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

Abstract

In this paper, the extensions of classes $\tilde{S},\tilde{C}$ and $\tilde{B}V$ are made by defining the classes ${\tilde{S}}_r$, ${\tilde{C}}_r$ and $\tilde{B}{V}_r$, $r=0,1,2,\dots$ It is also shown that class ${\tilde{S}}_r$ is a subclass of ${\tilde{C}}_r{\displaystyle \cap \tilde{B}{V}_r}$. Moreover, the results on $L^1$-convergence of r times differentiated trigonometric sine series have been obtained by considering the $r^{th}(r=0,1,2,\dots )$ derivative of modified sine sum under the new extended class ${\tilde{C}}_r{\displaystyle \cap \tilde{B}{V}_r}$.

1. Introduction

Consider the trigonometric sine series

$${\displaystyle {\displaystyle \sum _{k=1}^{\infty }}{a}_k\sin kx}$$

(1)

where $a_0,a_1,a_2,\dots$ are the real coefficients. The nth partial sum, $S_n$, of Series (1) is represented as

$$S_n(x)={\displaystyle {\displaystyle \sum _{k=1}^n}{a}_k\sin kx}=-{\displaystyle {\displaystyle \sum _{k=1}^n}{b}_k(\cos kx){}^{\prime }}$$

(2)

where the prime denotes derivatives and $b_k=\frac{a_k}{k}$. Also, $f(x)=\underset{n\to \infty }{{\displaystyle \lim }}{S}_n(x)$.

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 [9] introduced a class $\tilde{S}$, as follows:

Class $\tilde{S}$ [9]. A null sequence $\left\{a_k\right\}$ is said to belong to class $\tilde{S}$ if there exists a non-increasing sequence $\left\{B_k\right\}$ of numbers s.t.

$$\begin{array}{l}|\Delta b_k|\le B_k\forall k=1,2,3,\dots \\ {\displaystyle {\displaystyle \sum _{k=1}^{\infty }}k{B}_k}<\infty .\end{array}$$

where $b_k=\frac{a_k}{k},$ $\Delta b_k=b_k-b_{k+1}$ and proved the following result:

Theorem 1 [9].

If $\left\{a_k\right\}\in \tilde{S},$ then Series (1) is the Fourier series of some function$f\in L^1(0,\pi ).$

In $1989,$ Móricz [5] introduced new classes $\tilde{B}V$ and $\tilde{C}$ of the coefficient sequences for the sine series.

Class $\tilde{B}V$[5].

A null sequence $\left\{a_k\right\}$ belongs to $\tilde{B}V$ if

$${\displaystyle {\displaystyle \sum _{k=1}^{\infty }}k\left|\Delta b_k\right|}<\infty$$

(3)

Class $\tilde{C}$[5].

A null sequence $\left\{a_k\right\}$ belongs to class $\tilde{C}$ if for every $\epsilon >0$ there exists $\delta >0,$ independent of $n,$ and such that for all $n,$

$${\displaystyle \underset{0}{\overset{\delta }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\Delta b_k{D}_k^{\prime }\left(x\right)\right|dx\le \epsilon .$$

(4)

Here, $D_k^{\prime }\left(x\right)$ is the first derivative of Dirichlet kernel $\left(D_k(x)=\frac{\sin \left(k+\frac{1}{2}\right)x}{2\sin \frac{x}{2}}\right)$.

Equation (4) implies that, for $1\le n\le N$,

$${\displaystyle \underset{0}{\overset{\delta }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n}^N}\Delta b_k{D}_k^{\prime }\left(x\right)\right|dx\le 2\epsilon .$$

The following result was proved by Móricz [7].

Theorem 2 [5].

If$\left\{a_k\right\}\in \tilde{B}V,$then

$$\Vert u_n-f\Vert \to 0(n\to \infty )ifandonlyif\left\{a_k\right\}\in \tilde{C}.$$

where$u_n\left(x\right)=S_n\left(x\right)+b_{n+1}{D}_n^{\prime }\left(x\right)$.

The classes $\tilde{S},\tilde{B}V$ and $\tilde{C}$ seem to be more appropriate for the sine series than the classes $S$ ([7,8]) $BV$ [10], and $C$ [3] in the ordinary sense. Also, Móricz [5] has proved that $\tilde{S}\subset \tilde{B}V\cap \tilde{C}$.

