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<h1>Approximation by Matrix Transform in Generalized Grand Lebesgue Spaces with Variable Exponent</h1>
<p class="authors">
<span class="author">Ahmet Testici\(^1\) Daniyal M. Israfilov\(^{2\ast }\)</span>
</p>
<p class="date">March 11, 2021; accepted: June 4, 2021; published online: November 8, 2021.</p>
</div>
<div class="abstract"><p> In this work, the Lipschitz subclass of the generalized grand Lebesgue space with variable exponent is defined and the error of approximation by matrix transforms in this subclass is estimated. </p>
<p><b class="bf">MSC.</b> 41A10, 42A10 </p>
<p><b class="bf">Keywords.</b> trigonometric approximation, matrix transforms, grand variable exponent, Lebesgue spaces, Lipschitz classes, rate of approximation, Fourier series. </p>
</div>
<p>\(^1\)Department of Mathematics, Balikesir University, 10145 Balikesir, Turkey, e-mail: <span class="tt">testiciahmet@hotmail.com</span>. </p>
<p>\(^2\)Department of Mathematics, Balikesir University, 10145 Balikesir, Turkey, e-mail: <span class="tt">mdaniyal@balikesir.edu.tr</span>. </p>
<p>\(^\ast \)Institute of Mathematics and Mechanics, National Academy of Science of Azerbaijan, Az.-1141 Baku, Azerbaijan. </p>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \(\mathbb {T}:=\left[ 0,2\pi \right] \) and let \(p\left( \cdot \right) :\mathbb {T}\rightarrow \lbrack 0,\infty )\) be a Lebesgue measurable \(2\pi \) periodic function satisfying the conditions </p>
<div class="displaymath" id="a0000000003">
  \begin{equation*}  1\leq p_{-}:=\operatorname {ess\, inf}\limits _{x\in \mathbb {T}}\text{ }p\left( x\right) \leq \operatorname {ess\, sup}\limits _{x\in \mathbb {T}}\text{ }p\left( x\right) :=p^{+}{\lt}\infty ,\text{ \  } \end{equation*}
</div>
<div class="displaymath" id="a0000000004">
  \begin{equation*}  \left\vert p\left( x\right) -p\left( y\right) \right\vert \ln \left(\tfrac {2\pi }{ \left\vert x-y\right\vert } \right) \leq c,\qquad x,y\in \left[ 0,2\pi \right] \text{ and }\left\vert x-y\right\vert \leq 1/2,x\neq y \end{equation*}
</div>
<p>with some positive constant \(c=c\left( p\right) \). From now on, the class of such functions \(p\left( \cdot \right) \) we denote by \(\mathcal{P}\left( \mathbb {T}\right) \). We also denote \(\mathcal{P}_{0}\left( \mathbb {T}\right) :=\left\{  p\left( \cdot \right) \in \mathcal{P}\left( \mathbb {T}\right) :p_{-}{\gt}1\right\}  \). The Lebesgue space \(L^{p\left( \cdot \right) }\left( \mathbb {T}\right) \) with variable exponent is defined as the set of all Lebesgue measurable \(2\pi \) periodic functions \(f\) such that</p>
<div class="displaymath" id="a0000000005">
  \begin{equation*}  \rho _{p\left( \cdot \right) }\left( f\right) :=\int \limits _{0}^{2\pi }\left\vert f\left( x\right) \right\vert ^{p\left( x\right) }{\rm d}x{\lt}\infty \text{.} \end{equation*}
</div>
<p>If \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) \), then equipped with the norm</p>
<div class="displaymath" id="a0000000006">
  \begin{equation*}  \left\Vert f\right\Vert _{p\left( \cdot \right) }=\inf \left\{  \lambda {\gt}0:\rho _{p\left( \cdot \right) }\left( f\text{ }/\lambda \right) \leq 1\right\}  \end{equation*}
</div>
<p>\(L^{p\left( \cdot \right) }\left( \mathbb {T}\right) \) becomes a Banach space. Let \(\theta \geqslant 0\) and \(p\in \mathcal{P}_{0}\left( \mathbb {T}\right) \). The generalized grand Lebesgue space with variable exponent \(L^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) \) is the set of all measurable \(2\pi \) periodic functions \(f:\mathbb {T\rightarrow R}\) such that </p>
<div class="displaymath" id="a0000000007">
  \begin{equation*}  \left\Vert f\right\Vert _{p\left( \cdot \right) ,\theta }=\sup _{0{\lt}\varepsilon {\lt}p_{-}-1}\varepsilon ^{\frac{\theta }{p_{-}-\varepsilon }}\left\Vert f\right\Vert _{p\left( \cdot \right) -\varepsilon }{\lt}\infty \text{.} \end{equation*}
</div>
<p>It is easily seen that </p>
<div class="displaymath" id="a0000000008">
  \begin{equation*}  L^{p\left( \cdot \right) }\left( \mathbb {T}\right) \subset L^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) \subset L^{p\left( \cdot \right) -\varepsilon }\left( \mathbb {T}\right) \text{, } \qquad 0{\lt}\varepsilon {\lt}p_{-}-1\text{.} \end{equation*}
</div>
<p>When \(\theta =0\) and \(p\left( \cdot \right) =p=\)<i class="itshape">const</i>, the spaces \(L^{p,0}\left( \mathbb {T}\right) \) reduce to classical Lebesgue spaces \(L^{p}\left( \mathbb {T}\right) \) and when \(\theta =0\) and \(p\left( \cdot \right) \neq const\), the spaces \(L^{p\left( \cdot \right) ,0}\left( \mathbb {T}\right) \) reduce to the variable exponent Lebesgue spaces \(L^{p\left( \cdot \right) }\left( \mathbb {T}\right) \), investigated in detail in the monograph <span class="cite">
	[
	<a href="#diening" >8</a>
	, 
	<a href="#fio" >4</a>
	]
</span>. When \(p=const\) and \(\theta {\gt}0\) it was introduced by Iwaniec and Sbordone in <span class="cite">
	[
	<a href="#Sbor92" >19</a>
	]
</span> (for \(\theta =1\)) and by Greco, Iwaniec and Sbordone in <span class="cite">
	[
	<a href="#sbor97" >9</a>
	]
</span> (for \(\theta {\gt}1\)). </p>
<p>The grand and generalized grand Lebesgue spaces have been applied in the various fields; in particular in theory PDE <span class="cite">
	[
	<a href="#sbor98" >20</a>
	, 
	<a href="#sbor96" >32</a>
	, 
	<a href="#sbor98.1" >33</a>
	]
</span>. The fundamental problems of the spaces \(L^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) \) in view of potential theory, maximal and singular operator theory were investigated in the monograph <span class="cite">
	[
	<a href="#humberto" >26</a>
	]
</span>. </p>
<p>Regarding to grand spaces, it will be observed that these spaces in general are not separable; in particular, Lebesgue space is not dense in grand Lebesgue space. Similar situations are also valid in the case of variable exponents. The closure of \(L^{p\left( \cdot \right) }\left( \mathbb {T}\right) \) in \(L^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) \) doesn’t coincide with \(L^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) \) <span class="cite">
	[
	<a href="#kokimeskhi" >27</a>
	]
</span>. Henceforth, the closure of \(L^{p\left( \cdot \right) }\left( \mathbb {T}\right) \) in \(L^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) \) we denote by \(L_{0}^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) \). Then \(L_{0}^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) \) comprises the set of functions \(f\) such that</p>
<div class="displaymath" id="a0000000009">
  \begin{equation*}  \lim _{\varepsilon \rightarrow 0}\varepsilon ^{\frac{\theta }{p_{-}-\varepsilon }}\left\Vert f\right\Vert _{p\left( \cdot \right) -\varepsilon }=0\text{.} \end{equation*}
</div>
<p>Note that in the generalizations of classical Lebesgue spaces different problems of approximation theory are also investigated. In particular, in <span class="cite">
