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<h1>Approximation of continuous functions on hexagonal domains</h1>
<p class="authors">
<span class="author">Ali Guven\(^\ast \)</span>
</p>
<p class="date">September 6, 2017. Accepted: March 26, 2018. Published online: August 6, 2018.</p>
</div>
<p>\(^\ast \)Department of Mathematics, Faculty of Arts and Sciences, Balikesir University, 10145 Balikesir, Turkey. e-mail: <span class="tt">guvennali@gmail.com</span>. </p>

<div class="abstract"><p> Some approximation properties of hexagonal Fourier series are investigated. The order of approximation by Nörlund means of hexagonal Fourier series is estimated in terms of modulus of continuity. </p>
<p><b class="bf">MSC.</b> 41A25, 41A63, 42B08. </p>
<p><b class="bf">Keywords.</b> Hexagonal Fourier series, modulus of continuity, Nörlund means, order of approximation. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \(C_{2\pi }\) be the Banach space of \(2\pi \)-periodic continuous functions on the real line, equipped with the norm </p>
<div class="displaymath" id="a0000000003">
  \begin{equation*}  \left\Vert f\right\Vert _{C_{2\pi }}:=\underset {0\leq x\leq 2\pi }{\sup }\left\vert f\left( x\right) \right\vert . \end{equation*}
</div>
<p>The modulus of continuity of a function \(f\in C_{2\pi }\) is defined by </p>
<div class="displaymath" id="a0000000004">
  \begin{equation*}  \omega \left( f,\delta \right) :=\underset {0{\lt}\left\vert h\right\vert \leq \delta }{\sup }\left\Vert f-T_{h}\left( f\right) \right\Vert _{C_{2\pi }},\text{ }\left( \delta {\gt}0\right) , \end{equation*}
</div>
<p>where \(T_{h}\left( f\right) \left( x\right) :=f\left( x+h\right) .\) For \(0{\lt}\alpha \leq 1,\) we denote by \(H_{2\pi }^{\alpha }\) the Hölder class of functions \(f\in C_{2\pi }\) such that \(\omega \left( f,\delta \right) \ll \delta ^{\alpha },\) where \(A\ll B\) means that there exists a constant \(K{\gt}0\) such that \(A\leq KB\) holds. </p>
<p>Approximation of functions belonging the space \(C_{2\pi }\) by trigonometric polynomials is one of the most important topics in approximation theory and it has a very rich history. Especially, the order of approximation of functions in \(H_{2\pi }^{\alpha }\) classes was studied by several mathematicians. Linear summation methods of Fourier series are mostly used tools in these studies. </p>
<p>Let \(f\in C_{2\pi }\) has the Fourier series </p>
<div class="equation" id="FS">
<p>
  <div class="equation_content">
    \begin{equation}  f\left( x\right) \sim \underset {k=-\infty }{\overset {\infty }{\sum }}\widehat{f}_{k}e^{ikx}, \label{FS} \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p>with partial sums </p>
<div class="displaymath" id="a0000000005">
  \begin{equation*}  S_{n}\left( f\right) \left( x\right) :=\underset {k=-n}{\overset {n}{\sum }}\widehat{f}_{k}e^{ikx},\text{ }\left( n=0,1,...\right) . \end{equation*}
</div>
<p>We denote by \(\left( \sigma _{n}\left( f\right) \right) \) the sequence of Fejér means of \(\left( \ref{FS}\right) ,\) <i class="itshape">i.e.</i>, </p>
<div class="displaymath" id="a0000000006">
  \begin{equation*}  \sigma _{n}\left( f\right) \left( x\right) =\tfrac {1}{n+1}\underset {k=0}{\overset {n}{\sum }}S_{k}\left( f\right) \left( x\right) . \end{equation*}
</div>
<p>In 1912, S.N. Bernstein obtained the following estimate for the approximation order by Fejér means. </p>
<p><span class="scshape">Theorem A</span>. <span class="cite">
	[
	<a href="#bernstein" >2</a>
	]
</span>. <b class="bfseries"> </b><i class="itshape">Let </i>\(f\in H_{2\pi }^{\alpha }\)<i class="itshape"> </i>\(\left( 0{\lt}\alpha \leq 1\right) .\) <i class="itshape">Then the estimate </i></p>
<div class="equation" id="B">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{C_{2\pi }}\ll \left\{  \begin{array}{cc} \frac{1}{n^{\alpha }}, &  \alpha <1 \\[2mm] \frac{\log n}{n}, &  \alpha =1\end{array}\right. \label{B} \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p><i class="itshape">holds for </i>\(n\geq 2.\) </p>
<p>S.B. Stechkin extended Bernstein’s result as follows. </p>
<p><span class="scshape">Theorem B</span>. <span class="cite">
	[
	<a href="#stechkin" >14</a>
	]
</span>. <b class="bfseries"> </b><i class="itshape">Let </i>\(f\in C_{2\pi }.\)<i class="itshape"> Then the estimate </i></p>
<div class="equation" id="S">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{C_{2\pi }}\ll \tfrac {1}{n+1}\underset {k=0}{\overset {n}{\sum }}\omega \big( f,\tfrac {1}{k+1}\big) \label{S} \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p><i class="itshape">holds for every natural number </i>\(n.\) </p>
<p>Let \(p=\left( p_{n}\right) _{n=0}^{\infty }\) be a sequence of positive real numbers and let \(P_{n}=\underset {k=0}{\overset {n}{\sum }}p_{k}. \) Nörlund means of the series \(\left( \ref{FS}\right) \) with respect to the sequence \(p\) are defined by </p>
<div class="displaymath" id="a0000000007">
  \begin{equation*}  N_{n}\left( p;f\right) \left( x\right) =\tfrac {1}{P_{n}}\underset {k=0}{\overset {n}{\sum }}p_{n-k}S_{k}\left( f\right) \left( x\right) . \end{equation*}
</div>
<p>It is known that Nörlund summability method is regular if and only if \(p_{n}/P_{n}\rightarrow 0\) as \(n\rightarrow \infty \) <span class="cite">
	[
	<a href="#hardy" >8</a>
	, 
	p.