Motivated by the aforesaid authors, new extended classes ${\tilde{S}}_r,\tilde{B}{V}_r$, and ${\tilde{C}}_r$ ($r=0,1,2,\dots$) are defined in this paper as follows:

Class${\tilde{S}}_r$.

A sequence $\left\{a_k\right\}$ is said to belong to class ${\tilde{S}}_r$ ($r=0,1,2,\dots$) if $a_k\to 0$ as $k\to \infty$, and there exists a non-increasing sequence $\left\{B_k\right\}$ of numbers s.t.

$$\begin{array}{l}|\Delta b_k|\le B_k\forall k=1,2,3,\dots \\ {\displaystyle {\displaystyle \sum _{k=1}^{\infty }}{k}^{r+1}{B}_k}<\infty ,r=0,1,2,3,\dots \end{array}$$

where $b_k=\frac{a_k}{k},r=0,1,2,3,\dots$

$B_k\downarrow 0$ and ${\displaystyle {\displaystyle \sum _{k=1}^{\infty }}{k}^{r+1}}{B}_k<\infty ,$ implies that $k^{r+2}{B}_k=o(1)$ as $k\to \infty (r=0,1,2,\dots )$.

Remark 1.

For$r=0,$${\tilde{S}}_r=\tilde{S}.$

Remark 2.

Obviously, ${\tilde{S}}_{r+1}\subset {\tilde{S}}_r$, but the converse need not be true.

Example 1.

Consider a sequence $\Delta b_n=\frac{1}{n^{r+3}},$ $r=0,1,2,\dots$ and$n\in N.$

$$a_n=n{b}_n=n{\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\Delta b_k\le {\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\frac{k}{k^{r+3}}={\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\frac{1}{k^{r+2}}\to 0\mathrm{as}n\to \infty .$$

Choose $B_n=\frac{1}{n^{r+3}},r=0,1,2,\dots \forall n.$ Clearly, $B_n\downarrow 0$ as $n\to \infty$ and $|\Delta b_n|\le B_n\forall n.$

Consider the series

$${\displaystyle {\displaystyle \sum }_{n=1}^{\infty }}{n}^{r+1}{B}_n={\displaystyle {\displaystyle \sum }_{n=1}^{\infty }}{n}^{r+1}\frac{1}{n^{r+3}}\approx {\displaystyle {\displaystyle \sum }_{n=1}^{\infty }}\frac{1}{n^2}\mathrm{which}\mathrm{is}\mathrm{convergent}.$$

This implies $\left\{a_n\right\}\in {\tilde{S}}_r.$

But the series ${\displaystyle {\displaystyle \sum _{n=1}^{\infty }}{n}^{r+2}{B}_n\approx {\displaystyle {\displaystyle \sum _{n=1}^{\infty }}\frac{1}{n}}}$ is divergent.

This implies that $\left\{a_n\right\}$ does not belong to class ${\tilde{S}}_{r+1}$.

Class$\tilde{B}{V}_r$.

A null sequence $\left\{a_k\right\}$ belongs to $\tilde{B}{V}_r,(r=0,1,2,\dots )$ if

$${\displaystyle {\displaystyle \sum _{k=1}^{\infty }}{k}^{r+1}\left|\Delta b_k\right|}<\infty$$

Remark 3.

For$r=0,$$\tilde{B}{V}_r=\tilde{B}V.$

Remark 4.

Clearly,$\tilde{B}{V}_{r+1}\subset \tilde{B}{V}_r,(r=0,1,2,\dots )$, but the converse may not be true.

Class${\tilde{C}}_r$.

A null sequence $\left\{a_k\right\}$ belongs to class ${\tilde{C}}_r$ ($r=0,1,2,\dots$), if for every $\epsilon >0$, there exists $\delta >0,$ independent of $n,$ and such that for all $n,$

$${\displaystyle \underset{0}{\overset{\delta }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx\le \epsilon$$

Here, $D_k^{r+1}(x)$ is the ${(r+1)}^{th}$ derivative of Dirichlet kernel.