	[
	<a href="#Dankosam" >14</a>
	, 
	<a href="#sharap1" >34</a>
	, 
	<a href="#Sharapudinov 2" >35</a>
	, 
	<a href="#israfilovguven" >12</a>
	, 
	<a href="#guven3" >13</a>
	, 
	<a href="#jmaa" >15</a>
	, 
	<a href="#jafarovv" >23</a>
	]
</span> the approximation properties of different summation methods in the variable exponent Lebesgue spaces were studied. Similar investigations have been done also in the grand and generalized grand spaces, with constant \(p\) and variable exponent \(p\left( \cdot \right) \) in <span class="cite">
	[
	<a href="#kokinina1" >5</a>
	, 
	<a href="#kokinina" >6</a>
	, 
	<a href="#danik2" >16</a>
	, 
	<a href="#macaristan" >17</a>
	, 
	<a href="#jafarovgrand" >22</a>
	]
</span>. </p>
<p>Let \(f\in L^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) ,\) \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) \) and \(\theta {\gt}0\). For \(\alpha \in (0,1\)] we define the Lipschitz class</p>
<div class="displaymath" id="a0000000010">
  \begin{equation*}  \operatorname {Lip}\left( \mathbb {\alpha },p\left( \cdot \right) ,\theta \right) :=\left\{  f\in L_{0}^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) :\Omega \left( f,\delta \right) _{p\left( \cdot \right) ,\theta }=\mathcal{O}\left( \delta ^{\alpha }\right) ,\text{ \  }\delta {\gt}0\right\}  \text{,} \end{equation*}
</div>
<p>where \(\Omega \left( f,\delta \right) _{p\left( \cdot \right) ,\theta }\) is the modulus of smoothness for \(f\in L_{0}^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) \), defined as </p>
<div class="displaymath" id="a0000000011">
  \begin{equation*}  \Omega \left( f,\delta \right) _{p\left( \cdot \right) ,\theta }:=\sup _{\left\vert h\right\vert \leq \delta }\left\Vert \tfrac {1}{h}\int _{0}^{h}\left\vert f\left( x+t\right) -f\left( x\right) \right\vert {\rm d}t\right\Vert _{p\left( \cdot \right) ,\theta }\text{,}\  \  \delta {\gt}0\text{.} \end{equation*}
</div>
<p>Let \(f\in L^{1}\left( \mathbb {T}\right) \) and </p>
<div class="displaymath" id="a0000000012">
  \begin{equation*}  f\left( x\right) \sim \tfrac {a_{0}}{2}+\sum \limits _{k=1}^{\infty }\left( a_{k}\cos kx+b_{k}\sin kx\right) \end{equation*}
</div>
<p>be its Fourier series representation with the Fourier coefficients </p>
<div class="displaymath" id="a0000000013">
  \begin{equation*}  a_{k}:=\tfrac {1}{2\pi }\int \limits _{-\pi }^{\pi }f\left( t\right) \cos \left( kt\right) {\rm d}t\text{ and }b_{k}:=\tfrac {1}{2\pi }\int \limits _{-\pi }^{\pi }f\left( t\right) \sin \left( kt\right){\rm d}t \end{equation*}
</div>
<p>and let</p>
<div class="displaymath" id="a0000000014">
  \begin{equation*}  S_{n}\left( f\right) \left( x\right) =\sum \limits _{k=0}^{n}u_{k}\left( f\right) \left( x\right) ,\text{ \  }n=1,2,\ldots , \end{equation*}
</div>
<p>be the \(n\)th partial sums of the Fourier series of \(f,\) where</p>
<div class="displaymath" id="a0000000015">
  \begin{equation*}  u_{k}\left( f\right) \left( x\right) :=a_{0}/2\text{ and }u_{k}\left( f\right) \left( x\right) :=\left( a_{k}\cos kx+b_{k}\sin kx\right) ,\text{ }k=1,2,\ldots \end{equation*}
</div>
<p>Let \(A=\left( a_{n,k}\right) \) be a lower triangular infinite matrix of real elements \(a_{n,k}\) such that </p>
<div class="displaymath" id="a0000000016">
  \begin{equation*}  a_{n,k}\geq 0\quad \text{ for }k\leq n\text{, and\  }a_{n,k}=0\text{ for }k{\gt}n, \end{equation*}
</div>
<p>where \(k=0,1,2,\ldots ,\) and let </p>
<div class="displaymath" id="a0000000017">
  \begin{equation*}  s_{n}^{\left( A\right) }:=\sum \limits _{k=0}^{n}a_{n,k}=1\text{, }\quad n=0,1,2,\ldots \end{equation*}
</div>
<p>Unless otherwise indicate, we assume that \(A=\left( a_{n,k}\right) \) is a matrix that the summation of row elements equal to one throughout this work. For a given matrix \(A=(a_{n,k})\), the matrix transform of Fourier series of \(f\) is defined as</p>
<div class="displaymath" id="a0000000018">
  \begin{equation*}  T_{n}^{\left( A\right) }\left( f\right) \left( x\right) :=\sum \limits _{k=0}^{n}a_{n,n-k}S_{k}\left( f\right) \left( x\right),\quad n=0,1,2,\ldots \end{equation*}
</div>
<p>Let \(P_{n}=\sum _{k=0}^{n}p_{k}\) and \(\left( p_{n}\right) \) be a sequence of positive real numbers. If \(a_{n,k}=p_{k}/P_{n}\) , then \(T_{n}^{\left( A\right) }\left( f\right) \) coincides with the Nörlund means</p>
<div class="displaymath" id="a0000000019">
  \begin{equation*}  N_{n}\left( f\right) \left( x\right) =\tfrac {1}{P_{n}}\sum \limits _{k=0}^{n}p_{n-k}S_{k}\left( f\right) \left( x\right) ,\quad n=0,1,2,\ldots ,\text{ } \end{equation*}
</div>
<p>which in the case of \(p_{n}=1\), for all \(n=0,1,2,\ldots ,\) reduce to the Cesàro means </p>
<div class="displaymath" id="a0000000020">
  \begin{equation*}  \sigma _{n}\left( f\right) \left( x\right) =\tfrac {1}{n+1}\sum \limits _{k=0}^{n}S_{k}\left( f\right) \left( x\right) \text{.} \end{equation*}
</div>
<p>We consider the matrix transforms \(T_{n}^{\left( A\right) }\left( f\right) \) as approximating trigonometric polynomials to \(f\) and study approximation properties of \(T_{n}^{\left( A\right) }\left( f\right) \) in the generalized grand Lebesgue space with variable exponent, exactly in the Lipschitz classes defined above. The required conditions on the sequences \(\left( a_{n,k}\right) \) are crucial points to arrive better approach when matrix transform \(T_{n}^{\left( A\right) }\left( f\right) \) is constructed with respect to a given matrix \(A=(a_{n,k})\). We mention the notations and definitions of some classes of sequences consisting of nonnegative numbers to explain the amongst important relations. </p>
<p>A nonnegative sequence \(\left( a_{n,k}\right) \) is called <i class="itshape">almost monotone increasing (decreasing) sequence</i> if there exists a constant \(K_{1}:=K_{1}\left( a_{n,k}\right) \) (\(K_{2}:=K_{2}\left( a_{n,k}\right) )\) depending only the sequence \(\left( a_{n,k}\right) \) such that</p>
<div class="displaymath" id="a0000000021">
  \begin{equation*}  a_{n,k}\leq K_{1}a_{n,m}\text{ \  \  }\left( a_{n,m}\leq K_{2}a_{n,k}\right) \end{equation*}
</div>
<p>for all \(0\leq k\leq m\leq n\). If \(\left( a_{n,k}\right) \) is <i class="itshape">almost monotone decreasing sequence, </i>then<i class="itshape"> </i>we write \(\left( a_{n,k}\right) \in AMDS\) and if \(\left( a_{n,k}\right) \) is <i class="itshape">almost monotone increasing</i> <i class="itshape">sequence</i> then<i class="itshape"> </i>we write \(\left( a_{n,k}\right) \in AMIS\). </p>
<p>Let</p>
<div class="displaymath" id="a0000000022">
  \begin{equation*}  A_{n,k}=\tfrac {1}{k+1}\sum \limits _{j=n-k}^{n}a_{n,j}. \end{equation*}
</div>
<p>If \(\left( A_{n,k}\right) \in AMDS\), then \(\left( A_{n,k}\right) \) is called <i class="itshape">almost monotone decreasing upper mean sequence </i>and we write \(\left( A_{n,k}\right) \) \(\in AMDUMS\). If \(\left( A_{n,k}\right) \in AMIS\), then \(\left( A_{n,k}\right) \) is called <i class="itshape">almost monotone increasing upper mean sequence</i> and then we write \(\left( A_{n,k}\right) \in AMIUMS\). </p>
