Analysis Mathematica, 34(2008), 137–144

DOI: 10.1007/s10476-008-0204-8

Comments regarding the Sidon–Telyakovskiı class

LASZLO LEINDLER

Bolyai Institute, University of Szeged, Aradi vertanuk tere 1,

6720 Szeged, Hungary, e-mail: leindler@math.u-szeged.hu

Received November 3, 2006.

Dedicated to the memory of Professors K. Tandori and P. L. Ul’yanov

Ab s t r a c t . The aim of the paper is to analyze the relationship of some newlydefined classes of numerical sequences to the important Sidon–Telyakovskiı class.

1. Introduction

Several mathematicians have studied the convergence properties oftrigonometric series. They established their conditions for the coefficientsvia monotone, quasi-monotone, convex and quasi-convex sequences; and in1939 Sidon [11] introduced a respectable, but tricky sequence.

Telyakovskiı [12] redefined the Sidon’s sequence and gave a veryeffective, synoptic and flourishing definition. He denoted this class by S ,referring to Sidon.

His definition reads as follows: A null sequence a := {an} belongs tothe class S , in symbol: a ∈ S , if there exists a monotonically decreasingsequence {An} such that

∞X

n=1

An < ∞ and |∆ an| ≤ An hold for all n.

This research was partially supported by the Hungarian National Foundation for

Scientific Research under Grant # T 042 462.

0133–3852/$ 20.00c© 2008 Akademiai Kiado, Budapest

138 L. Leindler

The class S plays a very important role in many problems. Therefore,a great number of mathematicians have wanted to extend this class, anddefined “wider” classes and proved that the class S can be replaced by their“wider” class having the same conclusion.

In [3] we showed that some of these classes are identical with the class S .Furthermore, in [4] we proved that five other classes defined by different waysare wider than S , but they are identical among themselves.

Recently regarding the generalization of the classical result of Chaundy

and Jolliffe [1], several new classes of sequences were defined, graduallygeneralized the notion of monotonicity.

For example, in [5] we defined the class of sequences of rest boundedvariation, or briefly: the class of RBVS. Just after the appearance of ourpaper, Lee and Zhou [2] defined the class GBVS. Next, in [6] we introducedthe class γRBVS. Its definition reads as follows:

Let γ := {γn} be a positive sequence. A null sequence c := {cn} ofreal numbers is said to be a γRBVSequence if

(1.1)∞

X

n=m

|∆ cn| ≤ K(c)γm, m = 1, 2, . . . ,

where K(c) is a positive constant.

With

γm := m−1

2m−1X

n=m

cn,

assuming that all cn ≥ 0, in [9] we defined the class MRBVS, referring tothe word “mean”.

If γ ≡ c and cn > 0, then γRBVS ≡ RBVS. If

γm := maxm≤n<m+N0

|cn| (N0 ∈ N, fixed),

and if the inequality

(1.2)2mX

n=m

|∆ cn| ≤ K(c)cm

is satisfied, we get the class GBVS; and the special case N0 = 1 of GBVSwill be denoted by G1BVS. If γ := {γn ≥ 0} is an arbitrary nonnegativesequence and c satisfies (1.2) we obtain the class γGBVS (see [8]). In thesenotations the letter G refers to the word “group”.

Comments regarding the Sidon–Telyakovskiı class 139

Analogously to MRBVS, in [10] we defined the class MGBVS, that is,c ∈ MGBVS, if (1.2) holds with

γm = m−1

2m−1X

n=m

cn.

It is clear that MRBVS ⊂ MGBVS.In the cited papers, the interested readers can find the results proved

for these classes. Here we do not recall them.The aim of the present paper is to investigate the relation of the newer

classes MRBVS, MGBVS and G1BVS to the celebrated class S . They arecertainly not identical, but have common character.

We observe that very recently Tikhonov [13] also investigated the classG1BVS. He denoted it by GN, and called it the class of general monotonicsequences.