	
	64
	]
</span>. It is clear that \(N_{n}\left( p;f\right) \) coincides with \(\sigma _{n}\left( f\right) \) in the special case \(p_{n}=1\) \(\left( n=0,1,...\right) \). </p>
<p>In 1976, A.S.B. Holland, B. Sahney and J. Tzimbalario obtained a more general result than Theorem B. </p>
<p><span class="scshape">Theorem C</span>. <span class="cite">
	[
	<a href="#holland s t" >9</a>
	]
</span>. <b class="bfseries"> </b><i class="itshape">Let </i>\(p=\left( p_{n}\right) _{n=0}^{\infty }\)<i class="itshape"> be a sequence of positive real numbers such that </i>\(np_{n}\ll P_{n}.\)<i class="itshape"> Then for every </i>\(f\in C_{2\pi },\)<i class="itshape"> the inequality </i></p>
<div class="equation" id="HST">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-N_{n}\left( p;f\right) \right\Vert _{C_{2\pi }}\ll \tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega \big( f,\tfrac 1k\big) \label{HST} \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p><i class="itshape">holds. </i> </p>
<p>It is clear that in the case \(p_{n}=1\) \(\left( n=1,2,...\right) \) (<a href="#HST">4</a>) reduces to (<a href="#S">3</a>). Theorem C also extends a result of B. Sahney and D.S. Goel <span class="cite">
	[
	<a href="#sahney goel" >13</a>
	]
</span> which states that</p>
<div class="equation" id="SG">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-N_{n}\left( p;f\right) \right\Vert _{C_{2\pi }}\ll \tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {P_{k}}{k^{1+\alpha }} \label{SG} \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p>for \(f\in H_{2\pi }^{\alpha }\), where \(\left( p_{n}\right) \) is a non-increasing sequence of positive real numbers. </p>
<p>These theorems can be found in the survey <span class="cite">
	[
	<a href="#holland" >10</a>
	]
</span>. Also, we refer to the monographs <span class="cite">
	[
	<a href="#alexits" >1</a>
	]
</span>, <span class="cite">
	[
	<a href="#butzer nessel" >3</a>
	]
</span>, <span class="cite">
	[
	<a href="#devore lorentz" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#timan" >16</a>
	]
</span> and <span class="cite">
	[
	<a href="#zygmund" >18</a>
	]
</span> for more information and results about trigonometric approximation theory. </p>
<p>Approximation problems on cubes of the \(d-\)dimensional Euclidean space \(\mathbb {R}^{d}\) are studied by assuming that the functions are \(2\pi \)-periodic in each of their variables (see, for example <span class="cite">
	[
	<a href="#timan" >16</a>
	, 
	Sections
	
	5.3
	
	and
	
	6.3
	]
</span> and <span class="cite">
	[
	<a href="#zygmund" >18</a>
	, 
	vol.
	
	II,
	
	ch.
	
	XVII
	]
</span>). But, in the case of non-tensor product domain, for example for hexagonal domains in the Euclidean plane \(\mathbb {R}^{2},\) another definition of periodicity is needed. For such domains the most useful periodicity is the periodicity with respect to the lattices. </p>
<p>Let \(A\) be a non-singular \(d\times d\) matrix. The discrete subgroup \(A\mathbb {Z}^{d}=\big\{  Ak:k\in \mathbb {Z}^{d}\big\}  \) of the Euclidean space \(\mathbb {R}^{d}\) is called the lattice generated by \(A\), and the matrix \(A\) is called the generator matrix of this lattice. The lattice \(A^{-tr}\mathbb {Z}^{d}\), where \(A^{-tr}\) is the transpose of the inverse matrix \(A^{-1},\) is called the dual lattice of \(A\mathbb {Z}^{d}.\) A bounded set \(\Omega \subset \mathbb {R}^{d}\) is said to tile \(\mathbb {R}^{d}\) with the lattice \(A\mathbb {Z}^{d}\) if</p>
<div class="displaymath" id="a0000000008">
  \begin{equation*}  \underset {\alpha \in A\mathbb {Z}^{d}}{\sum }\chi _{\Omega }\left( x+\alpha \right) =1 \end{equation*}
</div>
<p>holds almost everwhere, that is, for almost every \(x\in \mathbb {R}^{d}\) there exists exactly one \(\alpha \in A\mathbb {Z}^{d}\) such that \(x+\alpha \in \Omega .\) In this case the set \(\Omega \) is called a spectral set for the lattice \(A\mathbb {Z}^{d}.\) One suppose that the spectral set \(\Omega \) contains \(0\) as an interior point and tiles \(\mathbb {R}^{d}\) with the lattice \(A\mathbb {Z}^{d}\) without overlapping and without gap, <i class="itshape">i.e.</i>, </p>
<div class="displaymath" id="a0000000009">
  \begin{equation*}  \underset {k\in \mathbb {Z}^{d}}{\sum }\chi _{\Omega }\left( x+Ak\right) =1 \end{equation*}
</div>
<p>for all \(x\in \mathbb {R}^{d}\) and \(\Omega +Ak\) and \(\Omega +Aj\) are disjoint if \(k\neq j.\) For example we can take \(\Omega =\big[ -\frac{1}{2},\frac{1}{2}\big) ^{d}\) for the standard lattice \(\mathbb {Z}^{d}\) (the lattice generated by the identity matrix)\(.\) </p>
<p>Let \(\Omega \) be the spectral set of the lattice \(A\mathbb {Z}^{d}.\) \(L^{2}\left( \Omega \right) \) becomes a Hilbert space with respect to the inner product </p>
<div class="displaymath" id="a0000000010">
  \begin{equation*}  \left\langle f,g\right\rangle _{\Omega }:=\tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }f\left( x\right) \overline{g\left( x\right) }dx, \end{equation*}
</div>