<p>There exist following embedding relations between these sequence classes </p>
<div class="displaymath" id="a0000000023">
  \begin{equation*}  NIS\subset AMDS\subset AMIUMS \end{equation*}
</div>
<p>and</p>
<div class="displaymath" id="a0000000024">
  \begin{equation*}  NDS\subset AMIS\subset AMDUMS, \end{equation*}
</div>
<p>where \(NIS\) is the class of nonnegative and nondecreasing sequences, \(NDS\) is the class of nonnegative and nonincreasing sequences <span class="cite">
	[
	<a href="#szal" >36</a>
	]
</span>. </p>
<p>We use also the notations </p>
<div class="displaymath" id="a0000000025">
  \begin{equation*}  \Delta a_{k}=a_{k}-a_{k+1}\text{, }\quad \Delta _{k}a_{n,k}=a_{n,k}-a_{n,k+1}\text{.} \end{equation*}
</div>
<p>We write \(m=\mathcal{O}\left( n\right) \) if there exists a positive constant \(K_{3}\) such that \(m\leq K_{3}n\). </p>
<p>Let now \(L^{p}\left( \mathbb {T}\right) \) be the classical Lebesgue space and let </p>
<div class="displaymath" id="a0000000026">
  \begin{equation*}  \omega \left( f,\delta \right) :=\sup _{0{\lt}h\leq \delta }\left\Vert f\left( \cdot +h\right) -f\right\Vert _{p}\text{,}\quad \delta {\gt}0 \end{equation*}
</div>
<p>is the modulus of continuity for \(f\in L^{p}\left( \mathbb {T}\right) \), where \(\left\Vert \cdot \right\Vert _{p}:=\rho _{p}^{1/p}\left( \cdot \right) \) for \(p\left( \cdot \right) :=\) \(p\) is <i class="itshape">const</i>. </p>
<p>In this case Lipschitz classes can be defined as </p>
<div class="displaymath" id="a0000000027">
  \begin{equation*}  \operatorname {Lip}\left( \alpha ,p\right) :=\left\{  f\in L^{p}\left( \mathbb {T}\right) :\omega \left( f,\delta \right) =\mathcal{O}\left( \delta ^{\alpha }\right) ,\text{ }\delta {\gt}0\right\}  \text{, } \end{equation*}
</div>
<p>where \(\alpha \in (0,1]\). </p>
<p>The rate of trigonometric approximation in the Lipschitz classes \(\operatorname {Lip}\left( \alpha ,p\right) \), \(\alpha \in (0,1]\) and \(1{\lt}p{\lt}\infty \), were investigated by a great number of authors. Initially, degree of approximation by \(\sigma _{n}\left( f\right) \), when \(f\in \operatorname {Lip}\left( \alpha ,p\right) \) was studied by Quade. He proved <span class="cite">
	[
	<a href="#quade" >31</a>
	]
</span> that if \(f\in \operatorname {Lip}\left( \alpha ,p\right) \) for \(\alpha \in (0,1]\) and \(1{\lt}p{\lt}\infty \), then \(\left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{p}=\mathcal{O}\left( n^{-\alpha }\right) \). There have been various generalizations of this result <span class="cite">
	[
	<a href="#mohapatra" >29</a>
	, 
	<a href="#chandra1" >2</a>
	, 
	<a href="#chandra" >3</a>
	, 
	<a href="#szal" >36</a>
	, 
	<a href="#guven1" >10</a>
	, 
	<a href="#guven2" >11</a>
	, 
	<a href="#israfilovguven" >12</a>
	, 
	<a href="#testici" >38</a>
	, 
	<a href="#jafarov" >25</a>
	, 
	<a href="#jafarov1" >21</a>
	, 
	<a href="#jafarov2" >24</a>
	, 
	<a href="#deger" >7</a>
	]
</span>. Especially, Chandra <span class="cite">
	[
	<a href="#chandra" >3</a>
	]
</span> generalizes the result of Quade and prove that if \(f\in \operatorname {Lip}\left( \alpha ,p\right) \) for \(\alpha \in (0,1]\) and \(1{\lt}p{\lt}\infty \), then \(\left\Vert f-N_{n}\left( f\right) \right\Vert _{p}=\mathcal{O}\left( n^{-\alpha }\right) \), when \(\left( p_{n}\right) \) is a monotonic sequence of positive real numbers such that \(\left( n+1\right) p_{n}=\mathcal{O}\left( P_{n}\right) \). Later, Leindler <span class="cite">
	[
	<a href="#leindler" >28</a>
	]
</span> generalizes the result of Chandra, imposing weaker assumptions on the sequence \(\left( p_{n}\right) \). Let’s give some results that are as close as possible to our work in the case of \(1{\lt}p{\lt}\infty \). </p>
<p><div class="theorem_thmwrapper " id="TheoremA">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> <span class="cite">
	[
	<a href="#leindler" >28</a>
	]
</span> <i class="itshape">Let</i> \(f\in \operatorname {Lip}\left( \alpha ,p\right) \) <i class="itshape">for</i> \(\alpha \in (0,1]\), \(1{\lt}p{\lt}\infty \) <i class="itshape">and</i> <i class="itshape">let</i> \(\left( p_{n}\right) \) <i class="itshape">be a sequence of positive real numbers</i>. <i class="itshape">If one of the conditions </i> </p>
<p>[i)] \(0{\lt}\alpha {\lt}1\) <i class="itshape">and</i> \(\left( p_{n}\right) \in AMDS,\) </p>
<p>[ii)] \(0{\lt}\alpha {\lt}1\), \(\left( p_{n}\right) \in AMIS\) <i class="itshape">and</i> \(\left( n+1\right) p_{n}=\mathcal{O}\left( P_{n}\right) ,\) </p>
<p>[iii)] \(\alpha =1\) <i class="itshape">and</i> \(\sum \limits _{k=1}^{n-1}k\left\vert \Delta p_{k}\right\vert =\mathcal{O}\left( P_{n}\right) ,\) </p>
<p>[iv)] \(\alpha =1\), \(\sum \limits _{k=0}^{n-1}\left\vert \Delta p_{k}\right\vert =\mathcal{O}\left( P_{n}\right) \) <i class="itshape">and</i> \(\left( n+1\right) p_{n}=\mathcal{O}\left( P_{n}\right) ,\) <br /><i class="itshape">holds,</i> <i class="itshape">then</i> </p>
<div class="displaymath" id="a0000000028">
  \begin{equation*}  \left\Vert f-N_{n}\left( f\right) \right\Vert _{p}=\mathcal{O}\left( n^{-\alpha }\right) . \end{equation*}
</div>

  </div>
</div> </p>
<p>Mittal and his collaborators <span class="cite">
	[
	<a href="#mittal" >30</a>
	]
</span> extend the results of Leindler by using matrix transforms of functions in \(\operatorname {Lip}\left( \alpha ,p\right) \), when the matrix \(A=\left( a_{n,k}\right) \) such that \(\left( a_{n,k}\right) \in NIS\) or \(\left( a_{n,k}\right) \in NDS\). In the weighted case Guven <span class="cite">
	[
	<a href="#guven2" >11</a>
	]
</span> proves this result using more general matrix transforms, namely for the matrices \(A=\left( a_{n,k}\right) \) such that \(\left( a_{n,k}\right) \in AMIS\) or \(\left( a_{n,k}\right) \in AMDS\). As a result we can deduce the following theorem from <i class="itshape">Theorems 1 and 2</i> proved in <span class="cite">
	[
	<a href="#guven2" >11</a>
	]
</span> for the matrix transforms defined as:</p>
<div class="displaymath" id="a0000000029">
  \begin{equation*}  T_{n}\left( f,A\right) :=\sum \limits _{k=0}^{n}a_{n,k}S_{k}\left( f\right) \left( x\right) ,\quad n=0,1,2,\ldots \end{equation*}
</div>
<p><div class="theorem_thmwrapper " id="TheoremB">
  <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="#guven2" >11</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p>  <i class="itshape">Let</i> \(f\in \operatorname {Lip}\left( \alpha ,p\right) \) <i class="itshape">for</i> \(\alpha \in (0,1]\), \(1{\lt}p{\lt}\infty \) <i class="itshape">and</i> \(A=\left( a_{n,k}\right) \) <i class="itshape">be a lower triangular matrix with non-negative entries such that </i>\(\left\vert s_{n}^{\left( A\right) }-1\right\vert =\mathcal{O}\left( n^{-\alpha }\right) \). <i class="itshape">If one of the conditions</i> </p>