Equation (4) implies, for $1\le n\le N,$

$${\displaystyle \underset{0}{\overset{\delta }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n}^N}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx\le 2\epsilon$$

Remark 5.

For$r=0,$${\tilde{C}}_r=\tilde{C}.$

Remark 6.

It is obvious that${\tilde{C}}_{r+1}\subset {\tilde{C}}_r$but the converse need not be true.

Example 2.

Define$\Delta b_n=\frac{1}{n^{r+3}},r=0,1,2,\dots$and$n=1,2,3,\dots$

$$a_n=n{b}_n=n{\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\Delta b_k\le {\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\frac{k}{k^{r+3}}={\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\frac{1}{k^{r+2}}\to 0\mathrm{as}n\to \infty .$$

Consider, the integral

$$\begin{array}{ll}{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\Delta b_k{D}_k^{r+2}\left(x\right)\right|dx& ={\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\frac{1}{n^{r+3}}{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}|D_k^{r+2}\left(x\right)|dx=O\left({\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\frac{1}{n^{r+3}}\left(n^{r+2}\log n\right)\right)\\ & =O\left({\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\frac{\log n}{n}\right)\end{array}$$

which is divergent.

However,

$$\begin{array}{ll}{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx& ={\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\frac{1}{n^{r+3}}{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}|D_k^{r+1}\left(x\right)|dx=O\left({\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\frac{1}{n^{r+3}}\left(n^{r+1}\log n\right)\right)\\ & =O\left({\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\frac{\log n}{n^2}\right)\mathrm{which}\mathrm{is}\mathrm{convergent}.\end{array}$$

Therefore $\left\{a_n\right\}\in {\tilde{C}}_r$.

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 ${\tilde{S}}_r$ is a subclass of ${\tilde{C}}_r{\displaystyle \cap \tilde{B}{V}_r}(r=0,1,2,\dots )$. Moreover, the theorems are presented concerning the $L^1$ convergence of trigonometric sine series using modified sine sum [11], defined as

$${\beta }_n\left(x\right)={\displaystyle {\displaystyle \sum }_{k=1}^n}\left(\frac{a_{k+1}}{k+1}+{\displaystyle {\displaystyle \sum }_{j=k}^n}{\Delta }^2\left(\frac{a_j}{j}\right)\right)k\sin kx$$

(5)

under the extended classes of numerical sequences.

2. Lemmas

Lemma 1.

[6] Let$n\ge 1$and$r$be a nonnegative integer$x\in \left[\epsilon ,\pi \right]$. Then,$\left|D_n^r(x)\right|\le \frac{C{n}^r}{x},$where$C$denotes a positive absolute constant.

Lemma 2.

[6]${\Vert D_n^r(x)\Vert }_{L^1}=O(n^r\log n),r=0,1,2,\dots$where$D_n^r(x)$represents the$r^{th}$derivative of the Dirichlet kernel.

3. Main Results

Theorem 3.

The following relation holds${\tilde{S}}_r\subset {\tilde{C}}_r\cap \tilde{B}{V}_r$for each$r\in \{0,1,2,\dots \}.$

Proof. 

It is plain that ${\tilde{S}}_r\subset \tilde{B}{V}_r$.

In order to prove that ${\tilde{S}}_r\subset {\tilde{C}}_r$ we take a sequence $\left\{a_k\right\}$ in ${\tilde{S}}_r$ and consider

$${\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx;\mathrm{where}{b}_k=\frac{a_k}{k}$$

If we apply summation by parts, we obtain

$$\begin{array}{c}{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx\\ \le \underset{N\to \infty }{\lim }[{\displaystyle {\displaystyle \sum }_{k=n}^{N-1}}\Delta B_k{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{j=0}^k}\frac{\Delta b_j}{B_j}{D}_j^{r+1}\left(x\right)\right|dx+B_N{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{K=0}^N}\frac{\Delta b_k}{B_k}{D}_k^{r+1}\left(x\right)\right|dx\\ +B_n{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{K=0}^{n-1}}\frac{\Delta b_k}{B_k}{D}_k^{r+1}\left(x\right)\right|dx]\end{array}$$

