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<h1>Further results on \(L^{1}\)-convergence of some modified complex trigonometric sums</h1>
<p class="authors">
<span class="author">Xhevat Z. Krasniqi\(^\ast \)</span>
</p>
<p class="date">December 20, 2015.</p>
</div>
<p>\(^\ast \)Department of Mathematics and Informatics, Faculty of Education, University of Prishtina "Hasan Prishtina", Avenue "Mother Theresa" 5, 10000 Prishtina, Kosovo, e-mail: <span class="tt">xhevat.krasniqi@uni-pr.edu</span>. </p>

<div class="abstract"><p> In this paper we have defined a new class of numerical sequences, which tend to zero, briefly denoted by \(\mathbb {K}^{2}\). Moreover, employing such class of numerical sequences we have studied \(L^{1}\)-convergence of some modified complex trigonometric sums introduced previously by others. </p>
<p><b class="bf">MSC.</b> 42A20, 42A32. </p>
<p><b class="bf">Keywords.</b> \(L^{1}\)-convergence, null sequence, trigonometric series, modified sums. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \((c_{k})\), \(k\in \{ 0,\pm 1,\pm 2,\dots \} \), be a sequence of complex numbers and let </p>
<div class="equation" id="eq11">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq11} \sum _{k=-\infty }^{\infty }c_{k}e^{ikx} \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> be a formal complex trigonometric series with its partial sums </p>
<div class="equation" id="eq12">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq12} S_{n}(x)=\sum _{k=-n}^{n}c_{k}e^{ikx}, \quad n\in \{  0,1,2,\dots \}  . \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> Let </p>
<div class="displaymath" id="a0000000003">
  \[ \| f\| _{L^{1}}=\frac{1}{2\pi }\int _{-\pi }^{\pi }|f(x)|dx \]
</div>
<p> be the \(L^{1}\)-norm of a function \(f\in L^1\). </p>
<p>The following interesting statement is a well-known one: If a trigonometric series converges in \(L^1\)-norm to a function \(f\in L^1\), then it is the Fourier series of the function \(f\). Riesz (see <span class="cite">
	[
	<a href="#N" >5</a>
	]
</span>, Vol. II, Chap. VIII, \(\S \) 22) gave a counter example showing that in the metric \(L^1\) we can not expect the converse of above mention statement to be true. This fact motivated the various authors to study the \(L^{1}\)-convergence of trigonometric series, introducing the so-called modified cosine and sine sums, since these modified sums approximate their limits better than the classical trigonometric series in the sense that they converge in \(L^1\)-norm to the sum of the trigonometric series whereas the classical series itself may not. C. S. Rees and C. V. Stanojevic <span class="cite">
	[
	<a href="#RS" >3</a>
	]
</span> for the first time introduced the following type of modified cosine sums </p>
<div class="displaymath" id="a0000000004">
  \[ f_{n}(x)=\tfrac {1}{2}\sum _{k=0}^{n}\triangle a_{k}+\sum _{k=1}^{n}\sum _{j=k}^{n}\triangle a_{j}\cos kx , \]
</div>
<p> and obtained a necessary and sufficient condition for the integrability of the limit of these sums, where \(\triangle a_{j}=a_{j}-a_{j+1}\). </p>
<p>Then several interesting properties (their integrability <span class="cite">
	[
	<a href="#R1" >2</a>
	]
</span> or \(L^{1}\)-convergence <span class="cite">
	[
	<a href="#XHK1" >15</a>
	]
</span>) of these sums were investigated imposing several conditions on the coefficients \(a_{k}\) in "old papers" <span class="cite">
	[
	<a href="#R" >1</a>
	]
</span>, <span class="cite">
	[
	<a href="#R1" >2</a>
	]
</span>, <span class="cite">
	[