<p>[i)] \(0{\lt}\alpha {\lt}1\), \(\left( a_{n,k}\right) \in AMDS\) <i class="itshape">and</i> \(\left( n+1\right) a_{n,0}=\mathcal{O}\left( 1\right) ,\) </p>
<p>[ii)] \(0{\lt}\alpha {\lt}1\), \(\left( a_{n,k}\right) \in AMIS\) <i class="itshape">and</i> \(\left( n+1\right) a_{n,r}=\mathcal{O}\left( 1\right) \) <i class="itshape">where</i> \(r\) <i class="itshape">is integer part of</i> \(n/2,\) </p>
<p>[iii)] \(\alpha =1\) <i class="itshape">and</i> \(\sum \limits _{k=1}^{n-1}\left\vert \Delta a_{n,k-1}\right\vert =\mathcal{O}\left( n^{-1}\right) ,\) </p>
<p><i class="itshape">holds,</i> <i class="itshape">then</i> </p>
<div class="displaymath" id="a0000000030">
  \begin{equation*}  \left\Vert f-T_{n}\left( f,A\right) \right\Vert _{p}=\mathcal{O}\left( n^{-\alpha }\right) . \end{equation*}
</div>

  </div>
</div> </p>
<p>This theorem was generalized to weighted and nonweighted Lebesgue space with variable exponent in <span class="cite">
	[
	<a href="#israfilovtestici" >18</a>
	, 
	<a href="#israfilovtestici2" >37</a>
	, 
	<a href="#guven3" >13</a>
	]
</span>, respectively. At the same time, it can be observed <span class="cite">
	[
	<a href="#szal" >36</a>
	]
</span> that theorem similar to <a href="#TheoremB">theorem 2</a> is also true under the assumption \(s_{n}^{\left( A\right) }=1\). </p>
<p><div class="theorem_thmwrapper " id="TheoremC">
  <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="#szal" >36</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p>  <i class="itshape">Let</i> \(f\in \operatorname {Lip}\left( \alpha ,p\right) \) <i class="itshape">for</i> \(\alpha \in (0,1]\), \(1{\lt}p{\lt}\infty \) <i class="itshape">and</i> \(A=\left( a_{n,k}\right) \) <i class="itshape">be a lower triangular matrix with non-negative entries and </i>\(s_{n}^{\left( A\right) }=1\). <i class="itshape">If one of the conditions</i> </p>
<p>[i)] \(0{\lt}\alpha {\lt}1\), \(\left( a_{n,k}\right) \in AMDUMS,\) </p>
<p>[ii)] \(0{\lt}\alpha {\lt}1\), \(\left( a_{n,k}\right) \in AMIUMS\) <i class="itshape">and</i> \(\left( n+1\right) a_{n,n}=\mathcal{O}\left( 1\right) ,\) </p>
<p>[iii)] \(\alpha =1\) <i class="itshape">and</i> \(\sum \limits _{k=0}^{n-2}\left\vert \Delta _{k}A_{n,k}\right\vert =\mathcal{O}\left( n^{-1}\right) ,\) </p>
<p><i class="itshape">holds,</i> <i class="itshape">then</i></p>
<div class="displaymath" id="a0000000031">
  \begin{equation*}  \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{p}=\mathcal{O}\left( n^{-\alpha }\right) . \end{equation*}
</div>

  </div>
</div> </p>
<h1 id="a0000000032">2 Main Results</h1>
<p>In this work, we estimate the error of trigonometric approximation by matrix transforms \(T_{n}^{\left( A\right) }\left( f\right) \) in \(f\in \operatorname {Lip}\left( \alpha ,p\left( \cdot \right) ,\theta \right) ,\) when \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) \), \(\alpha \in (0,1]\) and \(\theta {\gt}0\). We obtain the generalization of the above mentioned results. Theorem proved by us is stronger than the previous ones, because we prove it by imposing weaker assumptions. Main results are following. </p>
<p><div class="theorem_thmwrapper " id="theorem1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4</span>
  </div>
  <div class="theorem_thmcontent">
  <p><b class="bfseries"> </b><i class="itshape">Let</i> \(f\in \operatorname {Lip}\left( \mathbb {\alpha },p\left( \cdot \right) ,\theta \right) \) for \(\alpha \in (0,1],\) \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) ,\theta {\gt}0\) <i class="itshape">and</i> \(A=\left( a_{n,k}\right) \) <i class="itshape">be a lower triangular matrix with non-negative entries and </i>\(s_{n}^{\left( A\right) }=1\). <i class="itshape">If one of the conditions</i> </p>
<p>[i)] \(0{\lt}\alpha {\lt}1\), \(\left( a_{n,k}\right) \in AMDUMS,\) </p>
<p>[ii)] \(0{\lt}\alpha {\lt}1\), \(\left( a_{n,k}\right) \in AMIUMS\) <i class="itshape">and</i> \(\left( n+1\right) a_{n,n}=\mathcal{O}\left( 1\right) ,\) </p>
<p>[iii)] \(0{\lt}\alpha {\lt}1\) <i class="itshape">and</i> \(\sum \limits _{k=-1}^{n-1}\left\vert \Delta a_{n,k}\right\vert =\mathcal{O}\left( n^{-1}\right) \) <i class="itshape">where</i> \(a_{n,-1}=0,\) </p>
<p>[iv)] \(\alpha =1\) <i class="itshape">and</i> \(\sum \limits _{k=0}^{n-2}\left\vert \Delta _{k}A_{n,k}\right\vert =\mathcal{O}\left( n^{-1}\right) , \) <br /><i class="itshape">holds,</i> <i class="itshape">then</i> </p>
<div class="displaymath" id="a0000000033">
  \begin{equation*}  \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }=\mathcal{O}\left( n^{-\alpha }\right) . \end{equation*}
</div>

  </div>
</div> </p>
<p><a href="#theorem1">theorem 4</a> is more general than the corresponding theorems given in <span class="cite">
	[
	<a href="#israfilovtestici2" >37</a>
	]
</span>. In the case of \(p\left( \cdot \right) =\)<i class="itshape">const</i> and \(\theta =0\) it was proved in <span class="cite">
	[
	<a href="#szal" >36</a>
	]
</span>. </p>
<p>Let \(P_{n}=\sum _{k=0}^{n}p_{k}\), for a sequence \(\left( p_{n}\right) \) of positive real numbers and let </p>
<div class="displaymath" id="a0000000034">
  \begin{equation*}  P_{n,k}=\tfrac {1}{k+1}\sum _{j=n-k}^{n}p_{j}. \end{equation*}
</div>
<p>If \(a_{n,k}:=p_{k}/P_{n}\), then <a href="#theorem1">theorem 4</a> implies </p>
<p><div class="corollary_thmwrapper " id="corollary1">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">5</span>
  </div>
  <div class="corollary_thmcontent">
  <p><b class="bfseries"> </b><i class="itshape">Let</i> \(f\in \operatorname {Lip}\left( \mathbb {\alpha },p\left( \cdot \right) ,\theta \right) \) <i class="itshape">for</i> \(\alpha \in (0,1],\) \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) ,\theta {\gt}0\) <i class="itshape">and</i> \(\left( p_{n}\right) \) <i class="itshape">be a sequence of positive real numbers</i>. <i class="itshape">If one of the conditions</i> </p>
<p>[i)] \(0{\lt}\alpha {\lt}1\), \(\left( p_{k}\right) \in AMDUMS,\) </p>
<p>[ii)] \(0{\lt}\alpha {\lt}1\), \(\left( p_{k}\right) \in AMIUMS\) <i class="itshape">and</i> \(\left( n+1\right) p_{n}=\mathcal{O}\left( P_{n}\right) ,\) </p>
<p>[iii)] \(0{\lt}\alpha {\lt}1\) <i class="itshape">and</i> \(\sum \limits _{k=-1}^{n-1}\left\vert \Delta p_{k}\right\vert =\mathcal{O}\left( P_{n}/n\right) \) <i class="itshape">where</i> \(p_{-1}=0,\) </p>
<p>[iv)] \(\alpha =1\) <i class="itshape">and</i> \(\sum \limits _{k=0}^{n-2}\left\vert \Delta _{k}P_{n,k}\right\vert =\mathcal{O}\left( P_{n}/n\right) ,\) <br /><i class="itshape">holds,</i> <i class="itshape">then</i> </p>
<div class="displaymath" id="a0000000035">
  \begin{equation*}  \left\Vert f-N_{n}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }=\mathcal{O}\left( n^{-\alpha }\right) . \end{equation*}
</div>

  </div>
</div> </p>
<p>If \(p_{n}:=A_{n}^{\beta -1}\) for some \(\beta {\gt}0\), where</p>
<div class="displaymath" id="a0000000036">