<p>where \(\left\vert \Omega \right\vert \) is the \(d-\)dimensional Lebesgue measure of \(\Omega .\) A theorem of Fuglede states that the set \(\big\{  e^{2\pi i\left\langle \alpha ,x\right\rangle }:\alpha \in A^{-tr}\mathbb {Z}^{d}\big\}  \) is an orthonormal basis of the Hilbert space \(L^{2}\left( \Omega \right) \), where \(\left\langle \alpha ,x\right\rangle \) is the usual Euclidean inner product of \(\alpha \) and \(x\) <span class="cite">
	[
	<a href="#fuglede" >5</a>
	]
</span>. According to this theorem, Fourier series and approximation on the spectral set of the lattice \(A\mathbb {Z}^{d}\) can be studied by using the exponentials \(e^{2\pi i\left\langle \alpha ,x\right\rangle }\) \(\big( \alpha \in A^{-tr}\mathbb {Z}^{d}\big) .\) </p>
<p>A function \(f\) is said to be periodic with respect to the lattice \(A\mathbb {Z}^{d}\) if </p>
<div class="displaymath" id="a0000000011">
  \begin{equation*}  f\left( x+Ak\right) =f\left( x\right) \end{equation*}
</div>
<p>for all \(k\in \mathbb {Z}^{d}.\) </p>
<p>If we consider the standard lattice \(\mathbb {Z}^{d}\) and its spectral set \(\big[ -\frac{1}{2},\frac{1}{2}\big) ^{d},\) Fourier series with respect to this lattice coincide with usual multiple Fourier series of functions of \(d-\)variables. </p>
<p>We refer to <span class="cite">
	[
	<a href="#li sun xu" >11</a>
	]
</span> for more detailed information about Fourier analysis on lattices. </p>
<h1 id="a0000000012">2 Hexagonal Fourier series</h1>
<p>In the Euclidean plane \(\mathbb {R}^{2},\) besides the standard lattice \(\mathbb {Z}^{2}\) and the rectangular domain \(\big[ -\frac{1}{2},\frac{1}{2}\big) ^{2},\) the simplest lattice is the hexagon lattice and the simplest spectral set is the regular hexagon. </p>
<p>The generator matrix and the spectral set of the hexagonal lattice \(H\mathbb {Z}^{2}\) are given by </p>
<div class="displaymath" id="a0000000013">
  \begin{equation*}  H=\left[ \begin{array}{cc} \sqrt{3} &  0 \\ -1 &  2\end{array}\right] \end{equation*}
</div>
<p>and </p>
<div class="displaymath" id="a0000000014">
  \begin{equation*}  \Omega _{H}=\left\{  \left( x_{1},x_{2}\right) \in \mathbb {R}^{2}:-1\leq x_{2},\tfrac {\sqrt{3}}{2}x_{1}\pm \tfrac {1}{2}x_{2}{\lt}1\right\}  . \end{equation*}
</div>
<p>It is more convenient to use the homogeneous coordinates \(\left( t_{1},t_{2},t_{3}\right) \) that satisfy \(t_{1}+t_{2}+t_{3}=0\). If we define</p>
<div class="equation" id="homogeneous">
<p>
  <div class="equation_content">
    \begin{equation}  t_{1}:=-\tfrac {x_{2}}{2}+\tfrac {\sqrt{3}x_{1}}{2},\text{ }t_{2}:=x_{2},\text{ }t_{3}:=-\tfrac {x_{2}}{2}-\tfrac {\sqrt{3}x_{1}}{2}, \label{homogeneous} \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p>the hexagon \(\Omega _{H}\) becomes </p>
<div class="displaymath" id="a0000000015">
  \begin{equation*}  \Omega =\left\{  \left( t_{1},t_{2},t_{3}\right) \in \mathbb {R}^{3}:-1\leq t_{1},t_{2},-t_{3}{\lt}1,\text{ }t_{1}+t_{2}+t_{3}=0\right\}  . \end{equation*}
</div>
<p>We use bold letters \(\mathbf{t}\) for homogeneous coordinates and we denote by \(\mathbb {R}_{H}^{3}\) the plane \(t_{1}+t_{2}+t_{3}=0,\) that is </p>
<div class="displaymath" id="a0000000016">
  \begin{equation*}  \mathbb {R}_{H}^{3}=\left\{  \mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \mathbb {R}^{3}:t_{1}+t_{2}+t_{3}=0\right\}  . \end{equation*}
</div>
<p>Also we use the notation \(\mathbb {Z}_{H}^{3}\) for the set of points in \(\mathbb {R}_{H}^{3}\) with integer components, that is \(\mathbb {Z}_{H}^{3}=\mathbb {Z}^{3}\cap \mathbb {R}_{H}^{3}.\) </p>
<p>It follows from \((\ref{homogeneous})\) that the Jacobian determinant of the change of variables \(x=\left( x_{1},x_{2}\right) \rightarrow \mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \) is \(dx_{1}dx_{2}=\frac{2\sqrt{3}}{3}dt_{1}dt_{2}.\) </p>
<p>In the homogeneous coordinates, the inner product on \(L^{2}\left( \Omega \right) \) becomes</p>
<div class="displaymath" id="a0000000017">
  \begin{equation*}  \left\langle f,g\right\rangle _{H}=\tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }f\left( \mathbf{t}\right) \overline{g\left( \mathbf{t}\right) }d\mathbf{t}, \end{equation*}
</div>
<p>where \(\left\vert \Omega \right\vert \) denotes the area of \(\Omega ,\) and the orthonormal basis of \(L^{2}\left( \Omega \right) \) becomes </p>
<div class="displaymath" id="a0000000018">
  \begin{equation*}  \left\{  \phi _{\mathbf{j}}\left( \mathbf{t}\right) =e^{\frac{2\pi i}{3}\left\langle \mathbf{j},\mathbf{t}\right\rangle }:\mathbf{j}\in \mathbb {Z}_{H}^{3},\text{ }\mathbf{t}\in \mathbb {R}_{H}^{3}\right\}  . \end{equation*}
</div>
<p>Also, a function \(f\) is periodic with respect to the hexagonal lattice (or \(H\)-periodic) if and only if \(f\left( \mathbf{t}\right) =f\left( \mathbf{t}+\mathbf{s}\right) \) whenever \(\mathbf{s}\equiv \mathbf{0}\) \(\left( \operatorname {mod}3\right) ,\) where \(\mathbf{t}\equiv \mathbf{s}\) \(\left( \operatorname {mod}3\right) \) defined as </p>
<div class="displaymath" id="a0000000019">
  \begin{equation*}  t_{1}-s_{1}\equiv t_{2}-s_{2}\equiv t_{3}-s_{3}\text{ }\left( \operatorname {mod}3\right) . \end{equation*}
</div>
<p>It is clear that the functions \(\phi _{\mathbf{j}}\left( \mathbf{t}\right) \) are \(H-\)periodic. If the function \(f\) is \(H-\)periodic then </p>
