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# Extensions of Móricz Classes and Convergence of Trigonometric Sine Series in L1-Norm

Thapar Institute of Engineering and Technology, Patiala, Punjab 147004, India
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Author to whom correspondence should be addressed.
Mathematics 2018, 6(12), 292; https://doi.org/10.3390/math6120292
Received: 11 September 2018 / Revised: 14 November 2018 / Accepted: 21 November 2018 / Published: 29 November 2018
(This article belongs to the Special Issue Harmonic Analysis)

## Abstract

In this paper, the extensions of classes and $B ˜ V$ are made by defining the classes $S ˜ r$ , $C ˜ r$ and $B ˜ V r$ , $r = 0 , 1 , 2 , …$ It is also shown that class $S ˜ r$ is a subclass of $C ˜ r ∩ B ˜ V r$ . Moreover, the results on $L 1$ -convergence of r times differentiated trigonometric sine series have been obtained by considering the $r t h ( r = 0 , 1 , 2 , … )$ derivative of modified sine sum under the new extended class $C ˜ r ∩ B ˜ V r$ .
Keywords:

## 1. Introduction

Consider the trigonometric sine series
$∑ k = 1 ∞ a k sin k x$
where $a 0 , a 1 , a 2 , …$ are the real coefficients. The nth partial sum, $S n$, of Series (1) is represented as
$S n ( x ) = ∑ k = 1 n a k sin k x = − ∑ k = 1 n b k ( cos k x ) ′$
where the prime denotes derivatives and $b k = a k k$. Also, $f ( x ) = lim n → ∞ S n ( x )$.
Various conditions are given in the literature (see [1,2,3,4,5,6,7,8,9]), which guarantee that Series (1) is a Fourier series.
In $1984 ,$ Teljakovskii  introduced a class $S ˜$, as follows:
Class $S ˜$ . A null sequence ${ a k }$ is said to belong to class $S ˜$ if there exists a non-increasing sequence ${ B k }$ of numbers s.t.
$| Δ b k | ≤ B k ∀ k = 1 , 2 , 3 , … ∑ k = 1 ∞ k B k < ∞ .$
where $b k = a k k ,$ $Δ b k = b k − b k + 1$ and proved the following result:
Theorem 1 .
If ${ a k } ∈ S ˜ ,$ then Series (1) is the Fourier series of some function$f ∈ L 1 ( 0 , π ) .$
In $1989 ,$ Móricz  introduced new classes $B ˜ V$ and $C ˜$ of the coefficient sequences for the sine series.
Class $B ˜ V$.
A null sequence ${ a k }$ belongs to $B ˜ V$ if
$∑ k = 1 ∞ k | Δ b k | < ∞$
Class $C ˜$.
A null sequence ${ a k }$ belongs to class $C ˜$ if for every $ε > 0$ there exists $δ > 0 ,$ independent of $n ,$ and such that for all $n ,$
$∫ 0 δ | ∑ k = n ∞ Δ b k D k ′ ( x ) | d x ≤ ε .$
Here, $D k ′ ( x )$ is the first derivative of Dirichlet kernel $( D k ( x ) = sin ( k + 1 2 ) x 2 sin x 2 )$.
Equation (4) implies that, for $1 ≤ n ≤ N$,
$∫ 0 δ | ∑ k = n N Δ b k D k ′ ( x ) | d x ≤ 2 ε .$
The following result was proved by Móricz .
Theorem 2 .
If${ a k } ∈ B ˜ V ,$then