	<a href="#A" >6</a>
	]
</span> or in some "new papers" <span class="cite">
	[
	<a href="#L" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#K" >9</a>
	]
</span>, <span class="cite">
	[
	<a href="#XHK2" >13</a>
	]
</span>, <span class="cite">
	[
	<a href="#XHK" >14</a>
	]
</span>, <span class="cite">
	[
	<a href="#T" >16</a>
	]
</span>. </p>
<p>After introducing the sums \(f_{n}(x)\), B. Ram and S. Kumari <span class="cite">
	[
	<a href="#KR" >8</a>
	]
</span> seems to be motivated to introduce the set of the sums </p>
<div class="displaymath" id="a0000000005">
  \[ h_{n}(x)=\frac{a_0}{2}+\sum _{k=1}^{n}\sum _{j=k}^{n}\triangle \left(\frac{a_{j}}{j}\right)k\cos kx \]
</div>
<div class="displaymath" id="a0000000006">
  \[ g_{n}(x)=\sum _{k=1}^{n}\sum _{j=k}^{n}\triangle \left(\frac{a_{j}}{j}\right)k\sin kx \]
</div>
<p> and studied their \(L^{1}\)-convergence under condition that the coefficients \(a_n\) belong to the class \(\mathbb {R}\). </p>
<p><div class="definition_thmwrapper " id="a0000000007">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">1</span>
    <span class="definition_thmtitle"><span class="cite">
	[
	<a href="#TK" >12</a>
	]
</span></span>
  </div>
  <div class="definition_thmcontent">
  <p> If \(a_{k}\to 0\) as \(k\to \infty \) and </p>
<div class="displaymath" id="a0000000008">
  \[ \sum _{k=1}^{\infty }k^{2}\left|\triangle ^{2}\left(\frac{a_{k}}{k}\right)\right|{\lt}\infty  \]
</div>
<p> then it is said that \((a_{k})\) belongs to the class \(\mathbb {R}\), where \(\triangle ^{2}a_{j}=a_{j}-2a_{j+1}+a_{j+2}\). </p>

  </div>
</div> The above definition were introduced by T. Kano who verified a result which we will formulate it as follows. <div class="theorem_thmwrapper " id="the11">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2</span>
    <span class="theorem_thmtitle"><span class="cite">
	[
	<a href="#TK" >12</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p>  If \((a_{k})\in \mathbb {R}\) then the series <span class="rm">(<a href="#eq11">1</a>)</span> and <span class="rm">(<a href="#eq12">2</a>)</span> are Fourier series or equivalently they represent integrable functions. </p>

  </div>
</div> Among others, using this result B. Ram and S. Kumari proved the following result. <div class="theorem_thmwrapper " id="the12">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
    <span class="theorem_thmtitle"><span class="cite">
	[
	<a href="#KR" >8</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p>  Let \((a_{k})\in \mathbb {R}\). Then for \(x\in (0,\pi ]\) </p>
<div class="equation" id="eq13">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq13} \lim _{n\to \infty }t_{n}(x)=t(x),\quad t\in L(0,\pi ] , \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> and </p>
<div class="equation" id="eq14">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq14} \|  t_{n}-t\| _{L^1}\to 0,\, \, \text{as}\, \,  n\to \infty , \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> where \(t_{n}(x)\) represents either \(h_{n}(x)\) or \(g_{n}(x)\). </p>

  </div>
</div> </p>
<p>The authors of <span class="cite">
	[
	<a href="#K" >9</a>
	]
</span> have introduced the following modified complex trigonometric sums </p>
<div class="displaymath" id="a0000000009">
  \begin{align*}  g_{n}^{c}(x)& =S_{n}(x)+\frac{i}{2\sin x}[ c_{n}e^{i(n+1)x}- c_{-n}e^{-i(n+1)x}+c_{n+1}e^{inx}- c_{-(n+1)}e^{-inx}\\ & \quad +(c_n-c_{n+2})E_{n}(x)+(c_{-(n+2)}-c_{-n})E_{-n}(x)], \end{align*}
</div>
<p> where \(E_{n}(x)=\sum _{j=0}^{n}e^{ijx}\). </p>