Clearly $\left|\frac{\Delta b_k}{B_k}\right|\le 1$. Now, if we first apply Bernstein’s inequality [12] and then Sidon Fomin’s inequality ([1,7]), we get

$${\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=0}^n}\frac{\Delta b_k}{B_k}{D}_k^{\left(r+1\right)}\left(x\right)\right|dx\le M{\left(n+1\right)}^{r+2},r=0,1,2,\dots$$

$$\begin{array}{ll}{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx& \le \underset{N\to \infty }{\lim }\left\{{\displaystyle {\displaystyle \sum }_{k=n}^{N-1}}\Delta B_k{\left(k+1\right)}^{r+2}+B_N{\left(N+1\right)}^{r+2}\right\}+n^{r+2}{B}_n\\ & ={\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\left[{\left(k+1\right)}^{r+2}-k^{r+2}\right]B_k+n^{r+2}{B}_n\\ & =O\left({\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}{k}^{r+1}{B}_k\right)+n^{r+2}{B}_n\end{array}$$

So, by given hypothesis, we have

$${\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n}^{\infty }}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx\le \frac{\epsilon }{2}\mathrm{if}n\mathrm{is}\mathrm{large}\mathrm{enough}\mathrm{say}n\ge n_0.$$

(6)

For any $1\le n\le N,$ we can estimate as follows:

$$\begin{array}{ll}{\displaystyle \underset{0}{\overset{\delta }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n}^N}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx& \le {\displaystyle \underset{0}{\overset{\delta }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n}^{n_0}}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx+{\displaystyle \underset{0}{\overset{\delta }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n_0}^N}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx\\ & \le \frac{1}{2}\delta {\displaystyle {\displaystyle \sum }_{k=1}^{n_0}}k{\left(k+1\right)}^{r+1}\left|\Delta b_k\right|+\frac{\epsilon }{2}<\epsilon \end{array}$$

provided $\delta$ is small enough. This proves that $\left\{a_k\right\}\in {\tilde{C}}_r$. □

Theorem 4.

Let $\left\{a_k\right\}$ be a sequence of numbers belonging to the class $\tilde{C}{\displaystyle \cap \tilde{B}V}$ and if $\underset{n\to \infty }{{\displaystyle \lim }}{a}_n\log n=0,$ then

$$\Vert {\beta }_n-f\Vert =o\left(1\right),n\to \infty .$$

Proof. 

The modified trigonometric sine sum is given by

$$\begin{array}{ll}{\beta }_n\left(x\right)& ={\displaystyle {\displaystyle \sum }_{k=1}^n}\left(\frac{a_{k+1}}{k+1}+{\displaystyle {\displaystyle \sum }_{j=k}^n}{\Delta }^2\left(\frac{a_j}{j}\right)\right)k\sin kx\\ & ={\displaystyle {\displaystyle \sum }_{k=1}^n}{a}_k\sin kx+\left(\frac{a_{n+2}}{n+2}-\frac{a_{n+1}}{n+1}\right){\displaystyle {\displaystyle \sum }_{k=1}^n}k\sin kx\\ & =-{\displaystyle {\displaystyle \sum }_{k=1}^n}{b}_k{\left(\cos kx\right)}^{\prime }-\left(b_{n+2}-b_{n+1}\right)D_n^{\prime }\left(x\right)\end{array}$$

By using the summation by parts, we get

$${\beta }_n=-{\displaystyle {\displaystyle \sum }_{k=1}^n}\Delta b_k{D}_k^{\prime }\left(x\right)-b_n{D}_n^{\prime }\left(x\right)-\left(b_{n+2}-b_{n+1}\right)D_n^{\prime }\left(x\right)$$

Under the given hypothesis and Lemma 1, series ${\displaystyle {\displaystyle \sum _{k=1}^n}\Delta b_k}{D}_k^{\prime }(x)$ converges absolutely and $b_n{D}_n^{\prime }\left(x\right)\to 0$ as $n\to \infty$.

Hence $\underset{n\to \infty }{\lim }{\beta }_n(x)=f(x)$ exists in $(0,\pi )$.