where$u n ( x ) = S n ( x ) + b n + 1 D n ′ ( x )$.
The classes $S ˜ , B ˜ V$ and $C ˜$ seem to be more appropriate for the sine series than the classes $S$ ([7,8]) $B V$ , and $C$  in the ordinary sense. Also, Móricz  has proved that $S ˜ ⊂ B ˜ V ∩ C ˜$.
Motivated by the aforesaid authors, new extended classes $S ˜ r , B ˜ V r$, and $C ˜ r$ ($r = 0 , 1 , 2 , …$) are defined in this paper as follows:
Class$S ˜ r$.
A sequence ${ a k }$ is said to belong to class $S ˜ r$ ($r = 0 , 1 , 2 , …$) if $a k → 0$ as $k → ∞$, and there exists a non-increasing sequence ${ B k }$ of numbers s.t.
$| Δ b k | ≤ B k ∀ k = 1 , 2 , 3 , … ∑ k = 1 ∞ k r + 1 B k < ∞ , r = 0 , 1 , 2 , 3 , …$
where $b k = a k k , r = 0 , 1 , 2 , 3 , …$
$B k ↓ 0$ and $∑ k = 1 ∞ k r + 1 B k < ∞ ,$ implies that $k r + 2 B k = o ( 1 )$ as .
Remark 1.
For$r = 0 ,$$S ˜ r = S ˜ .$
Remark 2.
Obviously, $S ˜ r + 1 ⊂ S ˜ r$, but the converse need not be true.
Example 1.
Consider a sequence $Δ b n = 1 n r + 3 ,$ $r = 0 , 1 , 2 , …$ and
Choose $B n = 1 n r + 3 , r = 0 , 1 , 2 , … ∀ n .$ Clearly, $B n ↓ 0$ as $n → ∞$ and $| Δ b n | ≤ B n ∀ n .$
Consider the series
This implies ${ a n } ∈ S ˜ r .$
But the series $∑ n = 1 ∞ n r + 2 B n ≈ ∑ n = 1 ∞ 1 n$ is divergent.
This implies that ${ a n }$ does not belong to class $S ˜ r + 1$.
Class$B ˜ V r$.
A null sequence ${ a k }$ belongs to $B ˜ V r , ( r = 0 , 1 , 2 , … )$ if
$∑ k = 1 ∞ k r + 1 | Δ b k | < ∞$
Remark 3.
For$r = 0 ,$$B ˜ V r = B ˜ V .$
Remark 4.
Clearly,$B ˜ V r + 1 ⊂ B ˜ V r , ( r = 0 , 1 , 2 , … )$, but the converse may not be true.
Class$C ˜ r$.
A null sequence ${ a k }$ belongs to class $C ˜ r$ ($r = 0 , 1 , 2 , …$), if for every $ε > 0$, there exists $δ > 0 ,$ independent of $n ,$ and such that for all $n ,$
$∫ 0 δ | ∑ k = n ∞ Δ b k D k r + 1 ( x ) | d x ≤ ε$
Here, $D k r + 1 ( x )$ is the $( r + 1 ) t h$ derivative of Dirichlet kernel.
Equation (4) implies, for $1 ≤ n ≤ N ,$
$∫ 0 δ | ∑ k = n N Δ b k D k r + 1 ( x ) | d x ≤ 2 ε$
Remark 5.
For$r = 0 ,$$C ˜ r = C ˜ .$
Remark 6.
It is obvious that$C ˜ r + 1 ⊂ C ˜ r$but the converse need not be true.
Example 2.
Define$Δ b n = 1 n r + 3 , r = 0 , 1 , 2 , …$and$n = 1 , 2 , 3 , …$
Consider, the integral
which is divergent.
However,
Therefore ${ a n } ∈ C ˜ r$.
Lemmas related to the main results are given in Section 2. The Section 3 comprises the main results of this paper. Firstly, in this section, we have shown that the new extended class $S ˜ r$ is a subclass of $C ˜ r ∩ B ˜ V r ( r = 0 , 1 , 2 , … )$. Moreover, the theorems are presented concerning the $L 1$ convergence of trigonometric sine series using modified sine sum , defined as
under the extended classes of numerical sequences.