<div class="displaymath" id="a0000000020">
  \begin{equation*}  \underset {\Omega }{\int }f\left( \mathbf{t}+\mathbf{s}\right) d\mathbf{t=}\underset {\Omega }{\int }f\left( \mathbf{t}\right) d\mathbf{t},\text{ }\left( \mathbf{s}\in \mathbb {R}_{H}^{3}\right) . \end{equation*}
</div>
<p>For every natural number \(n,\) we define a subset of \(\mathbb {Z}_{H}^{3}\) by </p>
<div class="displaymath" id="a0000000021">
  \begin{equation*}  \mathbb {H}_{n}:=\left\{  \mathbf{j}=\left( j_{1},j_{2},j_{3}\right) \in \mathbb {Z}_{H}^{3}:-n\leq j_{1},j_{2},j_{3}\leq n\right\}  . \end{equation*}
</div>
<p>Note that, \(\mathbb {H}_{n}\) consists of all points with integer components inside the hexagon \(n\overline{\Omega }.\) Members of the set </p>
<div class="displaymath" id="a0000000022">
  \begin{equation*}  \mathcal{H}_{n}:=\text{span}\left\{  \phi _{\mathbf{j}}:\mathbf{j}\in \mathbb {H}_{n}\right\}  ,\text{ }\left( n\in \mathbb {N}\right) \end{equation*}
</div>
<p>are called hexagonal trigonometric polynomials. It is clear that the dimension of \(\mathcal{H}_{n}\) is \(\# \mathbb {H}_{n}=3n^{2}+3n+1.\) </p>
<p>The hexagonal Fourier series of an \(H-\)periodic function \(f\in L^{1}\left( \Omega \right) \) is </p>
<div class="equation" id="fourier">
<p>
  <div class="equation_content">
    \begin{equation}  f\left( \mathbf{t}\right) \sim \underset {\mathbf{j}\in \mathbb {Z}_{H}^{3}}{\sum }\widehat{f}_{\mathbf{j}}\phi _{\mathbf{j}}\left( \mathbf{t}\right) , \label{fourier} \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p>where </p>
<div class="displaymath" id="a0000000023">
  \begin{equation*}  \widehat{f}_{\mathbf{j}}=\tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }f\left( \mathbf{t}\right) e^{-\frac{2\pi i}{3}\left\langle \mathbf{j},\mathbf{t}\right\rangle }d\mathbf{t},\text{ }\left( \mathbf{j}\in \mathbb {Z}_{H}^{3}\right) . \end{equation*}
</div>
<p>The \(n\)th partial sum of the series \((\ref{fourier})\) is defined by </p>
<div class="displaymath" id="a0000000024">
  \begin{equation*}  S_{n}\left( f\right) \left( \mathbf{t}\right) :=\underset {\mathbf{j}\in \mathbb {H}_{n}}{\sum }\widehat{f}_{\mathbf{j}}\phi _{\mathbf{j}}\left( \mathbf{t}\right) \text{, }\left( n\in \mathbb {N}\right) . \end{equation*}
</div>
<p>The partial sums have the integral representation </p>
<div class="equation" id="partial">
<p>
  <div class="equation_content">
    \begin{equation}  S_{n}\left( f\right) \left( \mathbf{t}\right) =\tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }f\left( \mathbf{t}-\mathbf{s}\right) D_{n}\left( \mathbf{s}\right) d\mathbf{s}, \label{partial} \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p>where</p>
<div class="displaymath" id="a0000000025">
  \begin{equation*}  D_{n}\left( \mathbf{t}\right) :=\underset {\mathbf{j}\in \mathbb {H}_{n}}{\sum }\phi _{\mathbf{j}}\left( \mathbf{t}\right) \end{equation*}
</div>
<p>is the Dirichlet kernel of order \(n\). </p>
<p>It is known that (<span class="cite">
	[
	<a href="#sun" >15</a>
	]
</span>, <span class="cite">
	[
	<a href="#li sun xu" >11</a>
	]
</span>) the Dirichlet kernel can be expressed as </p>
<div class="equation" id="dirichlet">
<p>
  <div class="equation_content">
    \begin{equation}  D_{n}\left( \mathbf{t}\right) =\Theta _{n}\left( \mathbf{t}\right) -\Theta _{n-1}\left( \mathbf{t}\right) ,\text{ }\left( n\in \mathbb {N}\right) , \label{dirichlet} \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p>where</p>
<div class="equation" id="thetan">
<p>
  <div class="equation_content">
    \begin{equation}  \Theta _{n}\left( \mathbf{t}\right) :=\tfrac {\sin \frac{\left( n+1\right) \left( t_{1}-t_{2}\right) \pi }{3}\sin \frac{\left( n+1\right) \left( t_{2}-t_{3}\right) \pi }{3}\sin \frac{\left( n+1\right) \left( t_{3}-t_{1}\right) \pi }{3}}{\sin \frac{\left( t_{1}-t_{2}\right) \pi }{3}\sin \frac{\left( t_{2}-t_{3}\right) \pi }{3}\sin \frac{\left( t_{3}-t_{1}\right) \pi }{3}} \label{thetan} \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p>for \(\mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \mathbb {R}_{H}^{3}.\) </p>
<p>More general information about hexagonal Fourier series can be found in <span class="cite">
	[
	<a href="#li sun xu" >11</a>
	]
</span> and <span class="cite">
	[
	<a href="#xu" >17</a>
	]
</span>. </p>
<h1 id="a0000000026">3 Main result</h1>
<p>We denote by \(C_{H}\big( \overline{\Omega }\big) \) the set of complex valued \(H-\)periodic continuous functions defined on \(\mathbb {R}_{H}^{3}\). \(C_{H}\big( \overline{\Omega }\big) \) becomes a Banach space with respect to the uniform norm </p>
<div class="displaymath" id="a0000000027">
  \begin{equation*}  \left\Vert f\right\Vert _{C_{H}\big( \overline{\Omega }\big) }=\sup \left\{  \left\vert f\left( \mathbf{t}\right) \right\vert :\mathbf{t}\in \overline{\Omega }\right\}  . \end{equation*}
</div>
<p>The modulus of continuity of the function \(f\in C_{H}\big( \overline{\Omega }\big) \) is defined by </p>
<div class="displaymath" id="a0000000028">
  \begin{equation*}  \omega _{H}\left( f,\delta \right) :=\underset {0{\lt}\left\Vert \mathbf{h}\right\Vert \leq \delta }{\sup }\left\Vert f-T_{\mathbf{h}}\left( f\right) \right\Vert _{C_{H}\big( \overline{\Omega }\big) }, \end{equation*}