Later on, we shall use the notion L ≪ R for inequalities if there existsa positive constant K such that L ≤ KR holds. Naturally, the notation≪ between the terms of sequences, for example, an ≪ bn means that an ≤Kbn holds with the same constant K for every n.

2. Results

We will prove the following theorems.

Theor em 2.1. (i) If a sequence a := {an} belongs to the class

MGBVS and

(2.1) A :=∞

X

n=1

an < ∞,

then a belongs to the class S.(ii) If A = ∞, then a ∈ MRBVS ⊂ MGBVS, does not imply that

a ∈ S even under the additional condition

2nX

k=n

ak → 0.

(iii) The assumptions A < ∞ and a ∈ S do not imply a ∈ MGBVS,

consequently a 6∈ MRBVS.

Theor em 2.2. If a ∈ G1BVS and (2.1) holds, then a ∈ S.

Addendum 2.3. If a ∈ MRBVS and (2.1) holds, then a ∈ S.

140 L. Leindler

Addendum 2.3 is proved by means of Theorem 2.1 Part (i), but belowwe will give another easy proof for it.

3. An auxiliary result

Lemma 3.1 (see [6, Lemma 2.4]). If a sequence c := {cn} has the

property that

cn ≤ K(c)cm for all m ≤ n ≤ 2m,

then∞

X

n=1

c∗n ≪∞

X

n=1

cn , where c∗n := supk≥n

ck.

4. Proofs

Proo f o f Theor em 2.1. First, we prove statement (i). The assump-tion a ∈ MGBVS yields that

(4.1) ∆2m := max2m≤k≤2m+1

|∆ ak| ≤ K(c)2−m2

m+1−1X

k=2m

ak =: σm.

On the other hand, (2.1) implies that σm converges to zero. By (4.1), |∆ ak|also converges to zero.

Utilizing these facts, we define the following monotone decreasing se-quence:

αm := maxk≥2m

|∆ ak| = maxn≥m

∆2n , m = 1, 2, . . . .

Now, we can define the terms An of a sequence required in the definitionof S . Let

An := αm for 2m ≤ n < 2m+1, m = 0, 1, 2, . . . .

It is clear thatAn ≥ An+1 and |∆ an| ≤ An.

Finally, we show that

(4.2)∞

X

n=1

An =∞

X

m=0

2mαm < ∞.

Let 1 ≤ ν1 < ν2 < · · · < νm < · · · denote those natural numbers n for whichαn = ∆2n , that is, ανm

= ∆2νm . If νm+1 = νm + µ, then we can easily seethat

ανm+1 = ανm+2 = · · · = ανm+µ = ανm+1,

Comments regarding the Sidon–Telyakovskiı class 141

thus for all m ≥ 1,

νm+1X

ℓ=νm+1

2ℓαℓ = ανm+1

νm+1X

ℓ=νm+1

2ℓ < 2ανm+12νm+1 .

Hence, by (2.1) and (4.1), we obtain that

∞X

n=2ν1+1

An =∞

X

ℓ=ν1+1

2ℓαℓ =∞

X

m=1

νm+1X

ℓ=νm+1

2ℓαℓ < 2∞

X

m=1

2νm+1ανm+1<

< 2∞

X

m=1

2νm∆2νm ≤ 2K(c)∞

X

m=1

2νm+1

X

k=2νm

ak < ∞.

Herewith (4.2) is verified. Consequently, the sequence a satisfies the fourconditions needed in the definition of the sequences of S , namely an → 0clearly follows from (2.1), that is, a ∈ S is proved.

Next, we prove statement (ii). First we define the following sequence:

(4.3) an :=

0 if n = 2m,1

m2m if 2m < n < 2m+1,

m = 1, 2, . . . .

It is plain thatX

an = ∞

and

∞X

k=n

|∆ ak| ≪∞

X

k=m

2k

k≪

2m

m≪ n−1

2n−1X

k=n

an for 2m ≤ n < 2m+1,

that is, a ∈ MRBVS ⊂ MGBVS. Moreover, we have

2nX

k=n

ak → 0.