Next, consider

$$\begin{array}{ll}\Vert f\left(x\right)-{\beta }_n\left(x\right)\Vert & =\Vert {\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}{a}_k\sin kx-\left(\frac{a_{n+2}}{n+2}-\frac{a_{n+1}}{n+1}\right){\displaystyle {\displaystyle \sum }_{k=1}^n}k\sin kx\Vert \\ & ={\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|-{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}{b}_k{\left(\cos kx\right)}^{\prime }-\left(b_{n+1}-b_{n+2}\right)D_n^{\prime }\left(x\right)\right|dx\end{array}$$

(7)

By using Abel’s transformation, we have

$$\begin{array}{l}={\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|-{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\Delta b_k{D}_k^{\prime }\left(x\right)+b_{n+2}{D}_n^{\prime }\left(x\right)\right|dx\\ ={\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\Delta b_k{D}_k^{\prime }\left(x\right)\right|dx+\frac{n}{n+2}{a}_{n+2}\log n\end{array}$$

(8)

The second term of the above equation is of $o(1)$ as $a_n\log n=0asn\to \infty$. For the remaining part, let $\epsilon >0,$ then there exists $\delta >0$, such that

$${\displaystyle \underset{0}{\overset{\delta }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\Delta b_k{D}_k^{\prime }\left(x\right)\right|dx<\frac{\epsilon }{2}\mathrm{for}\mathrm{all}n\ge 0.$$

(9)

Then

$$\begin{array}{ll}{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\Delta b_k{D}_k^{\prime }\left(x\right)\right|dx& ={\displaystyle \underset{0}{\overset{\delta }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\Delta b_k{D}_k^{\prime }\left(x\right)\right|dx+{\displaystyle \underset{\delta }{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\Delta b_k{D}_k^{\prime }\left(x\right)\right|dx\\ & \le \frac{ϵ}{2}+{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\left|\Delta b_k\right|{\displaystyle \underset{\delta }{\overset{\pi }{{\displaystyle \int }}}}\left|D_k^{\prime }\left(x\right)\right|dx\\ & \le \frac{ϵ}{2}+C{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}k\left|\Delta b_k\right|{\displaystyle \underset{\delta }{\overset{\pi }{{\displaystyle \int }}}}dx/x^2\\ & \le \frac{ϵ}{2}+C{\delta }^{-1}{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}k\left|\Delta b_k\right|\le \epsilon \end{array}$$

(10)

This proves that $\Vert f(x)-{\beta }_n(x)\Vert =o(1)asn\to \infty$. □

Theorem 5.

Let $\left\{a_k\right\}$ be a sequence of numbers belonging to the class $\tilde{C}{\displaystyle \cap \tilde{B}V}$, and if $\underset{n\to \infty }{\lim }{a}_n\log n=0,$ then

$$\Vert S_n-f\Vert =o\left(1\right),n\to \infty .$$

Proof. 

$\Vert S_n-f\Vert \le \Vert S_n-{\beta }_n\Vert +\Vert {\beta }_n-f\Vert$

$$\begin{array}{l}\le \left|b_{n+1}\right|{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|D_n^{\prime }\left(x\right)\right|dx+\left|b_{n+2}\right|{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|D_n^{\prime }\left(x\right)\right|dx+o\left(1\right)\\ \le \left|a_{n+1}\right|\log n+\left|a_{n+2}\right|\log n\left(byLemma2\right)\\ =o\left(1\right),n\to \infty \end{array}$$

□

Theorem 6.

Let $\left\{a_k\right\}$ be a sequence of numbers belonging to the class ${\tilde{C}}_r{{\displaystyle \cap \tilde{B}V}}_r$ and if $n^r{a}_n\log n=0,asn\infty ,foreachr=0,1,2,\dots$ Then

$$\Vert {\beta }_n^r\left(x\right)-f^r\left(x\right)\Vert =o\left(1\right),n\to \infty$$

Here, $f^r(x)$ is the rth derivative of f(x), where$r=0,1,2,\dots$

Proof. 