## 2. Lemmas

Lemma 1.
 Let$n ≥ 1$and$r$be a nonnegative integer$x ∈ [ ε , π ]$. Then,$| D n r ( x ) | ≤ C n r x ,$where$C$denotes a positive absolute constant.
Lemma 2.
$‖ D n r ( x ) ‖ L 1 = O ( n r log n ) , r = 0 , 1 , 2 , …$where$D n r ( x )$represents the$r t h$derivative of the Dirichlet kernel.

## 3. Main Results

Theorem 3.
The following relation holds$S ˜ r ⊂ C ˜ r ∩ B ˜ V r$for each$r ∈ { 0 , 1 , 2 , … } .$
Proof.
It is plain that $S ˜ r ⊂ B ˜ V r$.
In order to prove that $S ˜ r ⊂ C ˜ r$ we take a sequence ${ a k }$ in $S ˜ r$ and consider
If we apply summation by parts, we obtain
$∫ 0 π | ∑ k = n ∞ Δ b k D k r + 1 ( x ) | d x ≤ lim N → ∞ [ ∑ k = n N − 1 Δ B k ∫ 0 π | ∑ j = 0 k Δ b j B j D j r + 1 ( x ) | d x + B N ∫ 0 π | ∑ K = 0 N Δ b k B k D k r + 1 ( x ) | d x + B n ∫ 0 π | ∑ K = 0 n − 1 Δ b k B k D k r + 1 ( x ) | d x ]$
Clearly $| Δ b k B k | ≤ 1$. Now, if we first apply Bernstein’s inequality  and then Sidon Fomin’s inequality ([1,7]), we get
$∫ 0 π | ∑ k = 0 n Δ b k B k D k ( r + 1 ) ( x ) | d x ≤ M ( n + 1 ) r + 2 , r = 0 , 1 , 2 , …$
So, by given hypothesis, we have
For any $1 ≤ n ≤ N ,$ we can estimate as follows:
$∫ 0 δ | ∑ k = n N Δ b k D k r + 1 ( x ) | d x ≤ ∫ 0 δ | ∑ k = n n 0 Δ b k D k r + 1 ( x ) | d x + ∫ 0 δ | ∑ k = n 0 N Δ b k D k r + 1 ( x ) | d x ≤ 1 2 δ ∑ k = 1 n 0 k ( k + 1 ) r + 1 | Δ b k | + ε 2 < ε$
provided $δ$ is small enough. This proves that ${ a k } ∈ C ˜ r$. □
Theorem 4.
Let ${ a k }$ be a sequence of numbers belonging to the class $C ˜ ∩ B ˜ V$ and if $lim n → ∞ a n log n = 0 ,$ then
Proof.
The modified trigonometric sine sum is given by
By using the summation by parts, we get
$β n = − ∑ k = 1 n Δ b k D k ′ ( x ) − b n D n ′ ( x ) − ( b n + 2 − b n + 1 ) D n ′ ( x )$
Under the given hypothesis and Lemma 1, series $∑ k = 1 n Δ b k D k ′ ( x )$ converges absolutely and $b n D n ′ ( x ) → 0$ as $n → ∞$.
Hence $lim n → ∞ β n ( x ) = f ( x )$ exists in $( 0 , π )$.
Next, consider
$‖ f ( x ) − β n ( x ) ‖ = ‖ ∑ k = n + 1 ∞ a k sin k x − ( a n + 2 n + 2 − a n + 1 n + 1 ) ∑ k = 1 n k sin k x ‖ = ∫ 0 π | − ∑ k = n + 1 ∞ b k ( cos k x ) ′ − ( b n + 1 − b n + 2 ) D n ′ ( x ) | d x$
By using Abel’s transformation, we have
$= ∫ 0 π | − ∑ k = n + 1 ∞ Δ b k D k ′ ( x ) + b n + 2 D n ′ ( x ) | d x = ∫ 0 π | ∑ k = n + 1 ∞ Δ b k D k ′ ( x ) | d x + n n + 2 a n + 2 log n$
The second term of the above equation is of $o ( 1 )$ as . For the remaining part, let $ε > 0 ,$ then there exists $δ > 0$, such that
Then
This proves that . □
Theorem 5.
Let ${ a k }$ be a sequence of numbers belonging to the class $C ˜ ∩ B ˜ V$, and if $lim n → ∞ a n log n = 0 ,$ then
$‖ S n − f ‖ = o ( 1 ) , n → ∞ .$
Proof.
$‖ S n − f ‖ ≤ ‖ S n − β n ‖ + ‖ β n − f ‖$
Theorem 6.
Let ${ a k }$ be a sequence of numbers belonging to the class $C ˜ r ∩ B ˜ V r$ and if Then