  \begin{equation*}  A_{0}^{\beta }=1,\text{ \  }A_{k}^{\beta }=\tfrac {\left( \beta +1\right) \left( \beta +2\right) \ldots \left( \beta +k\right) }{k!},\text{ \  }k\geq 1 \end{equation*}
</div>
<p>then \(N_{n}\left( f\right) \) coincides with the generalized Cesàro means </p>
<div class="displaymath" id="a0000000037">
  \begin{equation*}  \sigma _{n}^{\beta }\left( f\right) \left( x\right) :=\tfrac {1}{A_{n}^{\beta }}\sum \limits _{k=0}^{n}A_{n-k}^{\beta -1}S_{k}\left( f\right) \left( x\right) \text{ },\quad n=0,1,2,\ldots \end{equation*}
</div>
<p>Hence, <a href="#corollary1">corollary 5</a> implies </p>
<p><div class="corollary_thmwrapper " id="corollary2">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">6</span>
  </div>
  <div class="corollary_thmcontent">
  <p><b class="bfseries"> </b><i class="itshape">Let</i> \(f\in \operatorname {Lip}\left( \mathbb {\alpha },p\left( \cdot \right) ,\theta \right) \) <i class="itshape">for</i> \(\alpha \in (0,1],\) \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) ,\theta {\gt}0\) <i class="itshape">and</i> \(\beta {\gt}0.\) <i class="itshape">Then</i> </p>
<div class="displaymath" id="a0000000038">
  \begin{equation*}  \left\Vert f-\sigma _{n}^{\beta }\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }=\mathcal{O}\left( n^{-\alpha }\right) . \end{equation*}
</div>

  </div>
</div> </p>
<p><a href="#corollary2">corollary 6</a> in the case of \(p\left( \cdot \right) =\)<i class="itshape">const</i> and \(\theta =0\) was proved in <span class="cite">
	[
	<a href="#chandra" >3</a>
	]
</span> (nonweighted case) and in <span class="cite">
	[
	<a href="#guven1" >10</a>
	]
</span> (weighted case). </p>
<h1 id="a0000000039">3 Auxiliary Results</h1>
<p>Let</p>
<div class="displaymath" id="a0000000040">
  \begin{equation*}  E_{n}\left( f\right) _{p\left( \cdot \right) ,\theta }:=\inf \left\{  \left\Vert f-T_{n}\right\Vert _{p\left( \cdot \right) ,\theta }:T_{n}\in \Pi _{n}\right\}  \end{equation*}
</div>
<p>be the best approximation number of \(f\in \) \(L_{0}^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) \) in the class \(\Pi _{n}\) of trigonometric polynomials of degree not exceeding \(n\). </p>
<p><div class="lemma_thmwrapper " id="lemma1">
  <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="#israfilovtestici2" >37</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p>  <i class="itshape">If</i> \(f\in L_{0}^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) ,\) \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) \) <i class="itshape">and</i> \(\theta {\gt}0\), <i class="itshape">then</i> <i class="itshape">the inequality</i></p>
<div class="displaymath" id="a0000000041">
  \begin{equation*}  E_{n}\left( f\right) _{p\left( \cdot \right) ,\theta }=\text{ }\mathcal{O}\left( \Omega \left( f,1/n\right) _{p\left( \cdot \right) ,\theta }\right) , \quad n=1,2,\ldots , \end{equation*}
</div>
<p><i class="itshape">holds.</i> </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="lemma2">
  <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="#israfilovtestici2" >37</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p>  <i class="itshape">Let</i> \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) ,\theta {\gt}0\) <i class="itshape">,</i> \(\alpha \in (0,1].\) <i class="itshape">If</i> \(f\in \operatorname {Lip}\left( \mathbb {\alpha },p\left( \cdot \right),\theta \right)\), <i class="itshape">then the inequality</i> </p>
<div class="displaymath" id="a0000000042">
  \begin{equation*}  \left\Vert f-S_{n}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }=\mathcal{O}\left( n^{-\alpha }\right) ,\quad n=1,2,3\ldots , \end{equation*}
</div>
<p><i class="itshape">holds.</i> </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="lemma3">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">9</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#israfilovtestici2" >37</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p>  <i class="itshape">Let</i> \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) \) <i class="itshape">and</i> \(\theta {\gt}0.\) <i class="itshape">If</i> \(f\in \operatorname {Lip}\left( 1,p\left( \cdot \right) ,\theta \right) \), <i class="itshape">then the inequality</i></p>
<div class="displaymath" id="a0000000043">
  \begin{equation*}  \left\Vert S_{n}\left( f\right) -\sigma _{n}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }=\mathcal{O}\left( n^{-1}\right) ,\quad n=1,2,3\ldots , \end{equation*}
</div>
<p><i class="itshape">holds.</i> </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="lemma4">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">10</span>
  </div>
  <div class="lemma_thmcontent">
  <p><i class="itshape">Let</i> \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) \) <i class="itshape">and</i> \(\theta {\gt}0.\) <i class="itshape">If</i> \(f\in \operatorname {Lip}\left( \alpha ,p\left( \cdot \right) ,\theta \right) ,\) \(\alpha \in \left( 0,1\right) ,\) <i class="itshape">then the inequality</i></p>
<div class="displaymath" id="a0000000044">
  \begin{equation*}  \left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }=\mathcal{O}\left( n^{-\alpha }\right),\quad n=1,2,3\ldots , \end{equation*}
</div>
<p><i class="itshape">holds.</i> </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000045">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Let \(f\in \operatorname {Lip}\left( \alpha ,p\left( \cdot \right) ,\theta \right) \), \(\alpha \in \left( 0,1\right) \), \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) \) and \(\theta {\gt}0\). The conjugate function of \(f\in L^{1}\left( \mathbb {T}\right) \) is defined as</p>
<div class="displaymath" id="a0000000046">
  \begin{equation*}  \widetilde{f}\left( x\right) :=\tfrac {1}{\pi }\int \limits _{-\pi }^{\pi }\tfrac {f\left( t\right) }{2\tan \frac{t-x}{2}}{\rm d}t. \end{equation*}
</div>
<p>Since \(f\in L_{0}^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) ,\) \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) \) and \(\theta {\gt}0\), <i class="itshape">Theorem 2.1</i> given in <span class="cite">
	[
	<a href="#kokinina1" >5</a>
	]
</span> provides that there exists a constant \(c_{1}\left( p,\theta \right) {\gt}0,\) depending on \(p\left( \cdot \right) \) and \(\theta \) such that </p>
<div class="displaymath" id="a0000000047">
  \begin{equation*}  \left\Vert \widetilde{f}\right\Vert _{p\left( \cdot \right) ,\theta }\leq c_{1}\left( p,\theta \right) \left\Vert f\right\Vert _{p\left( \cdot \right) ,\theta }\text{.} \end{equation*}
</div>
<p>Using this inequality and the standard technics developed in <span class="cite">
	[
	<a href="#zygmund" >39</a>
	]
</span> we obtain the inequality</p>
<div class="equation" id="a">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert \sigma _{n}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }\leq c_{2}\left( p,\theta \right) \left\Vert f\right\Vert _{p\left( \cdot \right) ,\theta } \label{a} \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p>for some constant \(c_{2}\left( p,\theta \right) {\gt}0\). </p>