Consider the modified trigonometric sine sum as

$$\begin{array}{ll}{\beta }_n\left(x\right)& ={\displaystyle {\displaystyle \sum }_{k=1}^n}\left(\frac{a_{k+1}}{k+1}+{\displaystyle {\displaystyle \sum }_{j=k}^n}{\Delta }^2\left(\frac{a_j}{j}\right)\right)k\sin kx\\ & ={\displaystyle {\displaystyle \sum }_{k=1}^n}{a}_k\sin kx+\left(\frac{a_{n+2}}{n+2}-\frac{a_{n+1}}{n+1}\right){\displaystyle {\displaystyle \sum }_{k=1}^n}k\sin kx\end{array}$$

(11)

Taking r-times differentiation of ${\beta }_n(x),$ we get

$$\begin{array}{l}{\beta }_n^r\left(x\right)=S_n^r\left(x\right)+\left(\frac{a_{n+2}}{n+2}-\frac{a_{n+1}}{n+1}\right){\displaystyle {\displaystyle \sum }_{k=1}^n}{k}^{r+1}\sin \left(kx+\frac{r\pi }{2}\right)\\ ={\displaystyle {\displaystyle \sum }_{k=1}^n}{k}^r{a}_k\sin \left(kx+\frac{r\pi }{2}\right)+\left(\frac{a_{n+1}}{n+1}-\frac{a_{n+2}}{n+2}\right){\displaystyle {\displaystyle \sum }_{k=1}^n}{k}^{r+1}\cos \left(kx+\frac{\left(r+1\right)\pi }{2}\right)\\ =-{\displaystyle {\displaystyle \sum }_{k=1}^n}{k}^{r+1}{b}_k\cos \left(kx+\frac{\left(r+1\right)\pi }{2}\right)+\left(b_{n+1}-b_{n+2}\right)D_n^{r+1}\left(x\right)\end{array}$$

(12)

If we apply Abel’s transformation on the first term of above equation, we get

$$\begin{array}{ll}{\beta }_n^r\left(x\right)& =-{\displaystyle {\displaystyle \sum }_{k=1}^{n-1}}\Delta b_k{D}_k^{r+1}\left(x\right)-b_n{D}_n^{r+1}\left(x\right)+\left(b_{n+1}-b_{n+2}\right)D_n^{r+1}\left(x\right)\\ & =-{\displaystyle {\displaystyle \sum }_{k=1}^n}\Delta b_k{D}_k^{r+1}\left(x\right)-b_{n+2}{D}_n^{r+1}\left(x\right)\end{array}$$

(13)

The series ${\displaystyle {\displaystyle \sum _{k=1}^{\infty }}\Delta b_k{D}_k^{r+1}(x)}$ converges absolutely and $b_n{D}_n^{r+1}(x)\to 0$ as $n\to \infty$ using Lemma 1 and given hypothesis.

Therefore $\underset{n\to \infty }{{\displaystyle \lim }}{\beta }_n^r(x)=f^r(x)$ exists in $(0,\pi )$.

Next, consider

$$\begin{array}{l}\Vert f\left(x\right)-{\beta }_n\left(x\right)\Vert =\Vert {\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}{a}_k\sin kx-\left(\frac{a_{n+2}}{n+2}-\frac{a_{n+1}}{n+1}\right){\displaystyle {\displaystyle \sum }_{k=1}^n}k\sin kx\Vert \\ \begin{array}{ll}\Vert f^r\left(x\right)-{\beta }_n^r\left(x\right)\Vert & =\Vert {\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}{k}^r{a}_k\sin \left(kx+\frac{r\pi }{2}\right)\\ & -\left(\frac{a_{n+2}}{n+2}-\frac{a_{n+1}}{n+1}\right){\displaystyle {\displaystyle \sum }_{k=1}^n}{k}^{r+1}\sin \left(kx+\frac{r\pi }{2}\right)\Vert \end{array}\\ =\Vert {\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}{k}^r{a}_k\sin \left(kx+\frac{r\pi }{2}\right)+\left(\frac{a_{n+2}}{n+2}-\frac{a_{n+1}}{n+1}\right){\displaystyle {\displaystyle \sum }_{k=1}^n}{k}^{r+1}\cos \left(kx+\frac{\left(r+1\right)\pi }{2}\right)\Vert \\ ={\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|-{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}{k}^{r+1}{b}_k\cos \left(kx+\frac{\left(r+1\right)\pi }{2}\right)+\left(b_{n+2}-b_{n+1}\right)D_n^{r+1}\left(x\right)\right|dx\end{array}$$

