Furthermore the relations of two subclasses of the classS are analyzed

http://jipam.vu.edu.au/

Volume 4, Issue 2, Article 35, 2003

ADDITIONS TO THE TELYAKOVSKIˇI’S CLASS S

L. LEINDLER BOLYAIINSTITUTE,

UNIVERSITY OFSZEGED, ARADI VÉRTANÚK TERE1, H-6720 SZEGED, HUNGARY

leindler@math.u-szeged.hu

Received 17 March, 2003; accepted 07 April, 2003 Communicated by H. Bor

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

Key words and phrases: Cosine series, Fourier series, Inequalities, Classes of number sequences.

2000 Mathematics Subject Classification. 26D15, 42A20.

1. INTRODUCTION

In 1973, S.A. Telyakovskiˇı [3] defined the class Sof 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 class S plays a very important role in many problems.

The definition of the class S is the following: A null-sequence a := {a_n} belongs to the classS, or briefly a ∈ S, if there exists a monotonically decreasing sequence {A_n} such that P∞

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

an cosnx

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

0

a₀ 2 +

∞

X

n=1

an cosnx

dx≤C

∞

X

n=0

an,

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

whereCis an absolute constant.

Recently Ž. Tomovski [4] defined certain subclasses of S, and denoted them by S^r, r = 1,2, . . .as follows:

A null-sequence {a_n} belongs to S^r, 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} ∈Srthen the r-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 Theorem 1.1.

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 classS(α),if there exists a monotonically decreasing sequence

n A^(α)n

o

such that

∞

X

n=1

αnA^(α)_n <∞ and |∆an| ≤A^(α)_n . ClearlyS(α)withα_n =n^rincludesS^r.

In [2] we verified that if{a_n} ∈Sr,then{n^ra_n} ∈S,with a sequence{A_n}that satisfies the inequality

(1.2)

∞

X

n=1

An ≤(r+ 1)

∞

X

n=1

n^rA^(r)_n .

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. Letγ ≥β > 0andSα :=S(α)ifα_n = n^α.If{a_n} ∈S_γ then{n^βa_n} ∈ Sγ−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 Theorem 1.2 is not reversible in general.

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

n=1max_k≥n|∆a_k|<∞.

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

The reason, perhaps, is the appearing of the inconvenient addendsmaxk≥n|∆a_k|.

In the present note first we give a sufficient condition being of similar character as this defi- nition ofSbut withoutmaxk≥n|∆a_k|,which implies that{a_n} ∈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. RESULTS

Before formulating the first theorem we recall a definition.

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

c_n≤K(c)c_m

holds for anymandm≤n ≤2m.These sequences will be denoted byc∈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|∆an|<∞,

then{a_n} ∈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. Ifc:={cn} ∈LAM S andαn := sup_k≥nck,then for anyδ >−1 (2.2)

∞

X

n=1

n^δα_n ≤K(K(c), δ)

∞

X

n=1

n^δc_n.

Proof. Sincec∈LAM Sthus withK :=K(c)

(2.3) α₂ⁿ = sup

k≥2ⁿ

c_k ≤ sup

m≥n

K c₂^m ≤Ksup

m≥n

c₂^m. IfP

n^δc_n <∞,thenc_n →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)α₂k ≤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 with n = 0,and repeat it withn+pin place of n,if p≥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 sumP∞

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 2.5. Following the steps of the proof it is easy to see that withϕ_nin place ofn^δ,(2.2) also holds if{ϕ_n} ∈LAM Sand2ⁿϕ₂ⁿ is quasi geometrically increasing.

3. PROOFS

Proof of Theorem 2.1. Using Lemma 2.4 withcn =anandδ= 0,we immediately get that (3.1)

∞

X

n=1

maxk≥n |∆a_k|<∞,

namely the assumptiona_n→0yields thatsup|∆a_k|= max|∆a_k|,and thus (3.1) implies that

{a_n} ∈S.

Proof of Theorem 2.2. With respect to the equality

|∆(n^βa_n)|=|n^β(a_n−a_n+1)−a_n+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|, whereK is 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|,

then this sequenceA^(γ)n is clearly monotonically decreasing, andA^(γ)n ≥ |∆an|,furthermore by the assumptions of Theorem 1.2 and (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} ∈Sγis proved. The proof is complete.

Proof of Remark 2.3. 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, Remark 2.3 is verified, namely we can also see that the condition (2.1) cannot be

weakened in general.

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. [ONLINEhttp://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 applications, Math. Ineq. and Appl., 4(2) (2001), 231–238.

[5] Ž. TOMOVSKI, Some results onL¹-approximation of the r-th derivative of Fourier series, J. In- equal. Pure and Appl. Math., 3(1) (2002), Article 10. [ONLINEhttp://jipam.vu.edu.au/

v3n1/005_99.html]