Analysis Mathematica, 38(2012), 123–133

DOI: 10.1007/s10476-012-0203-7

On L1-convergence of sine series

LASZLO LEINDLER

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

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

Received November 25, 2010; in revised form July 1, 2011.

Ab s t r a c t . Our aim is to find the source why the logarithm sequences play thecrucial role in the L1-convergence of sine series. We define three new classes of sequences;one of them has the character of the logarithm sequences, the other two are the extensionsof the class defined by Zhou and named Logarithm Rest Bounded Variation Sequences.In terms of these classes, extended analogues of Zhou’s theorems are proved.

1. Introduction

In a very recent paper, S. P. Zhou [3] defined the notion of LogarithmRest Bounded Variation Sequence (LRBVS) which plays central role in hispaper. Among others, he gave necessary and sufficient condition for theL1-convergence of the series

(1.1)∞∑

n=1

an sinnx,

assuming that a := {an} ∈ LRBVS, but without the prior condition thatthe sum function of (1.1) is integrable.

Furthermore, he established two theorems showing that the sequencea ∈ LRBVS cannot be replaced by a Quasi-Monotone Sequence or a notslowly increasing sequence like {nα}, α > 0; herewith verifying that the

0133–3852/$ 20.00c© 2012 Akademiai Kiado, Budapest

124 L. Leindler

condition a ∈ LRBVS cannot be weakened further in L1-convergence forsine series.

In a certain sense, this means that a ∈ LRBVS is an ultimate condition.The aim of the present paper is to find the source why the logarithm

sequences play the crucial role in Zhou’s theorems.Analyzing the proofs of the theorems of [3] we collect those properties

of the sequence {logN n} which are cardinal in these results, and show thatif a sequence has these essential properties, then all of the relevant resultsof Zhou hold for this sequence, too.

These sequences will be called Log-Type Sequences, in symbol LTS.For some classical and newer attendances pertaining to this topic, we

refer to References in [3].We would like to call the attention of the interested readers to the paper

of Tikhonov, cited in [3] as [11], and here as [2], where the author alsoproved interesting theorems pertaining to L1-convergence of trigonometricseries, and gave necessary and sufficient conditions, too. Notwithstanding,we recall no theorem of this work, because the coefficients of the seriesconsidered by Tikhonov satisfy different type of monotonicity assumptionsthan that of Zhou, and thus they are not related to the aim of the presentpaper.

2. Notions and notations

To establish Zhou’s main theorem and our theorems, we remembersome notions and notations.

First, we recall some definitions of generalization of decreasing mono-tonicity related to our topic.

A sequence a := {an} of positive numbers will be called Almost Mono-tone Sequence, briefly a ∈ AMS, if an ≤ Kam, for all n ≥ m, where K =K(a) is a positive constant.

A nonnegative sequence a := {an} is called Quasi-Monotone Sequenceif for some α ≥ 0, the sequence {ann−α} is monotone decreasing, in symbols:a ∈ QMS.

Let γ := {γn} be a given positive sequence. A null sequence a := {an}(an → 0) of real numbers satisfying the inequalities

∞∑

n=m

|Δan| ≤ K(a)γm (Δan := an − an+1), m = 1, 2, . . . ,

with a positive constant K(a) is said to be a sequence of γ rest boundedvariation, in symbol, a ∈ γRBVS.

L1-convergence of sine series 125

If γ ≡ a, then γRBVS reduces to RBVS, that is, to a rest boundedvariation sequence.

We emphasize that if a ∈ γRBVS, then it may have infinitely manyzeros and negative terms, but this is not the case if a ∈ RBVS, see, e.g., [1].

A nonnegative bounded sequence a := {an} is called Logarithm RestBounded Variation Sequence , in symbols: a ∈ LRBVSN , if N is a positiveinteger and the sequence {an log−N n} belongs to RBVS.

As usual, if f ∈ L2π then its L1-norm will be denoted by

‖f‖ := ‖f‖L1 :=

∫ 2π

0|f | dx.

The nth partial sum of series (1.1) will be denoted by sn(x).We shall also use the notion L � R (L R) at inequalities if there

exists a positive constant K such that L ≤ KR (KL ≥ R) holds, where Kis not necessarily the same at each occurrence.

Now, we define a certain unification of the logarithm sequences.A positive nondecreasing sequence α := {αn} will be called Log-Type

Sequence, briefly LTS, if it satisfies the conditions:

(2.1) αn → ∞,

(2.2) αn2 � αn,

and

(2.3) |Δαn| � αn

n log n.