Since

|∆ a2m+1−1| =1

m2m,

for the sequence a given in (4.3), there exists no sequence {An} satisfyingthe crucial three conditions

(4.4) An ≥ An+1, |∆ an| ≤ An and∞

X

m=1

2mA2m < ∞

simultaneously, which are required to the relation a ∈ S . Thus, this sequencea 6∈ S.

142 L. Leindler

Finally, we verify relation (iii). To this effect, we consider the followingsequence:

an :=

2−2m if 2m < n ≤ 2m + 2m/m,0 if 2m + 2m/m < n ≤ 2m+1.

It is obvious that (2.1) holds, and with

An := 2−2m for 2m ≤ n < 2m+1, m = 0, 1, 2, . . . ,

the conditions in (4.4) are satisfied. Thus a ∈ S .At the same time, we have

2m+1

X

k=2m

|∆ ak| ≥ 2−2m

and

2−m2

m+1−1X

k=2m

ak ≪1

m2−2m.

Thus a 6∈ MGBVS, and consequently a 6∈ MRBVS.The proof of Theorem 2.1 is complete.

Proo f o f Theor em 2.2. The assumption a ∈ G1BVS clearly implies

an ≤2mX

ν=n

|∆ aν | + a2m+1 ≪ am + a2m+1 for m ≤ n ≤ 2m

and

a2m+1 ≤ am +2mX

ν=m

|∆ aν | ≪ 2am ≪ am.

Thus, we havean ≪ am for all m ≤ n ≤ 2m.

Herewith it is proved that if a ∈ G1BVS, then it is also locally almost

monotone, that is,a ∈ G1BVS =⇒ a ∈ LAMS.

Thus, we can use Lemma 3.1 with this a. If we define An to be

An := supk≥n

ak = maxk≥n

ak ≥ An+1,

then by Lemma 3.1, we have∞

X

n=1

An < ∞.

Furthermore, we also have

|∆ an| ≤2n

X

k=n

|∆ ak| ≪ an ≤ An.

The last three relations clearly imply a ∈ S , and the proof is complete.

Comments regarding the Sidon–Telyakovskiı class 143

Although the Addendum 2.3 is a consequence of Theorem 2.1, we givean easy direct proof.

Proo f o f Addendum 2.3. First we show that a ∈ MRBVS implies

(4.5) k−1

2kX

ν=k

aν ≪ ℓ−1

2ℓX

ν=ℓ

aν for all k ≥ ℓ.

If n ≥ ℓ, then we have

ℓ−1

2ℓX

ν=ℓ

aν ≫∞

X

ν=ℓ

|∆ aν | ≥∞

X

ν=n

|∆ aν | ≥ an.

Thus, we also have2k

X

n=k

ℓ−1

2ℓX

ν=ℓ

aν ≫2k

X

n=k

an,

whence (4.5) obviously follows.Let

An := supm≥n

m−1

2mX

ν=m

aν.

These An, by Lemma 3.1, clearly satisfy the conditions in (4.4), consequentlya ∈ S is proved.

Remarks. Inequality (4.5) shows that a ∈ MRBVS implies that the“means of the blocks”, that is, the following means

n−1

2nX

ν=n

aν

determine an almost monotone decreasing sequence, in symbol:

a ∈ MRBVS =⇒ a ∈ AMS.

Furthermore, considering (4.5), it is easy to verify that if {nan} ∈MRBVS, then the “blocks”

2nX

ν=n

aν

themselves are almost monotone decreasing.

144 L. Leindler: Comments regarding the Sidon–Telyakovskiı class

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Zameqanie o klasse Sidona–Tel�kovskogo

L. LE INDLER

Celь� raboty �vl�ets� analiz sv�zi vvedennyh klassov qislovyh posle-

dovatelьnostei s klassom Sidona–Tel�kovskogo.