear that

n^β|∆a_n| ≤A^(γ−β)_n +K n^β−1|a_n+1|, whereKis a constantK =K(β)>0independent ofn.

Hence, multiplying withn^−β,we get that (3.2) |∆a_n| ≤n^−βA^(γ−β)_n +K n⁻¹

∞

X

k=n+1

|∆a_k|,

thus if we define

A^(γ)_n :=n^−βA^(γ−β)_n +K n⁻¹

∞

X

k=n+1

|∆a_k|,

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then this sequenceA^(γ)n is clearly monotonically decreasing, andA^(γ)n ≥ |∆a_n|, furthermore by the assumptions of Theorem1.2and (3.2)

∞

X

n=1

n^γA^(γ)_n <∞, since

∞

X

n=1

n^γ−1

∞

X

k=n+1

|∆a_k| ≤K(γ)

∞

X

k=1

k^γ|∆a_k|<∞.

Thus{a_n} ∈Sγ is proved. The proof is complete.

Proof of Remark2.1. Let a_n = n^−β, then|∆n^βa_n| = 0, therefore{n^βa_n} ∈ S^γ−β holds e.g. withA^(γ−β)n =n^β−γ−2.On the other hand|∆a_n| ≥(n+1)^−β−1, thus, byγ ≥β,

(3.3)

∞

X

n=1

n^γ|∆an|=∞,

consequently, ifA^(γ)n ≥ |∆a_n|,then

∞

X

n=1

n^γA^(γ)_n =∞ also holds, therefore{a_n} 6∈Sγ.

In this case, by (3.3), the additional condition (2.1) does not maintain.

Herewith, Remark2.1is 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 Telyakovskiˇı’s classS,J. Inequal. Pure and Appl. Math., 2(3) (2001), Article 32. [ONLINE http://jipam.

vu.edu.au/v2n3/008_01.html]

[3] S.A. TELYAKOVSKIˇI, 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 applica- tions, 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.

[ONLINEhttp://jipam.vu.edu.au/v3n1/005_99.html]