</div>
<p>where \(T_{\mathbf{h}}\left( f\right) \left( \mathbf{t}\right) =f\left( \mathbf{t}+\mathbf{h}\right) \) and </p>
<div class="displaymath" id="a0000000029">
  \begin{equation*}  \left\Vert \mathbf{h}\right\Vert :=\max \left\{  \left\vert h_{1}\right\vert ,\left\vert h_{2}\right\vert ,\left\vert h_{3}\right\vert \right\}  \end{equation*}
</div>
<p>for \(\mathbf{h}=\left( h_{1},h_{2},h_{3}\right) \in \mathbb {R}_{H}^{3}.\) It is known that <span class="cite">
	[
	<a href="#xu" >17</a>
	]
</span> the modulus of continuity is a non-decreasing function and satisfies </p>
<div class="equation" id="modc">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{H}\left( f,\lambda \delta \right) \leq \left( 1+\lambda \right) \omega _{H}\left( f,\delta \right) \label{modc} \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p>for \(\lambda {\gt}0.\) </p>
<p>For \(0{\lt}\alpha \leq 1,\) we define the Hölder class \(H^{\alpha }( \overline{\Omega }) \) of \(H-\)periodic continuous functions as </p>
<div class="displaymath" id="a0000000030">
  \begin{equation*}  H^{\alpha }(\overline{\Omega }) :=\left\{  f\in C_{H}( \overline{\Omega }) :\omega _{H}\left( f,\delta \right) \ll \delta ^{\alpha },\text{ }\delta {\gt}0\right\}  . \end{equation*}
</div>
<p>The Fejér means of the series \(\left( \ref{fourier}\right) \) are defined by </p>
<div class="displaymath" id="a0000000031">
  \begin{equation*}  \sigma _{n}\left( f\right) \left( \mathbf{t}\right) =\tfrac {1}{n+1}\overset {n}{\underset {k=0}{\sum }}S_{k}\left( f\right) \left( \mathbf{t}\right) . \end{equation*}
</div>
<p>The following analogue of Theorem A for hexagonal Fourier series was proved in <span class="cite">
	[
	<a href="#guven2013" >6</a>
	]
</span>. </p>
<p><span class="scshape">Theorem D</span>. <i class="itshape">Let </i>\(f\in H^{\alpha }\left( \overline{\Omega }\right) \)<i class="itshape"> </i>\(\left( 0{\lt}\alpha \leq 1\right) .\) <i class="itshape">Then the estimate </i></p>
<div class="equation" id="G">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\ll \left\{  \begin{array}{cc} \frac{1}{n^{\alpha }}, &  \alpha <1 \\[2mm] \frac{\left( \log n\right) ^{2}}{n}, &  \alpha =1\end{array}\right. \label{G} \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p><i class="itshape">holds for </i>\(n\geq 2.\) </p>
<p>Let \(p=\left( p_{n}\right) _{n=0}^{\infty }\) be a sequence of positive real numbers and \(\left( N_{n}\left( p;f\right) \right) \) be the sequence of Nörlund means of the series (<a href="#fourier">7</a>) with respect to the sequence \(p,\) that is</p>
<div class="equation" id="NS">
<p>
  <div class="equation_content">
    \begin{equation}  N_{n}\left( p;f\right) \left( \mathbf{t}\right) =\tfrac {1}{P_{n}}\overset {n}{\underset {k=0}{\sum }}p_{n-k}S_{k}\left( f\right) \left( \mathbf{t}\right) ,\text{ }\left( n\in \mathbb {N}\right) . \label{NS} \end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<p>By considering \(\left( \ref{partial}\right) ,\) we get </p>
<div class="equation" id="NINT">
<p>
  <div class="equation_content">
    \begin{equation}  N_{n}\left( p;f\right) \left( \mathbf{t}\right) =\tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }f\left( \mathbf{t}-\mathbf{s}\right) F_{n}\left( p;\mathbf{s}\right) d\mathbf{s}, \label{NINT} \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<p>where </p>
<div class="displaymath" id="a0000000032">
  \begin{equation*}  F_{n}\left( p;\mathbf{t}\right) :=\tfrac {1}{P_{n}}\overset {n}{\underset {k=0}{\sum }}p_{n-k}D_{k}\left( \mathbf{t}\right) . \end{equation*}
</div>
<p>The aim of this work is to prove an analogue of Theorem C for hexagonal Fourier series. The main result is the following. </p>
<p><span class="scshape">Theorem 1</span>. <i class="itshape">Let </i>\(p=\left( p_{n}\right) \)<i class="itshape"> be a non-increasing sequence of positive real numbers. Then the estimate </i></p>
<div class="equation" id="Th1">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-N_{n}\left( p;f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\ll \tfrac {1}{P_{n}}\log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) \label{Th1} \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p><i class="itshape">holds for every </i>\(f\in C_{H}\big( \overline{\Omega }\big) \) <i class="itshape">and for every natural number </i>\(n.\) </p>
<p><div class="proof_wrapper" id="a0000000033">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> By (<a href="#NINT">14</a>), definition of \(\omega _{H}\left( f,\cdot \right) ,\) (<a href="#NS">13</a>) and (<a href="#thetan">10</a>) we have </p>
<div class="equation" id="fminusNn">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert f\left( \mathbf{t}\right) -N_{n}\left( p;f\right) \left( \mathbf{t}\right) \right\vert \ll \tfrac {1}{P_{n}}\int _\Omega \omega _{H}\left( f,\left\Vert \mathbf{s}\right\Vert \right) \left\vert p_{n}\! +\! \overset {n}{\underset {k=1}{\sum }}p_{n-k}\left( \Theta _{k}\left( \mathbf{s}\right) \! -\! \Theta _{k-1}\left( \mathbf{s}\right) \right) \right\vert d\mathbf{s}. \label{fminusNn} \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p>Since the function </p>