<p>Note that the above sums are indeed the complex form of the modified sine and cosine sums introduced in <span class="cite">
	[
	<a href="#KR1" >10</a>
	]
</span> and <span class="cite">
	[
	<a href="#KR2" >11</a>
	]
</span> respectively. </p>
<p>Also they have introduced the following definition. <div class="definition_thmwrapper " id="a0000000010">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">4</span>
  </div>
  <div class="definition_thmcontent">
  <p>A sequence \((c_{k})\) of complex numbers belongs to class \(J^{*}\) if \(\lim _{k\to \infty }c_k=0\), and there exists a sequence \((A_{k})\) such that </p>
<div class="equation" id="eq15">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq15} A_{k}\downarrow 0, \quad \text{as}\quad k\to \infty , \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<div class="equation" id="eq16">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq16} \sum _{k=1}^{\infty }kA_{k}< \infty , \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> and </p>
<div class="equation" id="eq17">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq17}\left|\triangle \left(\frac{c_{k}-c_{-k}}{k}\right)\right|\leq \frac{A_{k}}{k}, \quad \forall k. \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>

  </div>
</div> Throughout this paper we will denote by \(S_{n}(x)\) the partial sums of the series (<a href="#eq11">1</a>) and \(\lim _{n\to \infty }S_{n}(x)=f(x)\). <div class="theorem_thmwrapper " id="the12">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5</span>
    <span class="theorem_thmtitle"><span class="cite">
	[
	<a href="#KR" >8</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p>  Let \((c_{k})\) belongs to the class \(J^{*}\) then </p>
<div class="equation" id="eq18">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq18} \lim _{n\to \infty }g_{n}^{c}(x)=f(x),\quad \text{exists for}\quad |x|\in (0,\pi ], \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<div class="equation" id="eq19">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq19} f\in L^{1}(0,\pi ]\quad \text{and}\quad \|  g_{n}^{c}-f\|  _{L^{1}}\to 0,\, \, \text{as}\, \,  n\to \infty , \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p> and </p>
<div class="equation" id="eq20">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq20} \|  S_{n}-f\| _{L^1}\to 0,\, \, \text{as}\, \,  n\to \infty . \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>

  </div>
</div> </p>
<p>Here and in sequel we will use the notation \(\bigtriangleup ^{2}b_{k}=\bigtriangleup b_{k}-\bigtriangleup b_{k+1}, k\in \{ 1,2,\dots \} \). Now we are going to introduce the following class of sequences of complex numbers. </p>
<p><div class="definition_thmwrapper " id="a0000000011">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">6</span>
  </div>
  <div class="definition_thmcontent">
  <p>A sequence \((c_{k})\) of complex numbers belongs to class \(\mathbb {K}^{2}\) if \(\lim _{k\to \infty }c_k=0\), and there exists a sequence \((A_{k})\) such that </p>
<div class="equation" id="eq15">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq15} A_{k}\downarrow 0, \quad \text{as}\quad k\to \infty , \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<div class="equation" id="eq16">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq16} \sum _{k=1}^{\infty }k^{2}A_{k}< \infty , \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p> and </p>