<p>Let \(T_{n}^{0}\left( f\right) \) be the <i class="itshape">best approximation trigonometric polynomial</i> to \(f\) in \(L_{0}^{p\left( \cdot \right) ,\theta }\left( \mathbb {T}\right) ,\) that is \(\left\Vert f-T_{n}^{0}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }=E_{n}\left( f\right) _{p\left( \cdot \right) ,\theta }\) for \(n=0,1,2\ldots \) . Applying (<a href="#a">1</a>), the Minkowski<i class="itshape"> </i>inequality<i class="itshape"> </i>and <a href="#lemma1">lemma 7</a> we have </p>
<div class="displaymath" id="a0000000048">
  \begin{eqnarray*}  \left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta } & \leq & \left\Vert f-T_{n}^{0}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }+\left\Vert T_{n}^{0}\left( f\right) -\sigma _{n}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta } \\ & =& E_{n}\left( f\right) _{p\left( \cdot \right) ,\theta }+\left\Vert \sigma _{n}\left( T_{n}^{0}\left( f\right) -f\right) \right\Vert _{p\left( \cdot \right) ,\theta } \\ & =& E_{n}\left( f\right) _{p\left( \cdot \right) ,\theta }+\mathcal{O}(\left\Vert T_{n}^{0}\left( f\right) -f\right\Vert _{p\left( \cdot \right) ,\theta }) \\ & =& \mathcal{O}(E_{n}\left( f\right) _{p\left( \cdot \right) ,\theta })=\mathcal{O}\left( \Omega \left( f,1/n\right) _{p\left( \cdot \right) ,\theta }\right) \\ & =& \mathcal{O}\left( n^{-\alpha }\right) . \end{eqnarray*}
</div>
<p>Thus, lemma is proved. <div class="proof_wrapper" id="a0000000049">
  <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="lemma_thmwrapper " id="lemma5">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">11</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#szal" >36</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p>  <i class="itshape">Let</i> \(0{\lt}\alpha {\lt}1\) <i class="itshape">and</i> \(A=(a_{n,k})\) <i class="itshape">be infinite lower triangular matrix with non-negative entries and </i>\(s_{n}^{\left( A\right) }=1\). <i class="itshape">If one of the conditions</i> </p>
<p>[i)] \(\left( a_{n,k}\right) \in AMDUMS,\) </p>
<p>[ii)] \(\left( a_{n,k}\right) \in AMIUMS\) <i class="itshape">and</i> \(\left( n+1\right) a_{n,n}=\mathcal{O}\left( 1\right) ,\) <br /><i class="itshape">holds then</i></p>
<div class="displaymath" id="a0000000050">
  \begin{equation*}  \sum _{k=0}^{n}\left( k+1\right) ^{-\alpha }a_{n,n-k}=\mathcal{O}\left( \left( n+1\right) ^{-\alpha }\right) \text{.} \end{equation*}
</div>

  </div>
</div> </p>
<h1 id="a0000000051">4 Proof of Main Results</h1>
<p> <div class="proof_wrapper" id="a0000000052">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>[Proof of <a href="#theorem1">theorem 4</a>] <b class="bfseries"> </b>Let \(f\in \operatorname {Lip}\left( \mathbb {\alpha },p\left( \cdot \right) ,\theta \right) \), \(\alpha \in (0,1],\) \(p\left( \cdot \right) \in \mathcal{P}_{0}\left( \mathbb {T}\right) ,\theta {\gt}0\) and \(A=\left( a_{n,k}\right) \) be a lower triangular matrix with non-negative entries such that\(\mathit{\  }s_{n}^{\left( A\right) }=1\). We suppose that the conditions either i) or ii) holds. Since \(0{\lt}\alpha {\lt}1\) and </p>
<div class="displaymath" id="a0000000053">
  \begin{equation*}  f\left( x\right) -T_{n}^{\left( A\right) }\left( f\right) \left( x\right) =\sum \limits _{k=0}^{n}a_{n,n-k}\left( f\left( x\right) -S_{k}\left( f\right) \left( x\right) \right), \end{equation*}
</div>
<p> applying the Minkowski inequality, <a href="#lemma2">lemma 8</a> and <a href="#lemma5">lemma 11</a>, respectively we have that</p>
<div class="displaymath" id="a0000000054">
  \begin{eqnarray*}  \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta } & \leq & \sum \limits _{k=0}^{n}a_{n,n-k}\left\Vert f-S_{k}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta } \\ & \leq & c_{3}\sum \limits _{k=0}^{n}a_{n,n-k}\left( k+1\right) ^{-\alpha }=\mathcal{O}\left( n^{-\alpha }\right) . \end{eqnarray*}
</div>
<p>Therefore, we proved <a href="#theorem1">theorem 4</a> in the case of i) and ii). </p>
<p>Let \(0{\lt}\alpha {\lt}1\) and \(\sum \limits _{k=-1}^{n-1}\left\vert \Delta a_{n,k}\right\vert =\mathcal{O}\left( n^{-1}\right) \), where \(a_{n,-1}=0\). By Abel transformation (<span class="cite">
	[
	<a href="#bary" >1</a>
	, 
	p.
	
	1
	]
</span>) and definition of \(\sigma _{n}\left( f\right) \) we have </p>
<div class="displaymath" id="a0000000055">
  \begin{eqnarray*}  f\left( x\right) -T_{n}^{\left( A\right) }\left( f\right) \left( x\right) & =& \sum \limits _{k=0}^{n}a_{n,n-k}\left( f\left( x\right) -S_{k}\left( f\right) \left( x\right) \right) \\ & =& \sum \limits _{k=0}^{n-1}\left( a_{n,n-k}-a_{n,n-k-1}\right) \sum \limits _{j=0}^{k}\left( f\left( x\right) -S_{j}\left( f\right) \left( x\right) \right) \\ & & +a_{n,0}\sum \limits _{k=0}^{n}\left( f\left( x\right) -S_{k}\left( f\right) \left( x\right) \right) \\ & =& \sum \limits _{k=0}^{n-1}\left( a_{n,n-k}-a_{n,n-k-1}\right) \left( k+1\right) \left( f\left( x\right) -\sigma _{k}\left( f\right) \left( x\right) \right) \\ & & +a_{n,0}\left( n+1\right) \left( f\left( x\right) -\sigma _{n}\left( f\right) \left( x\right) \right) . \end{eqnarray*}
</div>
<p>Using Minkowski’s inequality, <a href="#lemma4">lemma 10</a> we obtain that </p>
<div class="displaymath" id="a0000000056">
  \begin{eqnarray*}  \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta } & \leq & \sum \limits _{k=0}^{n-1}\left\vert a_{n,n-k}-a_{n,n-k-1}\right\vert \left( k+1\right) \left\Vert f-\sigma _{k}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta } \\ & & +a_{n,0}\left( n+1\right) \left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta } \\ & \leq & c_{4}\left\{  \sum \limits _{k=0}^{n-1}\left\vert a_{n,n-k}\! -\! a_{n,n-k-1}\right\vert \left( k+1\right) ^{1-\alpha }\! \! +\! \! a_{n,0}\left( n\! \! +\! \! 1\right) ^{1-\alpha }\right\}  \end{eqnarray*}
</div>
<div class="displaymath" id="a0000000057">
  \begin{eqnarray*} & \leq & c_{5}\left( n+1\right) ^{1-\alpha }\left\{  \sum \limits _{k=0}^{n-1}\left\vert a_{n,n-k}-a_{n,n-k-1}\right\vert +a_{n,0}\right\}  \\ & =& c_{6}\left( n+1\right) ^{1-\alpha }\sum \limits _{k=-1}^{n-1}\left\vert \Delta a_{n,k}\right\vert =\mathcal{O}\left( n^{-\alpha }\right) . \end{eqnarray*}
</div>
<p>Hence, the iii) part of <a href="#theorem1">theorem 4</a> is also proved. Finally, we prove the last part of theorem. Let \(\alpha =1\) and \(\sum \limits _{k=0}^{n-2}\left\vert \Delta _{k}A_{n,k}\right\vert =\mathcal{O}\left( n^{-1}\right) \). Using twice Abel’s transformation we have </p>
<div class="displaymath" id="a0000000058">
  \begin{equation*}  f\left( x\right) -T_{n}^{\left( A\right) }\left( f\right) \left( x\right) =\sum \limits _{k=0}^{n}a_{n,n-k}\left( f\left( x\right) -S_{k}\left( f\right) \left( x\right) \right) \end{equation*}
</div>
<div class="displaymath" id="a0000000059">