If we apply Abel’s transformation, we obtain

$$\begin{array}{l}={\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|-{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\Delta b_k{D}_k^{r+1}\left(x\right)+b_{n+1}{D}_n^{r+1}\left(x\right)-b_{n+1}{D}_n^{r+1}\left(x\right)+b_{n+2}{D}_n^{r+1}\left(x\right)\right|dx\\ \le {\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx+\left|b_{n+2}\right|{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|D_n^{r+1}\left(x\right)\right|dx\\ \le {\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx+\frac{a_{n+2}}{n+2}{n}^{r+1}\log n\end{array}$$

The second term of the above equation are of o(1) as $n^r{a}_n\log n=0$ as $n\to \infty$. For the remaining part, let $\epsilon >0$, then there exists $\delta >0$, such that ${\displaystyle {\displaystyle \underset{0}{\overset{\delta }{\int }}}|{\displaystyle {\displaystyle \sum _{k=n+1}^{\infty }}\Delta b_k{D}_k^{r+1}(x)|dx<\epsilon }}/2$ for all $n\ge 0$. Then

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx={\displaystyle \underset{0}{\overset{\delta }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx+{\displaystyle \underset{\delta }{\overset{\pi }{{\displaystyle \int }}}}\left|{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\Delta b_k{D}_k^{r+1}\left(x\right)\right|dx\\ \le \frac{\epsilon }{2}+{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}\left|\Delta b_k\right|{\displaystyle \underset{\delta }{\overset{\pi }{{\displaystyle \int }}}}\left|D_k^{r+1}\left(x\right)\right|dx\\ \le \frac{\epsilon }{2}+C{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}{k}^{r+1}\left|\Delta b_k\right|{\displaystyle \underset{\delta }{\overset{\pi }{{\displaystyle \int }}}}dx/x^{r+2}\\ \le \frac{\epsilon }{2}+C{\delta }^{-\left(r+1\right)}{\displaystyle {\displaystyle \sum }_{k=n+1}^{\infty }}{k}^{r+1}\left|\Delta b_k\right|\le \epsilon \left(\mathrm{by}\mathrm{given}\mathrm{hypothesis}\right)\end{array}$$

Therefore, ${\Vert f^r(x)-{\beta }_n^r(x)\Vert }_{L^1}=o(1)$ as $n\to \infty$. □

Remark 7.

For $r=0$, Theorem 6 reduces to Theorem 4.

Theorem 7.

Let $\left\{a_k\right\}$ be a sequence of numbers belonging to the class ${\tilde{C}}_r{{\displaystyle \cap \tilde{B}V}}_r$ and if $n^r{a}_n\log n=o(1)$ as $n\to \infty$. Then

$$\Vert S_n^r\left(x\right)-f^r\left(x\right)\Vert =o\left(1\right),n\to \infty .$$

where$r=0,1,2\dots .$

Proof. 

$\Vert S_n^r-f^r\Vert \le \Vert S_n^r-{\beta }_n^r\Vert +\Vert {\beta }_n^r-f^r\Vert$

$$\begin{array}{l}\le \left|b_{n+2}\right|{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|D_n^{r+1}\left(x\right)\right|dx+\left|b_{n+1}\right|{\displaystyle \underset{0}{\overset{\pi }{{\displaystyle \int }}}}\left|D_n^{r+1}\left(x\right)\right|dx+o\left(1\right)\\ \le \frac{\left|a_{n+2}\right|}{n+2}{n}^{r+1}\log n+\frac{\left|a_{n+1}\right|}{n+1}{n}^{r+1}\log n\left(byLemma2\right)\\ =o\left(1\right)asn\to \infty \end{array}$$

(14)

□

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 $\left\{a_k\right\}\in {\tilde{S}}_r(r=0,1,2,3,\dots )$ and if $n^r{a}_n\log n=o(1)$ as $n\to \infty$. Then

(i) 

$\Vert {\beta }_n^r\left(x\right)-f^r\left(x\right)\Vert =o\left(1\right),n\to \infty .$

(ii) 

$\Vert S_n^r\left(x\right)-f^r\left(x\right)\Vert =o\left(1\right),n\to \infty .$