By means of this sequence, we define two new classes of sequences.Let γ := {γn} be a given positive sequence. If

α := {αn} ∈ LTS and{ anαn

}∈ γRBVS,

then the sequence a := {an} will be called γ Log-Type Rest BoundedVariation Sequence, in symbols: a ∈ γLTRBVS.

If γn = an/αn , then the sequence a will be said simply Log-Type RestBounded Variation Sequence, in symbol: LTRBVS.

In other words, a ∈ LTRBVS if

α ∈ LTS and{ anαn

}∈ RBVS.

By definitions, the following embedding relations clearly follow:

(2.4) LRBVSN ⊂ LTRBVS ⊂ γLTRBVS.

We mention that Zhou [3] proved that the class RBVS is a propersubset of LRBVSN . Thus, by (2.4), RBVS is also a proper subset of theclasses defined newly.

126 L. Leindler

3. Theorems

First, we recall Zhou’s main theorem.

Theor em Z. Let a := {an} ∈ LRBVSN . Write

(3.1) g(x) =∞∑

n=1

an sinnx

at x, where it converges. Then

(3.2) limn→∞ ‖g − sn(g)‖ = 0

if and only if

(3.3)∞∑

n=1

ann

< ∞.

We generalize this theorem as follows.

Theor em 1. Let a ∈ LTRBVS, that is,

α := {αn} ∈ LTS and{ anαn

}∈ RBVS.

Then the assertions (3.2) and (3.3) are equivalent.

It is clear that if αn = (log n)N , then Theorem 1 reduces to Theorem Z.

The implication (3.3) ⇒ (3.2) has a further generalization as follows.

Theor em 2. Let

α := {αn} ∈ LTS and γ := {γn} ∈ AMS.

If an/αn ∈ γRBVS, that is, a ∈ γLTRBVS, and

(3.4)∞∑

n=1

αnγnn−1 < ∞,

then (3.2) holds.

It is easy to see that if γn = an/αn, then (3.4) reduces to (3.3).Consequently, Theorem 2, by (2.4), is a generalization of the implication(3.3) ⇒ (3.2) stated in Theorem 1.

The following two theorems are certain analogues of Zhou’s theoremsproved also in [3].

L1-convergence of sine series 127

Theor em 3. If α := {αn} is an increasing sequence and satisfies thecondition

(3.5)n2∑

k=n

αk

k� αn2 (log n � αn2

αn→ ∞),

then there exists a positive sequence a := {an} such that {an/αn} ∈ RBVSand (3.3) holds, but the condition (3.2) does not preserve.

Theor em 4. Let

α := {αn} ∈ LTS and{ anαn

}∈ RBVS,

that is, a ∈ LTRBVS. Then the condition

(3.6)∞∑

n=2

|Δan| log n < ∞

and condition (3.3) are equivalent.

We mention that in the special case αn = logN n, Theorem 4 reducesto Theorem 4 of [3].

4. Lemmas

The following assertions were proved in [3] implicitly or as a Proposi-tion.

Lemma 1. If a := {an} is a nonnegative sequence, then the condition∞∑

n=1

|Δan| log n < ∞

implies (3.3).

Lemma 2. If g(x) ∈ L2π , then the coefficients of series (1.1) are itsFourier coefficients, and (3.3) is satisfied .

5. Proofs of Theorems 1–4

Proo f o f Theor em 1. More or less we follow the proof of Theorem Z.First, we verify the implication (3.3) ⇒ (3.2). If m > n we can write

(5.1) sm(x)− sn(x) =m∑

k=n+1

akαk

αk sin kx =:∑

m,n.

128 L. Leindler

By Abel’s transformation we obtain that

(5.2)∑

m,n = − an+1

αn+1

n∑

k=1

αk sin kx+amαm

m∑

k=1

αk sin kx+

+m−1∑

ν=n+1

Δaναν

ν∑

k=1

αk sin kx =: I1(x) + I2(x) + I3(x).

Sincen∑

k=1

αk sin kx =n−1∑

k=1

Δαk

k∑

�=1

sin �x+ αn

n∑

�=1

sin �x,

we estimate as follows∫ π

0

∣∣∣n∑

k=1

αk sin kx∣∣∣ dx �

�n−1∑

k=1

|Δαk|∫ π

0

∣∣∣k∑

�=1

sin �x∣∣∣ dx+ αn

∫ π

0

∣∣∣n∑

�=1

sin �x∣∣∣ dx � αn log n .

Consequently, we have

(5.3)∫ π

0|I1(x)| dx � an+1

αn+1αn log n � an+1 log n,

(5.4)∫ π

0|I2(x)| dx � am logm,

and

(5.5)

∫ π

0|I3(x)| dx �

m−1∑

ν=n+1

∣∣∣Δaναν

∣∣∣∫ π

0

∣∣∣ν∑

k=1

αk sin kx∣∣∣ dx �

�m−1∑

ν=n+1

∣∣∣Δaναν

∣∣∣αν log ν =: An,m.