  \begin{eqnarray*} & =& \sum \limits _{k=0}^{n-1}\left( S_{k+1}\left( f\right) \left( x\right) -S_{k}\left( f\right) \left( x\right) \right) \sum \limits _{j=0}^{k}a_{n,n-j}+\left( f\left( x\right) -S_{n}\left( f\right) \left( x\right) \right) \sum \limits _{k=0}^{n}a_{n,n-k} \\ & =& \sum \limits _{k=0}^{n-1}\left( S_{k+1}\left( f\right) \left( x\right) -S_{k}\left( f\right) \left( x\right) \right) \sum \limits _{j=n-k}^{n}a_{n,j}+\left( f\left( x\right) -S_{n}\left( f\right) \left( x\right) \right) s_{n}^{\left( A\right) } \\ & =& f\left( x\right) -S_{n}\left( f\right) \left( x\right) +\sum \limits _{k=0}^{n-1}u_{k+1}\left( f\right) \left( x\right) \left( k+1\right) A_{n,k} \\ & =& f\left( x\right) -S_{n}\left( f\right) \left( x\right) +\sum \limits _{k=0}^{n-2}\left( A_{n,k}-A_{n,k+1}\right) \sum \limits _{j=0}^{k}\left( j+1\right) u_{j+1}\left( f\right) \left( x\right) \\ & & +A_{n,n-1}\sum \limits _{k=0}^{n-1}\left( k+1\right) u_{k+1}\left( f\right) \left( x\right) \\ & =& f\left( x\right) -S_{n}\left( f\right) \left( x\right) +\sum \limits _{k=0}^{n-2}\left( A_{n,k}-A_{n,k+1}\right) \sum \limits _{j=0}^{k}\left( j+1\right) u_{j+1}\left( f\right) \left( x\right) \\ & & +\tfrac {1}{n}\sum \limits _{j=1}^{n}a_{n,j}\sum \limits _{k=0}^{n-1}\left( k+1\right) u_{k+1}\left( f\right) \left( x\right) . \end{eqnarray*}
</div>
<p>Hence, applying the Minkowski inequality we get </p>
<div class="displaymath" id="a0000000060">
  \begin{equation*}  \left\Vert f\! -\! T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }\leq \left\Vert f\! -\! S_{n}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }\! +\! \sum \limits _{k=0}^{n-2}\left\vert A_{n,k}\! -\! A_{n,k+1}\right\vert \left\Vert \sum \limits _{j=1}^{k+1}ju_{j}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta } \end{equation*}
</div>
<div class="displaymath" id="a0000000061">
  \begin{equation*}  +\tfrac {1}{n}s_{n}^{\left( A\right) }\left\Vert \sum \limits _{k=1}^{n}ku_{k}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta } \end{equation*}
</div>
<div class="displaymath" id="a0000000062">
  \begin{equation*}  = \left\Vert f-S_{n}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }+\sum \limits _{k=0}^{n-2}\left\vert \Delta _{k}A_{n,k}\right\vert \left\Vert \sum \limits _{j=1}^{k+1}ju_{j}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta } \end{equation*}
</div>
<div class="equation" id="b">
<p>
  <div class="equation_content">
    \begin{equation}  +\tfrac {1}{n}\left\Vert \sum \limits _{k=1}^{n}ku_{k}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }. \label{b} \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p>Since</p>
<div class="displaymath" id="a0000000063">
  \begin{eqnarray*}  S_{n}\left( f\right) \left( x\right) -\sigma _{n}\left( f\right) \left( x\right) & =& \sum \limits _{k=0}^{n}\left[ u_{k}\left( f\right) \left( x\right) -\tfrac {1}{n+1}\sum \limits _{\nu =0}^{k}u_{\nu }\left( f\right) \left( x\right) \right] \\ & =& \sum \limits _{k=0}^{n}\left( 1-\tfrac {n+1-k}{n+1}\right) u_{k}\left( f\right) \left( x\right) \\ & =& \tfrac {1}{n+1}\sum \limits _{k=0}^{n}ku_{k}\left( f\right) \left( x\right) , \end{eqnarray*}
</div>
<p>using <a href="#lemma3">lemma 9</a> we have</p>
<div class="equation" id="c">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert \sum \limits _{k=1}^{n}ku_{k}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }=\left( n+1\right) \left\Vert S_{n}\left( f\right) -\sigma _{n}\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }=\mathcal{O}\left( 1\right) . \label{c} \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p>Finally, combining  (<a href="#b">2</a>), (<a href="#c">3</a>) and <a href="#lemma2">lemma 8</a> for \(\alpha =1\) we obtain that </p>
<div class="displaymath" id="a0000000064">
  \begin{equation*}  \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{p\left( \cdot \right) ,\theta }\leq \mathcal{O}\big( n^{-1}\big) +\sum \limits _{k=0}^{n-2}\left\vert \Delta _{k}A_{n,k}\right\vert \mathcal{O}\left( 1\right) +\tfrac {1}{n}\mathcal{O}\left( 1\right) =\mathcal{O}\big( n^{-1}\big) . \end{equation*}
</div>
<p>Thereby, all parts of <a href="#theorem1">theorem 4</a> are proved. <div class="proof_wrapper" id="a0000000065">
  <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="a0000000066">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgements
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The authors would like to thank the anonymous referee. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
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</dd>
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  <dd><p><a href ="https://doi.org/10.1007/BF00375119"> <i class="sc">T. Iwaniec, C. Sbordone</i>, <i class="itshape">On integrability of the Jacobian under minimal hypotheses</i>, Arch. Rational Mechanics Anal., <b class="bfseries">119</b> (1992), pp. 129–143, <a href="https://doi.org/10.1007/BF00375119">https://doi.org/10.1007/BF00375119</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="sbor98">20</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/BF02819450"> <i class="sc">T. Iwaniec, C. Sbordone</i>, <i class="itshape">Riesz transform and elliptic pde’s with VMO coefficients, </i>J. Analyse Math., <b class="bfseries">74</b> (1998), pp. 183–212, <a href="https://doi.org/10.1007/BF02819450">https://doi.org/10.1007/BF02819450</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="jafarov1">21</a></dt>
  <dd><p><a href ="http://www.hjms.hacettepe.edu.tr/uploads/7fb9ca70-022b-4bc7-817e-3e39804a0ea3.pdf"> <i class="sc">S.Z. Jafarov</i>, <i class="itshape">Approximation by Fejér sums of Fourier trigonometric series in weighted Orlicz spaces</i>, Hacettepe Journal of Mathematics and Statistics, <b class="bfseries">42</b> (2013) no. 3, pp. 259–268, <a href="http://www.hjms.hacettepe.edu.tr/uploads/7fb9ca70-022b-4bc7-817e-3e39804a0ea3.pdf">http://www.hjms.hacettepe.edu.tr/uploads/7fb9ca70-022b-4bc7-817e-3e39804a0ea3.pdf</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="jafarovgrand">22</a></dt>
  <dd><p><i class="sc">S.Z. Jafarov</i>, <i class="itshape">Approximation by trigonometric polynomials in subspace of variable exponent grand Lebesgue spaces</i>, Global J. Math., <b class="bfseries">8</b> (2016) no. 2, pp. 836–843. </p>
</dd>
  <dt><a name="jafarovv">23</a></dt>
  <dd><p><a href ="https://doi.org/10.1080/17476933.2017.1379999"> <i class="sc">S.Z. Jafarov</i>, <i class="itshape">Approximation of the functions in weighted Lebesgue spaces with variable exponent</i>, Complex Var. Elliptic Equ., <b class="bfseries">63</b> (2018) no. 10, pp. 1444–1458, <a href="https://doi.org/10.1080/17476933.2017.1379999">https://doi.org/10.1080/17476933.2017.1379999</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="jafarov2">24</a></dt>