<div class="displaymath" id="a0000000034">
  \begin{equation*}  \mathbf{t}\rightarrow \omega _{H}\left( f,\left\Vert \mathbf{t}\right\Vert \right) \left\vert p_{n}+\underset {k=1}{\overset {n}{\sum }}p_{n-k}\left( \Theta _{k}\left( \mathbf{t}\right) -\Theta _{k-1}\left( \mathbf{t}\right) \right) \right\vert \end{equation*}
</div>
<p>is symmetric with respect to variables \(t_{1},t_{2}\) and \(t_{3}\), where \(\mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \Omega ,\) it is sufficient to estimate the integral </p>
<div class="displaymath" id="a0000000035">
  \begin{equation*}  I_{n}:= \int _\Delta \omega _{H}\left( f,\left\Vert \mathbf{t}\right\Vert \right) \left\vert p_{n}+\underset {k=1}{\overset {n}{\sum }}p_{n-k}\left( \Theta _{k}\left( \mathbf{t}\right) -\Theta _{k-1}\left( \mathbf{t}\right) \right) \right\vert d\mathbf{t,} \end{equation*}
</div>
<p>where</p>
<div class="displaymath" id="a0000000036">
  \begin{align*}  \Delta & :=\left\{  \mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \mathbb {R}_{H}^{3}:0\leq t_{1},t_{2},-t_{3}\leq 1\right\}  \\ & =\left\{  \left( t_{1},t_{2}\right) :t_{1}\geq 0,\text{ }t_{2}\geq 0,\text{ }t_{1}+t_{2}\leq 1\right\}  , \end{align*}
</div>
<p>which is one of the six equilateral triangles in \(\overline{\Omega }.\) By considering the formula </p>
<p>\((\ref{thetan})\), we obtain </p>
<div class="displaymath" id="a0000000037">
  \begin{align*}  I_{n} =& \underset {\Delta }{\int }\omega _{H}\left( f,\left\Vert \mathbf{t}\right\Vert \right) \left\vert p_{n}+\underset {k=1}{\overset {n}{\sum }}p_{n-k}\left( \Theta _{k}\left( \mathbf{t}\right) -\Theta _{k-1}\left( \mathbf{t}\right) \right) \right\vert d\mathbf{t} \\ =& \underset {\Delta }{\int }\omega _{H}\left( f,t_{1}+t_{2}\right) \bigg\vert p_{n}+\underset {k=1}{\overset {n}{\sum }} p_{n-k}\Big( \tfrac {\sin \frac{\left( k+1\right) \left( t_{1}-t_{2}\right) \pi }{3}\sin \frac{\left( k+1\right) \left( t_{2}-t_{3}\right) \pi }{3}\sin \frac{\left( k+1\right) \left( t_{3}-t_{1}\right) \pi }{3}}{\sin \frac{\left( t_{1}-t_{2}\right) \pi }{3}\sin \frac{\left( t_{2}-t_{3}\right) \pi }{3}\sin \frac{\left( t_{3}-t_{1}\right) \pi }{3}} \\ & \hspace{5.5cm}-\tfrac {\sin \frac{k\left( t_{1}-t_{2}\right) \pi }{3}\sin \frac{k\left( t_{2}-t_{3}\right) \pi }{3}\sin \frac{k\left( t_{3}-t_{1}\right) \pi }{3}}{\sin \frac{\left( t_{1}-t_{2}\right) \pi }{3}\sin \frac{\left( t_{2}-t_{3}\right) \pi }{3}\sin \frac{\left( t_{3}-t_{1}\right) \pi }{3}}\Big) \bigg\vert d\mathbf{t}. \end{align*}
</div>
<p>If we use the change of variables</p>
<div class="displaymath" id="a0000000038">
  \begin{equation*}  s_{1}:=\tfrac {t_{1}-t_{3}}{3}=\tfrac {2t_{1}+t_{2}}{3},\text{ }s_{2}:=\tfrac {t_{2}-t_{3}}{3}=\tfrac {t_{1}+2t_{2}}{3} \end{equation*}
</div>
<p>as in <span class="cite">
	[
	<a href="#xu" >17</a>
	]
</span>, we get </p>
<div class="displaymath" id="a0000000039">
  \begin{align*}  I_{n}= &  3\underset {\widetilde{\Delta }}{\int }\omega _{H}\left( f,s_{1}\! +\!  s_{2}\right) \bigg\vert p_{n}\! \! +\! \! \underset {k=1}{\overset {n}{\sum }}p_{n-k}\Big( \tfrac {\sin \left( \left( k+1\right) \left( s_{1}-s_{2}\right) \pi \right) \sin \left( \left( k+1\right) s_{2}\pi \right) \sin \left( \left( k+1\right) \left( -s_{1}\pi \right) \right) }{\sin \left( \left( s_{1}-s_{2}\right) \pi \right) \sin \left( s_{2}\pi \right) \sin \left( -s_{1}\pi \right) } \\ & \hspace{5 cm}-\tfrac {\sin \left( k\left( s_{1}-s_{2}\right) \pi \right) \sin \left( ks_{2}\pi \right) \sin \left( k\left( -s_{1}\pi \right) \right) }{\sin \left( \left( s_{1}-s_{2}\right) \pi \right) \sin \left( s_{2}\pi \right) \sin \left( -s_{1}\pi \right) }\Big) \bigg\vert ds_{1}ds_{2}, \end{align*}
</div>
<p>where \(\widetilde{\Delta }\) is the image of \(\Delta \) in the plane, that is</p>
<div class="displaymath" id="a0000000040">
  \begin{equation*}  \widetilde{\Delta }:=\left\{  \left( s_{1},s_{2}\right) :0\leq s_{1}\leq 2s_{2},\text{ }0\leq s_{2}\leq 2s_{1},\text{ }s_{1}+s_{2}\leq 1\right\}  . \end{equation*}
</div>
<p>Since the integrated function is symmetric with respect to \(s_{1}\) and \(s_{2},\) we have</p>
<div class="displaymath" id="a0000000041">
  \begin{align*}  I_{n}=6\underset {\Delta ^{\ast }}{\int }\left( s_{1}+s_{2}\right) ^{\alpha }\bigg\vert p_{n}+\underset {k=1}{\overset {n}{\sum }}p_{n-k}\Big( \tfrac {\sin \left( \left( k+1\right) \left( s_{1}-s_{2}\right) \pi \right) \sin \left( \left( k+1\right) s_{2}\pi \right) \sin \left( \left( k+1\right) \left( -s_{1}\pi \right) \right) }{\sin \left( \left( s_{1}-s_{2}\right) \pi \right) \sin \left( s_{2}\pi \right) \sin \left( -s_{1}\pi \right) } \\ -\tfrac {\sin \left( k\left( s_{1}-s_{2}\right) \pi \right) \sin \left( ks_{2}\pi \right) \sin \left( k\left( -s_{1}\pi \right) \right) }{\sin \left( \left( s_{1}-s_{2}\right) \pi \right) \sin \left( s_{2}\pi \right) \sin \left( -s_{1}\pi \right) }\Big) \bigg\vert ds_{1}ds_{2}, \end{align*}
</div>
<p>where \(\Delta ^{\ast }\) is the half of \(\widetilde{\Delta }:\)</p>
<div class="displaymath" id="a0000000042">