$‖ β n r ( x ) − f r ( x ) ‖ = o ( 1 ) , n → ∞$
Here, $f r ( x )$ is the rth derivative of f(x), where$r = 0 , 1 , 2 , …$
Proof.
Consider the modified trigonometric sine sum as
Taking r-times differentiation of $β n ( x ) ,$ we get
$β n r ( x ) = S n r ( x ) + ( a n + 2 n + 2 − a n + 1 n + 1 ) ∑ k = 1 n k r + 1 sin ( k x + r π 2 ) = ∑ k = 1 n k r a k sin ( k x + r π 2 ) + ( a n + 1 n + 1 − a n + 2 n + 2 ) ∑ k = 1 n k r + 1 cos ( k x + ( r + 1 ) π 2 ) = − ∑ k = 1 n k r + 1 b k cos ( k x + ( r + 1 ) π 2 ) + ( b n + 1 − b n + 2 ) D n r + 1 ( x )$
If we apply Abel’s transformation on the first term of above equation, we get
$β n r ( x ) = − ∑ k = 1 n − 1 Δ b k D k r + 1 ( x ) − b n D n r + 1 ( x ) + ( b n + 1 − b n + 2 ) D n r + 1 ( x ) = − ∑ k = 1 n Δ b k D k r + 1 ( x ) − b n + 2 D n r + 1 ( x )$
The series $∑ k = 1 ∞ Δ b k D k r + 1 ( x )$ converges absolutely and $b n D n r + 1 ( x ) → 0$ as $n → ∞$ using Lemma 1 and given hypothesis.
Therefore $lim n → ∞ β n r ( x ) = f r ( x )$ exists in $( 0 , π )$.
Next, consider
$‖ f ( x ) − β n ( x ) ‖ = ‖ ∑ k = n + 1 ∞ a k sin k x − ( a n + 2 n + 2 − a n + 1 n + 1 ) ∑ k = 1 n k sin k x ‖ ‖ f r ( x ) − β n r ( x ) ‖ = ‖ ∑ k = n + 1 ∞ k r a k sin ( k x + r π 2 ) − ( a n + 2 n + 2 − a n + 1 n + 1 ) ∑ k = 1 n k r + 1 sin ( k x + r π 2 ) ‖ = ‖ ∑ k = n + 1 ∞ k r a k sin ( k x + r π 2 ) + ( a n + 2 n + 2 − a n + 1 n + 1 ) ∑ k = 1 n k r + 1 cos ( k x + ( r + 1 ) π 2 ) ‖ = ∫ 0 π | − ∑ k = n + 1 ∞ k r + 1 b k cos ( k x + ( r + 1 ) π 2 ) + ( b n + 2 − b n + 1 ) D n r + 1 ( x ) | d x$
If we apply Abel’s transformation, we obtain
$= ∫ 0 π | − ∑ k = n + 1 ∞ Δ b k D k r + 1 ( x ) + b n + 1 D n r + 1 ( x ) − b n + 1 D n r + 1 ( x ) + b n + 2 D n r + 1 ( x ) | d x ≤ ∫ 0 π | ∑ k = n + 1 ∞ Δ b k D k r + 1 ( x ) | d x + | b n + 2 | ∫ 0 π | D n r + 1 ( x ) | d x ≤ ∫ 0 π | ∑ k = n + 1 ∞ Δ b k D k r + 1 ( x ) | d x + a n + 2 n + 2 n r + 1 log n$
The second term of the above equation are of o(1) as $n r a n log n = 0$ as $n → ∞$. For the remaining part, let $ε > 0$, then there exists $δ > 0$, such that $∫ 0 δ | ∑ k = n + 1 ∞ Δ b k D k r + 1 ( x ) | d x < ε / 2$ for all $n ≥ 0$. Then
Therefore, $‖ f r ( x ) − β n r ( x ) ‖ L 1 = o ( 1 )$ as $n → ∞$. □
Remark 7.
For $r = 0$, Theorem 6 reduces to Theorem 4.
Theorem 7.
Let ${ a k }$ be a sequence of numbers belonging to the class $C ˜ r ∩ B ˜ V r$ and if $n r a n log n = o ( 1 )$ as $n → ∞$. Then
where$r = 0 , 1 , 2 … .$
Proof.
$‖ S n r − f r ‖ ≤ ‖ S n r − β n r ‖ + ‖ β n r − f r ‖$
Remark 8.
For $r = 0$, Theorem 7 reduces to Theorem 5.
Remark 9.
Combining Theorem 6 and Theorem 7 with Theorem 3, the following result holds:
Corollary 1.
If ${ a k } ∈ S ˜ r ( r = 0 , 1 , 2 , 3 , … )$ and if $n r a n log n = o ( 1 )$ as $n → ∞$. Then
(i)
(ii)