  <dd><p><a href ="https://dergipark.org.tr/tr/download/article-file/732343"> <i class="sc">S.Z. Jafarov</i>, <i class="itshape">Linear methods of summing Fourier series and approximation in weighted Orlicz spaces</i>, Turkish J. Math., <b class="bfseries">42</b> (2018), pp. 2916–2925, <a href="https://dergipark.org.tr/tr/download/article-file/732343">https://dergipark.org.tr/tr/download/article-file/732343</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="jafarov">25</a></dt>
  <dd><p><a href ="https://dergipark.org.tr/tr/download/article-file/948899"> <i class="sc">S.Z. Jafarov</i>, <i class="itshape">Approximation by matrix transforms in weighted Orlicz spaces</i>, Turkish J. Math., <b class="bfseries">44</b> (2020), pp. 179–193, <a href="https://dergipark.org.tr/tr/download/article-file/948899">https://dergipark.org.tr/tr/download/article-file/948899</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="humberto">26</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/978-3-319-21018-6"> <i class="sc">V.M. Kokilashvili, A. Meskhi, H. Rafeiro</i>, et al., <i class="itshape">Integral Operators in Non-standard Function Spaces</i>, volumes I and II, Birkhäuser, Basel, 2016 <a href="https://doi.org/10.1007/978-3-319-21018-6">https://doi.org/10.1007/978-3-319-21018-6</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="kokimeskhi">27</a></dt>
  <dd><p><a href ="https://doi.org/10.1515/gmj-2014-0047"> <i class="sc">V.M. Kokilashvili, A. Meskhi</i>, <i class="itshape">Maximal and Calderón–Zygmund operators in grand variable exponent Lebesgue spaces</i>, Georgian Math. J., <b class="bfseries">21</b> (2014) no. 4, pp. 447–461, <a href="https://doi.org/10.1515/gmj-2014-0047">https://doi.org/10.1515/gmj-2014-0047</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="leindler">28</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.jmaa.2004.07.049"> <i class="sc">L. Leindler</i>, <i class="itshape">Trigonometric approximation in L</i>\(_{p}\)<i class="itshape">-norm,</i> J. Math. Anal. Appl., <b class="bfseries">302</b> (2005) no. 1, pp. 129–136, <a href="https://doi.org/10.1016/j.jmaa.2004.07.049">https://doi.org/10.1016/j.jmaa.2004.07.049</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="mohapatra">29</a></dt>
  <dd><p><a href ="https://doi.org/10.1017/S144678870002317X"> <i class="sc">R.N. Mohapatra, D.C. Russell</i>, <i class="itshape">Some direct and inverse theorems in approximation of functions,</i> J. Austral. Math. Soc. (Ser. A), <b class="bfseries">34</b> (1983), pp. 143–154, <a href="https://doi.org/10.1017/S144678870002317X">https://doi.org/10.1017/S144678870002317X</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="mittal">30</a></dt>
  <dd><p><i class="sc">M. L. Mittal, B. E. Rhoades, V.N. Mishra, U. Singh</i>, <a href ="https://doi.org/10.1016/j.jmaa.2006.03.053"> <i class="itshape">Using infinite matrices to approximate functions of class image using trigonometric polynomials</i>, J. Math. Anal. Appl., <b class="bfseries">326</b> (2007) no. 1, pp. 667–676, <a href="https://doi.org/10.1016/j.jmaa.2006.03.053">https://doi.org/10.1016/j.jmaa.2006.03.053</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="quade">31</a></dt>
  <dd><p><a href ="http://doi.org/10.1215/S0012-7094-37-00342-9"> <i class="sc">E.S. Quade</i>, <i class="itshape">Trigonometric approximation in the mean</i>, Duke Math. J., <b class="bfseries">3</b> (1937), pp. 529–543, <a href="http://doi.org/10.1215/S0012-7094-37-00342-9">http://doi.org/10.1215/S0012-7094-37-00342-9</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="sbor96">32</a></dt>
  <dd><p><a href ="https://lematematiche.dmi.unict.it/index.php/lematematiche"> <i class="sc">C. Sbordone</i>, <i class="itshape">Grand Sobolev spaces and their applications to variational problems,</i> Le Mathematiche, LI(2) (1996), pp. 335–347 <a href="https://lematematiche.dmi.unict.it/index.php/lematematiche">https://lematematiche.dmi.unict.it/index.php/lematematiche</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="sbor98.1">33</a></dt>
  <dd><p><i class="sc">C. Sbordone</i>, <i class="itshape">Nonlinear elliptic equations with right hand side in nonstandard spaces, </i>Rend. Sem. Math. Fis. Modena, Supplemento al XLVI (1998), pp. 361–368. </p>
</dd>
  <dt><a name="sharap1">34</a></dt>
  <dd><p><a href ="https://iopscience.iop.org/article/10.1070/IM2013v077n02ABEH002641"> <i class="sc">I.I. Sharapudinov</i>, <i class="itshape">Approximation of functions in </i>\(L_{2\pi }^{p\left( x\right) }\)<i class="itshape"> by trigonometric polynomials</i>, Izvestiya RAN: Ser. Math., 77(2), (2013), 197–224; English transl., Izvestiya: Mathematics, <b class="bfseries">77</b> (2013) no. 2, pp. 407–434, <a href="https://iopscience.iop.org/article/10.1070/IM2013v077n02ABEH002641">https://iopscience.iop.org/article/10.1070/IM2013v077n02ABEH002641</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Sharapudinov 2">35</a></dt>
  <dd><p><i class="sc">I.I. Sharapudinov</i>, <a href ="https://iopscience.iop.org/article/10.1070/SM8509"> <i class="itshape">Approximation of functions by De Vallée Poussin means in the Lebesgue and Sobolev spaces with variable exponent</i>, Sbornik Math., <b class="bfseries">207</b> (2016) no. 7, pp. 1010–1036, <a href="https://iopscience.iop.org/article/10.1070/SM8509">https://iopscience.iop.org/article/10.1070/SM8509</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="szal">36</a></dt>
  <dd><p><a href ="https://cmuc.karlin.mff.cuni.cz/pdf/cmuc0904/szal.pdf"> <i class="sc">B. Szal</i>, <i class="itshape">Trigonometric approximation by Nörlund type means in</i> \(L^{p}\)<i class="itshape">-norm</i>, Comment. Math. Univ. Carolin., <b class="bfseries">50</b> (2009) no. 4, pp. 575–589, <a href="https://cmuc.karlin.mff.cuni.cz/pdf/cmuc0904/szal.pdf">https://cmuc.karlin.mff.cuni.cz/pdf/cmuc0904/szal.pdf</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="israfilovtestici2">37</a></dt>
  <dd><p><a href ="https://doi.org/10.1080/00036811.2019.1622680"> <i class="sc">A. Testici, D.M. Israfilov</i>, <i class="itshape">Approximation by matrix transform in generalized grand Lebesgue spaces with variable exponent</i>, Applicable Analysis, <b class="bfseries">100</b> (2019) no. 4, pp. 819–834, <a href="https://doi.org/10.1080/00036811.2019.1622680">https://doi.org/10.1080/00036811.2019.1622680</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="testici">38</a></dt>
  <dd><p><a href ="https://doi.org/10.31801/cfsuasmas.460449"> <i class="sc">A. Testici</i>, <i class="itshape">Approximation by Nörlund and Riesz means in weighted Lebesgue space with variable exponent</i>, Commun. Fac. Sci. Univ. Ank. Series A1, <b class="bfseries">68</b> (2019), pp. 2014–2025, <a href="https://doi.org/10.31801/cfsuasmas.460449">https://doi.org/10.31801/cfsuasmas.460449</a>. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="zygmund">39</a></dt>
  <dd><p><i class="sc">A. Zygmund</i>, <i class="itshape">Trigonometric Series</i>, vol. I and II, Cambridge University Press, 1959. </p>
</dd>
</dl>


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