<div class="equation" id="eq17">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq17}\max \left\{ \left|\triangle ^2\left(\frac{c_{k}}{k}\right)\right| , \left|\triangle ^2\left(\frac{c_{-k}}{k}\right)\right|\right\}  \leq \frac{A_{k}}{k^2},\quad k\in \{  1,2,\dots \}  . \end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>

  </div>
</div> Next example shows that the there exist sequences that belong or not belong to the class \(\mathbb {K}^{2}\). </p>
<p>Let \(\left(c_k\right)\) be a sequence defined by its general term \(c_k:=\frac{1}{n^2}\) , \(n\in \{ 1,2,\dots \} \). Then, \(\left|\bigtriangleup ^2\left(\frac{c_{\pm k}}{k}\right)\right|\leq \frac{4}{k^3}=\frac{A_{k}}{k^2}\), \(A_{k}=\frac{4}{k}\downarrow 0\), and \(\sum _{k=1}^{\infty }k^2A_{k}=+\infty \), which means that \(\left(c_k\right)\not\in \mathbb {K}^2\). </p>
<p>On the other hand, let \(\left(\overline{c}_k\right)\) be a sequence defined by its general term \(\overline{c}_k=\frac{1}{n^5}\) , \(n\in \{ 1,2,\dots \} \). Then, \(\left|\bigtriangleup ^2\left(\frac{c_{\pm k}}{k}\right)\right|\leq \frac{4}{k^6}=\frac{A_{k}}{k^2}\), \(A_{k}=\frac{4}{k^4}\downarrow 0\), and \(\sum _{k=1}^{\infty }k^2A_{k}{\lt}+\infty \), which means that \(\left(\overline{c}_k\right)\in \mathbb {K}^2\). </p>
<p>The aim of this paper is to study \(L^{1}\)-convergence of sums \(g^{c}_{n}(x)\) under condition that the coefficients \(c_k\) belong to the class \(\mathbb {K}^{2}\). </p>
<p>Closing this section, we recall the well-known equality named as Abel’s transformation: Let \(n\) be a positive integer, and \(a_{1}, a_2,\dots , a_{n}\) and \(b_{1}, b_2,\dots , b_{n}\) be two sequences. If \(S_k=a_1+a_2+\cdots +a_n\), then </p>
<div class="displaymath" id="a0000000012">
  \[ \sum _{k=1}^{n}a_kb_k=\sum _{k=1}^{n-1}S_k(b_k-b_{k+1})+S_nb_n. \]
</div>
<h1 id="a0000000013">2 Helpful Lemmas</h1>
<p>Let </p>
<div class="displaymath" id="a0000000014">
  \[ \widetilde{D}_{n}(x)=\sum _{j=1}^{n}\cos (jx)=\frac{\sin \left(n+\frac{1}{2}\right)x}{2\sin \frac{x}{2}} \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000015">
  \[ \widetilde{D}_{n}(x)=\sum _{j=1}^{n}\sin (jx)=\frac{\cos \frac{x}{2}-\cos \left(n+\frac{1}{2}\right)x}{2\sin \frac{x}{2}} \]
</div>
<p> be the Dirichlet’s and conjugate Dirichlet’s kernels respectively. </p>
<p>The following statements are needed for the proof of the main result. <div class="lemma_thmwrapper " id="the14">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">7</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#BBR" >7</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> Let \(r\) be a non-negative integer and \(0{\lt}\varepsilon {\lt}\pi \). Then there exists \(M_{r\varepsilon }{\gt}0\) such that for all \(\varepsilon \leq |x|\leq \pi \) and all \(n\geq 1\), </p>
<ol class="enumerate">
  <li><p>\(|E_{n}^{(r)}(x)|\leq \frac{M_{r\varepsilon }n^r}{|x|}\), </p>
</li>
  <li><p>\(|E_{-n}^{(r)}(x)|\leq \frac{M_{r\varepsilon }n^r}{|x|}\), </p>
</li>
  <li><p>\(|D_{n}^{(r)}(x)|\leq \frac{2M_{r\varepsilon }n^r}{|x|}\), </p>
</li>
  <li><p>\(|\widetilde{D}_{n}^{(r)}(x)|\leq \frac{M_{r\varepsilon }n^r}{|x|}\), </p>
</li>
</ol>

  </div>
</div> <div class="lemma_thmwrapper " id="the16">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">8</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#K" >9</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> For \(n\geq 1\), we have </p>
<ol class="enumerate">
  <li><p>\(\left\| \frac{E_{n}(x)}{2\sin x}\right\| _{L^1}=o(n)\) as \(n\to \infty \), </p>
</li>
  <li><p>\(\left\| \frac{E_{-n}(x)}{2\sin x}\right\| _{L^1}=o(n)\) as \(n\to \infty \), </p>
</li>