  \begin{equation*}  \Delta ^{\ast }:=\left\{  \left( s_{1},s_{2}\right) \in \widetilde{\Delta }:s_{1}\leq s_{2}\right\}  =\left\{  \left( s_{1},s_{2}\right) :s_{1}\leq s_{2}\leq 2s_{1},\text{ }s_{1}+s_{2}\leq 1\right\}  . \end{equation*}
</div>
<p>The change of variables</p>
<div class="displaymath" id="a0000000043">
  \begin{equation*}  s_{1}:=\tfrac {u_{1}-u_{2}}{2},\text{ }s_{2}:=\tfrac {u_{1}+u_{2}}{2} \end{equation*}
</div>
<p>transforms the triangle \(\Delta ^{\ast }\) to the triangle </p>
<div class="displaymath" id="a0000000044">
  \begin{equation*}  \Gamma :=\left\{  \left( u_{1},u_{2}\right) :0\leq u_{2}\leq \tfrac {u_{1}}{3},\text{ }0\leq u_{1}\leq 1\right\}  , \end{equation*}
</div>
<p>hence we have </p>
<div class="displaymath" id="a0000000045">
  \begin{equation*}  I_{n}=3\underset {\Gamma }{\int }\omega _{H}\left( f,u_{1}\right) \left\vert p_{n}+\underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}, \end{equation*}
</div>
<p>where </p>
<div class="displaymath" id="a0000000046">
  \begin{align*}  D_{k}^{\ast }\left( u_{1},u_{2}\right) :=& \tfrac {\sin \left( \left( k+1\right) u_{2}\pi \right) \sin \left( \left( k+1\right) \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \left( k+1\right) \left( \frac{u_{1}-u_{2}}{2}\pi \right) \right) }{\sin \left( u_{2}\pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) } \\ & -\tfrac {\sin \left( ku_{2}\pi \right) \sin \left( k\frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( k\left( \frac{u_{1}-u_{2}}{2}\pi \right) \right) }{\sin \left( u_{2}\pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) }. \end{align*}
</div>
<p>By elementary trigonometric identities, we obtain</p>
<div class="equation" id="expression1">
<p>
  <div class="equation_content">
    \begin{equation}  D_{k}^{\ast }\left( u_{1},u_{2}\right) =D_{k,1}^{\ast }\left( u_{1},u_{2}\right) +D_{k,2}^{\ast }\left( u_{1},u_{2}\right) +D_{k,3}^{\ast }\left( u_{1},u_{2}\right) , \label{expression1} \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p>where </p>
<div class="displaymath" id="a0000000047">
  \begin{align*}  D_{k,1}^{\ast }\left( u_{1},u_{2}\right) & :=2\cos \left( \big( k+\tfrac {1}{2}\big) u_{2}\pi \right) \tfrac {\sin \left( \frac{1}{2}u_{2}\pi \right) \sin \left( \left( k+1\right) \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \left( k+1\right) \frac{u_{1}-u_{2}}{2}\pi \right) }{\sin \left( u_{2}\pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) }, \\ D_{k,2}^{\ast }\left( u_{1},u_{2}\right) & :=2\cos \left( \big( k+\tfrac {1}{2}\big) \tfrac {u_{1}+u_{2}}{2}\pi \right) \tfrac {\sin \left( ku_{2}\pi \right) \sin \left( \frac{1}{2}\frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \left( k+1\right) \frac{u_{1}-u_{2}}{2}\pi \right) }{\sin \left( u_{2}\pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) }, \end{align*}
</div>
<p>and </p>
<div class="displaymath" id="a0000000048">
  \begin{align*}  D_{k,3}^{\ast }\left( u_{1},u_{2}\right) & :=2\cos \left( \big( k+\tfrac {1}{2}\big) \tfrac {u_{1}-u_{2}}{2}\pi \right) \tfrac {\sin \left( ku_{2}\pi \right) \sin \left( k\frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{1}{2}\frac{u_{1}-u_{2}}{2}\pi \right) }{\sin \left( u_{2}\pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) }. \end{align*}
</div>
<p>Since </p>
<div class="displaymath" id="a0000000049">
  \begin{equation*}  \sin 2x+\sin 2y+\sin 2z=-4\sin x\sin y\sin z \end{equation*}
</div>
<p>for \(x+y+z=0,\) we also get the expression </p>
<div class="equation" id="expression2">
<p>
  <div class="equation_content">
    \begin{equation}  D_{k}^{\ast }\left( u_{1},u_{2}\right) =H_{k,1}\left( u_{1},u_{2}\right) +H_{k,2}\left( u_{1},u_{2}\right) +H_{k,3}\left( u_{1},u_{2}\right) , \label{expression2} \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p>where</p>
<div class="displaymath" id="a0000000050">
  \begin{align*}  H_{k,1}\left( u_{1},u_{2}\right) & :=\tfrac {1}{2}\tfrac {\cos \left( \left( 2k+1\right) u_{2}\pi \right) }{\sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) }, \\ H_{k,2}\left( u_{1},u_{2}\right) & :=-\tfrac {1}{2}\tfrac {\cos \left( \left( 2k+1\right) \frac{u_{1}+u_{2}}{2}\pi \right) }{\sin \left( u_{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) }, \\ H_{k,3}\left( u_{1},u_{2}\right) & :=\tfrac {1}{2}\tfrac {\cos \left( \left( 2k+1\right) \frac{u_{1}-u_{2}}{2}\pi \right) }{\sin \left( u_{2}\pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) }. \end{align*}
</div>
<p>By considering the fact \(\left( n+1\right) p_{n}\ll P_{n}\) and by (<a href="#modc">11</a>) we get </p>
<div class="displaymath" id="a0000000051">
  \begin{eqnarray*}  \underset {\Gamma }{\int }p_{n}\omega _{H}\left( f,u_{1}\right) du_{1}du_{2} & \leq & p_{n}\omega _{H}\left( f,1\right) \ll \tfrac {P_{n}}{n}\omega _{H}\left( f,1\right) \\ & =& \tfrac {P_{n}}{n}\omega _{H}\big( f,n\tfrac {1}{n}\big) \ll \tfrac {P_{n}}{n}n\omega _{H}\big( f,\tfrac {1}{n}\big) \\ & =& \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{n}P_{n}\omega _{H}\big( f,\tfrac {1}{n}\big) \leq \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) , \end{eqnarray*}
</div>