## Author Contributions

All authors have contributed in obtaining the new results presented in this article. All authors read and approved the final manuscript. Investigation, S.K.C.; Supervision, J. K. and S.S.B.

## Funding

This research received no external funding.

## Conflicts of Interest

The authors declare that they have no conflicts of interest.

## References

1. Fomin, G.A. On linear method for summing Fourier series. Mat. Sb (Russ.) 1964, 64, 144–152. [Google Scholar]
2. Fomin, G.A. A class of trigonometric series. Math. Notes 1978, 23, 117–123. [Google Scholar] [CrossRef]
3. Garrett, J.W.; Stanojevic, C.V. Necessary and Sufficient conditions for L1-convergence of trigonometric series. Proc. Am. Math. Soc. 1976, 60, 68–71. [Google Scholar]
4. Kano, T. Coefficients of some trigonometric series. J. Fac. Sci. Shinshu Univ. 1968, 3, 153–162. [Google Scholar]
5. Móricz, F. On the integrability and L1-convergence of sine series. Studia Math. 1989, 335, 187–200. [Google Scholar] [CrossRef]
6. Sheng, S.Y. The extension of the theorems of Č.V. Stanojevic and V.B. Stanojevic. Proc. Am. Math. Soc. 1990, 110, 895–904. [Google Scholar] [CrossRef]
7. Sidon, S. Hinreichende Bedingungen fur den Fourier-charakter einer trigonometrischen Reihe. J. Lond. Math. Soc. 1939, 14, 158–160. [Google Scholar] [CrossRef]
8. Telyakovskii, S.A. On a sufficient condition of Sidon for integrability of trigonometric series. Mat. Zametki 1973, 14, 317–328. [Google Scholar] [CrossRef]
9. Telyakovskii, S.A. On the integrability of sine series. Trudy Mat. Inst. Steklov. 1984, 163, 229–233. [Google Scholar]
10. Bary, N.K. A Treatise on Trigonometric Series; Pergamon Press: London, UK, 1964; Volumes I, II. [Google Scholar]
11. Chouhan, S.K.; Kaur, J.; Bhatia, S.S. Convergence and Summability of Fourier sine and cosine series with its applications. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 2018, 1–8. [Google Scholar] [CrossRef]
12. Zygmund, A. Trigonometric Series; Cambridge University Press: Cambridge, UK, 1959. [Google Scholar]
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