  <li><p>\(\left\| \frac{e^{inx}}{2\sin x}\right\| _{L^1}=o(\log n)\) as \(n\to \infty \). </p>
</li>
</ol>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="the15">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">9</span>
  </div>
  <div class="lemma_thmcontent">
  <p> Let \(r\) be a non-negative integer and \(0{\lt}\varepsilon {\lt}\pi \). Then there exists \(M_{r\varepsilon }{\gt}0\) such that for all \(\varepsilon \leq |x|\leq \pi \) and all \(n\geq 1\), </p>
<ol class="enumerate">
  <li><p>\(|\overline{E}_{n}'(x)|\leq \frac{M_{r\varepsilon }n^2}{|x|}\), </p>
</li>
  <li><p>\(|\overline{E}_{-n}'(x)|\leq \frac{M_{r\varepsilon }n^2}{|x|}\), </p>
</li>
</ol>
<p> where \(\overline{E}_{n}(x)=\sum _{m=1}^{n}{E}_{m}(x)\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000016">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> (i) Under conditions of this Lemma and Lemma <a href="#the14">7</a> we have </p>
<div class="displaymath" id="a0000000017">
  \begin{eqnarray*} |\overline{E}_{n} ’(x)|& \leq & \sum _{m=1}^{n}|{E}_{m} ’(x)|\leq \frac{M_{r\varepsilon }}{|x|}\sum _{m=1}^{n}m\\ & =& \frac{M_{r\varepsilon }}{|x|}\cdot \frac{n(n+1)}{2}\leq \frac{M_{r\varepsilon }n^2}{|x|}, \end{eqnarray*}
</div>
<p> for \(0{\lt}\varepsilon \leq |x|\leq \pi \). </p>
<p>(ii) Similarly we have obtained </p>
<div class="displaymath" id="a0000000018">
  \begin{eqnarray*} |\overline{E}_{-n} ’(x)|\leq \sum _{m=1}^{n}|{E}_{-m} ’(x)|\leq \frac{M_{r\varepsilon }}{|x|}\sum _{m=1}^{n}m\leq \frac{M_{r\varepsilon }n^2}{|x|}, \end{eqnarray*}
</div>
<p> for \(0{\lt}\varepsilon \leq |x|\leq \pi \). <div class="proof_wrapper" id="a0000000019">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<h1 id="a0000000020">3 Main Results</h1>
<p> The following theorem presents the main result. <div class="theorem_thmwrapper " id="the31">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">10</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \((c_{k})\) belongs to the class \(\mathbb {K}^{2}\). Then </p>
<div class="equation" id="eq31">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq31} \lim _{n\to \infty }g_{n}^{c}(x)=f(x),\quad \text{exists for}\quad |x|\in (0,\pi ], \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<div class="equation" id="eq32">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq32} f\in L^{1}(0,\pi ]\quad \text{and}\quad \|  g_{n}^{c}-f\|  _{L^{1}}\to 0,\, \, \text{as}\, \,  n\to \infty , \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> and </p>
<div class="equation" id="eq33">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq33} \|  S_{n}(f)-f\| _{L^1}\to 0,\, \, \text{as}\, \,  n\to \infty . \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000021">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Firstly, we will show that \(f(x)\) exists on \((0,\pi ]\). Indeed, it is clear that we can write </p>
<div class="displaymath" id="a0000000022">
  \[ S_{n}(x)=\sum _{k=-n}^{n}c_{k}e^{ikx}=c_{0}+\sum _{k=1}^{n}\left(\frac{c_{k}}{k}ke^{ikx} +\frac{c_{-k}}{k}ke^{-ikx}\right) \]
</div>
<p>Applying twice the Abel’s transformation we obtain </p>
<div class="displaymath" id="eq34">
  \begin{eqnarray} \label{eq34} S_{n}(x)\! \! \!  & =& \! \! \!  c_{0}-i\left[ \sum _{k=1}^{n-2}\bigtriangleup ^{2}\left(\frac{c_{k}}{k}\right)\overline{E}’_{k}(x)+\bigtriangleup \left(\frac{c_{n-1}}{n-1}\right)\overline{E}’_{n}(x)\right] -i\frac{c_{n}}{n}{E}’_{n}(x)\nonumber \\ & & +i\left[ \sum _{k=1}^{n-2}\bigtriangleup ^{2}\left(\frac{c_{-k}}{k}\right)\overline{E}’_{-k}(x)+\bigtriangleup \left(\frac{c_{-(n-1)}}{n-1}\right)\overline{E}’_{-n}(x)\right] +i\frac{c_{-n}}{n}{E}’_{-n}(x)\nonumber . \end{eqnarray}