Denote

Rn :=∞∑

ν=n

∣∣∣Δaναν

∣∣∣.

Then, using the conditions {an/αn} ∈ RBVS and (2.3), we get

(5.6) An,m =m−1∑

ν=n+1

(Rν −Rν+1)αν log ν =

L1-convergence of sine series 129

=

∣∣∣∣Rn+1αn+1 log(n+ 1)−Rmαm logm+

+m−1∑

ν=n+1

Rν+1(αν+1 log(ν + 1)− αν log ν)

∣∣∣∣ �

� an+1 log(n+ 1) + am logm+m−1∑

ν=n+1

aν+1

αν+1

(|Δαν | log(ν + 1) +

αν

ν

)�

� an+1 log(n+ 1) + am logm+m−1∑

ν=n+1

aν+1

ν.

Next, we use the assumptions {an/αn} ∈ RBVS and (2.2), whence it followsthat

(5.7) an � aν for all√n ≤ ν ≤ n.

Namely, by {an/αn} ∈ RBVS, we have

anαn

� aναν

,

and, by (2.2), we have αn � αν , thus we conclude that

anαν

� aναν

.

By (3.3) and (5.7), it is easy to show that an log n tends to zero, namely

(5.8) an log n � an

n∑

k≥√n

1

k�

n∑

k≥√n

akk

→ 0.

Collecting the estimates (5.1)–(5.8), and applying Cauchy’s criterion,the implication (3.3) ⇒ (3.2) is verified.

It follows from Lemma 2 that assertion (3.2) implies (3.3).The proof of Theorem 1 is complete.

Proo f o f Theor em 2. The proof proceeds on the line of that ofTheorem 1, up to the estimate given (5.5). But now in (5.6) we utilize theassumption {an/αn} ∈ γRBVS, which conveys that

anαn

≤ Rn � γn,

whence it follows that

(5.9) an � αnγn, n = 1, 2, . . . .

130 L. Leindler

Putting these estimates into (5.6) gives

(5.10) An,m � αn+1γn+1 log(n + 1) + αmγm logm+m−1∑

ν=n+1

γν+1αν

ν.

Next, we verify that αnγn log n → 0. Indeed, by the assumptions wehave γ ∈ AMS, (3.4), (2.1) and (2.2),

(5.11) αnγn log n � αnγn

n∑

k≥√n

1

k�

n∑

k≥√n

1

kαkγk → 0,

where we used the following estimates: αn � αk due to (2.1) and (2.2), andγn � γk , which follows from γ ∈ AMS.

If we put the estimates (5.9) into (5.3) and (5.4), too, then the modifiedestimates (5.3), (5.4) and (5.10), by (5.11) and (3.4), imply the assertion ofTheorem 2.

Proo f o f Theor em 3. It is clear that the sequence α satisfying (3.5)does not belong to LTS, and requires more than αn2/αn → ∞, thus it isnot the best counterexample.

In this proof, for simplicity, we shall write√n instead of the integer

part of√n.

Next, we define three sequences {nk}, {mk} and {ρn} as follows. Let

n1 = 1, n2 = 2, for k ≥ 2 mk+1 = 2n2k and nk+1 = (mk+1)

2.

Furthermore, let

ρ1 = 1 and ρn := max{ρn−1,

αn

α√n

}if n ≥ 2.

We define the sequence {an} by means of these sequences . Let

an :=

⎧⎪⎨

⎪⎩

αn/(ρnαmk+1log n) if 2nk ≤ n < mk+1,

αn/(ρnk+1αmk+1

log nk+1) if mk+1 ≤ n < nn+1,

(log nk+1)−1 if nk+1 ≤ n < 2nk+1.

It is easy to see that the sequence {an/αn} is decreasing for 2nk ≤ n <nk+1. Since

αnk+1≥ ρnk+1

αmk+1,

we haveank+1

αnk+1

=1

αnk+1log nk+1

≤ 1

ρnk+1αmk+1

log nk+1=

ank+1− 1

αnk+1− 1

.

Herewith we have proved that the complete sequence {an/αn} is decreasing,consequently {an/αn} ∈ RBVS.

L1-convergence of sine series 131

Now, we turn to the proof of (3.3). To this effect, we set

∞∑

n=2n2

ann

=∞∑

k=2

(mk+1−1∑

n=2nk

+

nk+1−1∑

n=mk+1

+

2nk+1−1∑

n=nk+1

)ann

=:∑

1 +∑

2 +∑

3.