</div>
<p>Based on Lemmas <a href="#the14">7</a> and <a href="#the15">9</a> we clearly have </p>
<div class="displaymath" id="eq35">
  \begin{eqnarray} \label{eq35} \! \! \! \!  & & \! \! \! \!  |S_{n}(x)|\leq \nonumber \\ \! \! \! \!  & \leq & \! \! \! \!  |c_{0}|+\sum _{k=1}^{n-2}\Bigg[\left|\bigtriangleup ^{2}\left(\frac{c_{k}}{k}\right)\right| |\overline{E}’_{k}(x)|+\left|\bigtriangleup ^{2}\left(\frac{c_{-k}}{k}\right)\right| |\overline{E}’_{-k}(x)|\Bigg] \nonumber \\ & & +\Bigg[\left| \bigtriangleup \left( \frac{c_{n-1}}{n-1}\right) \right| |\overline{E}’_{n}(x)|+\left| \bigtriangleup \left( \frac{c_{-(n-1)}}{n-1} \right) \right| |\overline{E}’_{-n}(x)| \Bigg] \nonumber \\ & & +\frac{|c_{n}|}{n}|{E}’_{n}(x)|+\frac{|c_{-n}|}{n}|{E}’_{-n}(x)|\nonumber \\ \! \! \! \!  & \leq & \! \! \! \!  |c_{0}|+\frac{M_{r\varepsilon }}{|x|}\Bigg\{ \sum _{k=1}^{n-2}k^2 \Bigg[\left|\bigtriangleup ^{2}\left(\frac{c_{k}}{k}\right)\right| +\left|\bigtriangleup ^{2}\left(\frac{c_{-k}}{k}\right)\right| \Bigg] \nonumber \\ & & +n^2\Bigg[\left| \bigtriangleup \left( \frac{c_{n-1}}{n-1}\right) \right| +\left| \bigtriangleup \left( \frac{c_{-(n-1)}}{n-1} \right) \right| \Bigg] +|c_{n}|+|c_{-n}| \Bigg\} \nonumber \\ \! \! \! \!  & \leq & \! \! \! \!  |c_{0}|+\frac{2M_{r\varepsilon }}{|x|}\Bigg\{ \sum _{k=1}^{n-2}A_{k}+2\sum _{k=n-1}^{\infty }k^2 \Bigg[\left|\bigtriangleup ^{2}\left(\frac{c_{k}}{k}\right)\right| +\left|\bigtriangleup ^{2}\left(\frac{c_{-k}}{k}\right)\right| \Bigg] +2\overline{M} \Bigg\} \nonumber \\ \! \! \! \!  & \leq & \! \! \! \!  |c_{0}|+\frac{2M_{r\varepsilon }}{|x|}\Bigg\{ 5\sum _{k=1}^{\infty }k^2A_{k}+2\overline{M} \Bigg\}  {\lt}+\infty ,\nonumber \end{eqnarray}
</div>
<p> since \((c_{k})\in \mathbb {K}^2\), where \(\overline{M}\) is a positive constant. </p>
<p>Subsequently, \(\lim _{n \to \infty }S_{n}(x)=\lim _{n\to \infty }g_{n}^{c}(x)=f(x)\) exists, because of the boundedness of the functions \(\frac{e^{inx}}{\sin x}\), \(\frac{E_{n}(x)}{\sin x}\), \(\frac{E_{-n}(x)}{\sin x}\) on \((0,\pi ]\), and thus (<a href="#eq31">14</a>) holds true. </p>
<p>Now we are going to prove (<a href="#eq32">15</a>). Indeed, for \(x\not=0\) we have </p>
<div class="displaymath" id="a0000000023">
  \begin{eqnarray*}  f(x)-g_{n}^{c}(x)\! \! \! \!  & =& \! \! \! \!  \sum _{k=n+1}^{\infty }\left(\frac{c_{k}}{k}ke^{ikx} +\frac{c_{-k}}{k}ke^{-ikx}\right) \\ & & -\frac{i}{2\sin x}[ c_{n}e^{i(n+1)x}- c_{-n}e^{-i(n+1)x}+c_{n+1}e^{inx}\\ & & - c_{-(n+1)}e^{-inx}+(c_n-c_{n+2})E_{n}(x)+(c_{-(n+2)}-c_{-n})E_{-n}(x)]. \end{eqnarray*}
</div>
<p> Again, applying twice the Abel’s transformation to the above equality we obtain </p>
<div class="displaymath" id="a0000000024">
  \begin{eqnarray*}  \! \! \! \!  & & \! \! \! \!  f(x)-g_{n}^{c}(x)= \nonumber \\ \! \! \! \!  & =& \! \! \! \!  -i\lim _{p\to \infty }\Bigg\{ \sum _{k=n+1}^{p-2}\left[\bigtriangleup ^{2}\left(\frac{c_{k}}{k}\right)\overline{E}’_{k}(x)-\bigtriangleup ^{2}\left(\frac{c_{-k}}{k}\right)\overline{E}’_{-k}(x)\right] \\ & & +\bigtriangleup \left(\frac{c_{p-1}}{p-1}\right)\overline{E}’_{p-1}(x) -\bigtriangleup \left(\frac{c_{-(p-1)}}{p-1}\right)\overline{E}’_{-(p-1)}(x) +\frac{c_p}{p}{E}’_{p}(x) \\ & & -\frac{c_{-p}}{p}{E}’_{-p}(x)-\bigtriangleup \left(\frac{c_{n}}{n}\right)\overline{E}’_{n}(x) +\bigtriangleup \left(\frac{c_{-n}}{n}\right)\overline{E}’_{-n}(x)- \frac{c_{n+1}}{n+1}{E}’_{n}(x)\\ & & +\frac{c_{-(n+1)}}{n+1}{E}’_{-n}(x)\Bigg\} -\frac{i}{2\sin x}[ c_{n}e^{i(n+1)x}- c_{-n}e^{-i(n+1)x}+c_{n+1}e^{inx}\\ & & -c_{-(n+1)}e^{-inx}+(c_n-c_{n+2})E_{n}(x)+(c_{-(n+2)}-c_{-n})E_{-n}(x)]\\ \! \! \! \!  & =& \! \! \! \!  -i\Bigg\{ \sum _{k=n+1}^{\infty }\left[\bigtriangleup ^{2}\left(\frac{c_{k}}{k}\right)\overline{E}’_{k}(x)-\bigtriangleup ^{2}\left(\frac{c_{-k}}{k}\right)\overline{E}’_{-k}(x)\right] -\bigtriangleup \left(\frac{c_{n}}{n}\right)\overline{E}’_{n}(x)\\ & & +\bigtriangleup \left(\frac{c_{-n}}{n}\right)\overline{E}’_{-n}(x)- \frac{c_{n+1}}{n+1}{E}’_{n}(x)+\frac{c_{-(n+1)}}{n+1}{E}’_{-n}(x)\Bigg\} \\ & & -\frac{i}{2\sin x}[ c_{n}e^{i(n+1)x}- c_{-n}e^{-i(n+1)x}+c_{n+1}e^{inx}-c_{-(n+1)}e^{-inx}\\ & & +(c_n-c_{n+2})E_{n}(x)+(c_{-(n+2)}-c_{-n})E_{-n}(x)]. \end{eqnarray*}
</div>
<p>Hence, using Lemmas <a href="#the14">7</a> and <a href="#the15">9</a> we get </p>
<div class="displaymath" id="a0000000025">
  \begin{eqnarray*}  \! \! \! \!  & & \! \! \! \!  |f(x)-g_{n}^{c}(x)| \leq \\ \! \! \! \!  & \leq & \! \! \! \!  \sum _{k=n+1}^{\infty }\left[\left|\bigtriangleup ^{2}\left(\frac{c_{k}}{k}\right)\right| |\overline{E}’_{k}(x)|+\left|\bigtriangleup ^{2}\left(\frac{c_{-k}}{k}\right)\right| |\overline{E}’_{-k}(x)|\right] \\ & & +\left|\bigtriangleup \left(\frac{c_{n}}{n}\right)\right| |\overline{E}’_{n}(x)| +\left|\bigtriangleup \left(\frac{c_{-n}}{n}\right)\right| |\overline{E}’_{-n}(x)|+\left| \frac{c_{n+1}}{n+1}\right| |{E}’_{n}(x)|\\ & & +\left|\frac{c_{-(n+1)}}{n+1}\right| |{E}’_{-n}(x)|\Bigg\} +\frac{1}{2|\sin x|}[ |c_{n}|+|c_{-n}|+|c_{n+1}|\\ & & +|c_{-(n+1)}|+(|c_n|+|c_{n+2}|)|E_{n}(x)|+(|c_{-(n+2)}|+|c_{-n}|)|E_{-n}(x)|]\\ \! \! \! \!  & \leq & \! \! \! \!  \frac{M_{r\varepsilon }}{|x|}\Bigg\{ \sum _{k=n+1}^{\infty }k^2\left[\left|\bigtriangleup ^{2}\left(\frac{c_{k}}{k}\right)\right| +\left|\bigtriangleup ^{2}\left(\frac{c_{-k}}{k}\right)\right| \right] \\ & & +n^2\left[\left|\bigtriangleup \left(\frac{c_{n}}{n}\right)\right| +\left|\bigtriangleup \left(\frac{c_{-n}}{n}\right)\right|\right] +(n+1)\left(\left| c_{n+1}\right| +\left| c_{-(n+1)}\right|\right) \Bigg\}  \\ & & +\frac{|c_{n}|+|c_{-n}|+|c_{n+1}|+|c_{-(n+1)}|}{2|\sin x|} \end{eqnarray*}
</div>
<div class="displaymath" id="a0000000026">
  \begin{eqnarray*} & & +(|c_n|+|c_{n+2}|)\left|\frac{E_{n}(x)}{2\sin x}\right| +(|c_{-(n+2)}|+|c_{-n}|)\left|\frac{E_{-n}(x)}{2\sin x}\right|\\ \! \! \! \!  & \leq & \! \! \! \!  \frac{M_{r\varepsilon }}{|x|}\Bigg[2\sum _{k=n}^{\infty }A_{k} +(n+1)\left(\left| c_{n+1}\right| +\left| c_{-(n+1)}\right|\right) \Bigg] \\ & & +\frac{|c_{n}|+|c_{-n}|+|c_{n+1}|+|c_{-(n+1)}|}{2|\sin x|} \end{eqnarray*}
</div>
<div class="displaymath" id="a0000000027">
  \begin{eqnarray*} & & +(|c_n|+|c_{n+2}|)\left|\frac{E_{n}(x)}{2\sin x}\right| +(|c_{-(n+2)}|+|c_{-n}|)\left|\frac{E_{-n}(x)}{2\sin x}\right| . \end{eqnarray*}
</div>
<p>Therefore, using Lemma <a href="#the16">8</a> we obtain </p>
<div class="displaymath" id="a0000000028">
  \begin{eqnarray*}  \|  f-g_{n}^{c}\| _{L^{1}}\! \! \! \!  & \leq & \! \! \! \!  M_{r\varepsilon }\Bigg[2\sum _{k=n}^{\infty }A_{k}\int _{0}^{\pi }\frac{dx}{|x|} +(n+1)\left(\left| c_{n+1}\right| +\left| c_{-(n+1)}\right|\right) \int _{0}^{\pi }\frac{dx}{|x|} \Bigg] \\ & & +\frac{|c_{n}|+|c_{-n}|+|c_{n+1}|+|c_{-(n+1)}|}{2}\int _{0}^{\pi }\frac{dx}{|\sin x|}\\ & & +(|c_n|+|c_{n+2}|)\int _{0}^{\pi }\left|\frac{E_{n}(x)}{2\sin x}\right| dx \\ & & +(|c_{-(n+2)}|+|c_{-n}|)\int _{0}^{\pi }\left|\frac{E_{-n}(x)}{2\sin x}\right| dx\\ \! \! \! \!  & \leq & \! \! \! \!  M_{r\varepsilon }\Bigg[2\sum _{k=n}^{\infty }A_{k}\log k +(n+1)\left(\left| c_{n+1}\right| +\left| c_{-(n+1)}\right|\right) o\left( \log n\right) \Bigg] \\ & & +\frac{|c_{n}|+|c_{-n}|+|c_{n+1}|+|c_{-(n+1)}|}{2}o\left( \log n\right) \\ & & +(|c_n|+|c_{n+2}|)o(n) +(|c_{-(n+2)}|+|c_{-n}|)o(n). \end{eqnarray*}
</div>
<p> Now we note that </p>
<div class="displaymath" id="a0000000029">
  \[ \sum _{k=n}^{\infty }A_{k}\log k\leq \sum _{k=n}^{\infty }k^2A_{k}=o(1), \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000030">
  \begin{eqnarray*}  (n+1)\left| c_{\pm (n+1)}\right| \log n\! \! \! \!  & \leq & \! \! \! \!  (n+1)^3\left| \frac{c_{\pm (n+1)}}{n+1}\right|\\ \! \! \! \!  & \leq & \! \! \! \!  (n+1)^3\sum _{k=n+1}^{\infty }\left| \bigtriangleup \left(\frac{c_{\pm k}}{k}\right)\right|\\ \! \! \! \!  & \leq & \! \! \! \!  (n+1)^3\sum _{k=n+1}^{\infty }\sum _{j=k}^{\infty }\left| \bigtriangleup ^2\left(\frac{c_{\pm j}}{j}\right)\right|\\ \! \! \! \!  & =& \! \! \! \!  (n+1)^3\sum _{j=n+1}^{\infty }(j-n)\left| \bigtriangleup ^2\left(\frac{c_{\pm j}}{j}\right)\right| \end{eqnarray*}
</div>
<div class="displaymath" id="a0000000031">
  \begin{eqnarray*}  \! \! \! \!  & \leq & \! \! \! \!  \sum _{j=n+1}^{\infty }j^4\left| \bigtriangleup ^2\left(\frac{c_{\pm j}}{j}\right)\right|\\ \! \! \! \!  & \leq & \! \! \! \!  \sum _{j=n+1}^{\infty }j^4\frac{A_{j}}{j^2}=\sum _{j=n+1}^{\infty }j^2A_{j}=o(1) \end{eqnarray*}
</div>
<p> as \(n\to \infty \). </p>
<p>Subsequently, we get </p>
<div class="displaymath" id="a0000000032">
  \[ \|  f-g_{n}^{c}\| _{L^{1}}=o(1)\quad \text{as}\quad n\to \infty . \]
</div>
<p>Using the latest equality and the fact that \(g_{n}^{c}(x)\) is a polynomial it follows that \(f\in L^1(0,\pi ]\). </p>
<p>Finally, we will prove (<a href="#eq33">16</a>). Namely, using some facts used above we have </p>
<div class="displaymath" id="a0000000033">
  \begin{eqnarray*} & & \int _{0}^{\pi }|f(x)-{S}_{n}(x)|dx\leq \\ & & \leq \int _{0}^{\pi }|f(x)-g^c_{n}(x)|dx+\int _{0}^{\pi }|g^c_{n} (x)-{S}_{n}(x)|dx\\ & & \leq \int _{0}^{\pi }|f(x)-g^c_{n}(x)|dx+\int _{0}^{ \pi }\bigg|\frac{i}{2\sin x}[ c_{n}e^{i(n+1)x}- c_{-n}e^{-i(n+1)x}+c_{n+1}e^{inx}\\ & & \hspace{.5cm}-c_{-(n+1)}e^{-inx}+(c_n-c_{n+2})E_{n}(x)+(c_{-(n+2)}-c_{-n})E_{-n}(x)]\bigg| dx\\ & & \leq \int _{0}^{\pi }|f(x)-g^c_{n}(x)|dx+[ |c_{n}|+| c_{-n}|+|c_{n+1}|+|c_{-(n+1)}|]\int _{0}^{ \pi }\frac{dx}{2|\sin x|}\\ & & \hspace{.5cm}+(|c_n|+|c_{n+2}|)\int _{0}^{ \pi }\bigg|\frac{E_{n}(x)}{2\sin x}\bigg| dx +(|c_{-(n+2)}|+|c_{-n}|)\int _{0}^{ \pi }\bigg|\frac{E_{-n}(x)}{2\sin x}\bigg| dx \\ & & \leq \int _{0}^{\pi }|f(x)-g^c_{n}(x)|dx+[ |c_{n}|+| c_{-n}|+|c_{n+1}|+|c_{-(n+1)}|]o\left( \log n\right)\\ & & \hspace{.5cm}+[|c_n|+|c_{n+2}|+|c_{-(n+2)}|+|c_{-n}|]o\left( n\right) \\ & & =o(1)+o(1)+o(1)=o(1), \, \,  n\to \infty . \end{eqnarray*}
</div>
<p>The proof of theorem is completed. </p>
<p><div class="proof_wrapper" id="a0000000034">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="acknowledgement_thmwrapper " id="a0000000035">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgements
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The author would like to thank the anonymous referee for her/his remarks which improved the presentation of this paper. Also, many thanks goes for my ex-supervisor, Professor Naim L. Braha, for his advices. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
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</dl>


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