It is easy to estimate∑

1 and∑

3. Namely, by (3.5), ρn log n, thus wehave

∑1 =

∞∑

k=2

mk+1−1∑

n=2nk

αn

nρnαmk+1log n

�∞∑

k=2

mk+1−1∑

n=2nk

1

n log2 n< ∞

and

∑3 =

∞∑

k=2

2nk+1−1∑

n=nk+1

1

n log nk+1�

∞∑

k=2

1

log nk+1�

∞∑

k=2

1

n2k

< ∞.

Finally, in the estimation of∑

2 we utilize the full strength of the condition(3.5). By (3.5) and

αnk+1� ρnk+1

αmn+1,

we get

∑2 =

∞∑

k=2

nk+1−1∑

n=mk+1

αn

n· 1

ρnk+1αmk+1

log nk+1�

�∞∑

k=2

αnk+1

ρnk+1αmk+1

log nk+1�

∞∑

k=2

1

log nk+1< ∞.

Herewith we proved (3.3).Next, we show that

‖f − sn(g)‖ �→ 0.

Utilizing the following function

φn(x) :=n∑

k=1

(sin(n + k)x

k− sin(n− k)x

k

)= 2cosnx

n∑

k=1

sin kx

k

and the well-known inequality

∣∣∣n∑

k=1

sin kx

k

∣∣∣ � 1,

we obtain that

(5.12) I :=

∫ 2π

0φnk+1

(x)(s2nk+1(x)− snk+1

(x)) dx �

�∫ 2π

0|s2nk+1

(x)− snk+1(x)| dx,

132 L. Leindler

while for k ≥ 2,

(5.13) I =

nk+1∑

i=1

ank+1+i

i 1

log nk+1

nk+1∑

i=1

1

i 1.

By (5.12) and (5.13), we have

‖s2nk+1− snk+1

‖ 1,

this shows that (3.2) does not hold.The proof of Theorem 3 is complete.

Proo f o f Theor em 4. The implication (3.6) ⇒ (3.3) follows fromLemma 1.

The proof of the assertion (3.3) ⇒ (3.6) is similar to that of Theorem 1,and we shall use some parts from its proof. Denote

Rk =∞∑

n=k

∣∣∣Δanαn

∣∣∣.

Sinceak − ak+1 = αk+1Δ

akαk

− akαk

(αk+1 − αk),

by (2.3), we get

(5.14)m∑

k=2

|Δak| log k �m∑

k=2

∣∣∣∣Δakαk

∣∣∣∣αk+1 log k +m∑

k=2

akαk

|Δαk| log k �

�n∑

k=2

(Rk −Rk+1)αk+1 log k +m∑

k=2

akk.

Now, we repeat the same consideration as we did in (5.6), thus we have

(5.15)m∑

k=2

(Rk −Rk+1)αk+1 log k � am+1 log(m+ 1) +m∑

k=2

ak+1

k.

The assumptions are the same as in Theorem 1, thus (5.8) can be used toget that

(5.16) am+1 log(m+ 1) → 0.

Summing up (5.14)–(5.16), we obtain that

m∑

k=2

|Δak| log k �m+1∑

k=2

akk

,

herewith the implication (3.3) ⇒ (3.6) is verified.The proof of Theorem 4 is complete.

L1-convergence of sine series 133

References

[1] L. Leindler, Embedding results regarding strong approximation of sine series, Acta

Sci. Math. (Szeged), 71(2005), 91–103.

[2] S. Tikhonov, On L1-convergence of Fourier series, J. Math. Anal Appl., 347(2008),

416–427.

[3] S. P. Zhou, What condition can correctly generalizes monotonicity in L1-convergence

of sine series?, Science China Chinise Ed., 40(2010), 801–812 (in Chinese) (obviously

the author has read only the English translation of [3] as the referee of a recent joint

paper of Zhou).

O shodimosti sinus-r�dov v L1

L. LE �INDLER

Naxa cel� – opredelit� priqinu, v silu kotoro�i logarifmiqeskie posle-dovatel�nosti igra�t rexa�wu� rol� v L1-shodimosti sinus-r�dov. Dl� �togomy vvodim v rassmotrenie tri novyh klassa posledovatel�noste�i; odin iz nihlogarifmiqeskogo tipa, a drugie dva �vl��ts� rasxireni�mi klassov, kotoryebyli vvedeny �u i poluqili nazvanie posledovatel�noste�i ograniqenno�i varia-cii logarifmiqeskogo ostatka. V terminah �tih novyh klassov poluqeny obob-wennye analogi teorem �u.