<p>since the sequence \(\left( P_{n}/n\right) \) non-increasing and \(\omega _{H}\left( f,\cdot \right) \) is non-decreasing. Hence, </p>
<div class="equation" id="A">
<p>
  <div class="equation_content">
    \begin{equation}  I_{n}\ll I_{n}^{\ast }+\underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f, \tfrac {1}{k}\big) , \label{A} \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p>where </p>
<div class="displaymath" id="a0000000052">
  \begin{equation*}  I_{n}^{\ast }:=\underset {\Gamma }{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}. \end{equation*}
</div>
<p>If we partition the triangle \(\Gamma \) as \(\Gamma =\Gamma _{1}\cup \Gamma _{2}\cup \Gamma _{3},\) where </p>
<div class="displaymath" id="a0000000053">
  \begin{align*}  \Gamma _{1} & :=\left\{  \left( u_{1},u_{2}\right) \in \Gamma :u_{1}\leq \tfrac {1}{n+1}\right\}  , \\ \Gamma _{2} & :=\left\{  \left( u_{1},u_{2}\right) \in \Gamma :u_{1}\geq \tfrac {1}{n+1},\text{ }u_{2}\leq \tfrac {1}{3\left( n+1\right) }\right\}  , \\ \Gamma _{3} & :=\left\{  \left( u_{1},u_{2}\right) \in \Gamma :u_{1}\geq \tfrac {1}{n+1},\text{ }u_{2}\geq \tfrac {1}{3\left( n+1\right) }\right\}  , \end{align*}
</div>
<p>we have </p>
<div class="displaymath" id="a0000000054">
  \begin{equation*}  I_{n}^{\ast }=I_{n,1}^{\ast }+I_{n,2}^{\ast }+I_{n,3}^{\ast }, \end{equation*}
</div>
<p>where </p>
<div class="displaymath" id="a0000000055">
  \begin{equation*}  I_{n,j}^{\ast }:=\underset {\Gamma _{j}}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2},\text{ }\left( j=1,2,3\right) . \end{equation*}
</div>
<p>We shall need the well known inequalities</p>
<div class="equation" id="sin1">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert \tfrac {\sin nt}{\sin t}\right\vert \leq n,\text{ }\left( n\in \mathbb {N}\right) , \label{sin1} \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<p>and</p>
<div class="equation" id="sin2">
<p>
  <div class="equation_content">
    \begin{equation}  \sin t\geq \tfrac {2}{\pi }t,\text{ }\left( 0\leq t\leq \tfrac {\pi }{2}\right) \label{sin2} \end{equation}
  </div>
  <span class="equation_label">21</span>
</p>
</div>
<p>to estimate integrals \(I_{n,1}^{\ast },I_{n,2}^{\ast }\) and \(I_{n,3}^{\ast }. \) </p>
<p>By (<a href="#expression1">17</a>) and (<a href="#sin1">20</a>) we obtain</p>
<div class="displaymath" id="a0000000056">
  \begin{align*}  I_{n,1}^{\ast }& =\underset {\Gamma _{1}}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \\ & \leq \underset {\Gamma _{1}}{\int }\omega _{H}\left( f,u_{1}\right) \left( \underset {k=1}{\overset {n}{\sum }}\left( k+1\right) ^{2}p_{n-k}\right) du_{1}du_{2} \\ & \leq \left( n+1\right) ^{2}P_{n}\underset {\Gamma _{1}}{\int }\omega _{H}\left( f,u_{1}\right) du_{1}du_{2} \\ & =\left( n+1\right) ^{2}P_{n}\underset {0}{\overset {1/(3(n+1)) }{\int }}\underset {3u_{2}}{\overset {1/(n+1)}{\int }}\omega _{H}\left( f,u_{1}\right) du_{1}du_{2} \\ & \leq \left( n+1\right) ^{2}P_{n}\omega _{H}\left( f,\tfrac {1}{n+1}\right) \underset {0}{\overset {1/(3\left( n+1\right)) }{\int }}\underset {3u_{2}}{\overset {1/(n+1)}{\int }}du_{1}du_{2} \\ & \leq P_{n}\omega _{H}\left( f,\tfrac {1}{n}\right) =\underset {k=1}{\overset {n}{\sum }}\tfrac {1}{n}P_{n}\omega _{H}\big( f,\tfrac {1}{n}\big) . \end{align*}
</div>
<p>Since the sequence \(\left( P_{n}/n\right) \) is non-increasing we get </p>
<div class="equation" id="B2">
<p>
  <div class="equation_content">
    \begin{equation}  I_{n,1}^{\ast }\leq \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \label{B2} \end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p>We write the rectangle \(\Gamma _{2}\) as \(\Gamma _{2}=\Gamma _{2}^{\prime }\cup \Gamma _{2}^{\prime \prime },\) where </p>
<div class="displaymath" id="a0000000057">
  \begin{equation*}  \Gamma _{2}^{\prime }:=\left\{  \left( u_{1},u_{2}\right) \in \Gamma _{2}:u_{2}\leq \tfrac {p_{n}}{3\left( n+1\right) P_{n}}\right\}  \end{equation*}
</div>
<p>and </p>
<div class="displaymath" id="a0000000058">
  \begin{equation*}  \Gamma _{2}^{\prime \prime }:=\left\{  \left( u_{1},u_{2}\right) \in \Gamma _{2}:u_{2}\geq \tfrac {p_{n}}{3\left( n+1\right) P_{n}}\right\}  \end{equation*}
</div>
<p>to estimate \(I_{n,2}^{\ast }.\) </p>
<p>By (<a href="#sin2">21</a>) we obtain </p>
<div class="displaymath" id="a0000000059">
  \begin{eqnarray*} & & \! \! \! \! \! \! \! \! \! \! \underset {\Gamma _{2}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k,1}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\leq \\ & \leq & \overset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\underset {0}{\int }}\underset {\frac{1}{n+1}}{\overset {1}{\int }}\omega _{H}\left( f,u_{1}\right) \left( \underset {k=1}{\overset {n}{\sum }}p_{n-k}\left\vert D_{k,1}^{\ast }\left( u_{1},u_{2}\right) \right\vert \right) du_{1}du_{2} \\ & \ll & P_{n}\overset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\underset {0}{\int }}\underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{1}du_{2}=\tfrac {p_{n}}{3\left( n+1\right) }\underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{1} \\ & =& \tfrac {p_{n}}{3\left( n+1\right) }\underset {1}{\overset {n+1}{\int }}\omega _{H}\big( f,\tfrac {1}{t}\big) dt=\tfrac {p_{n}}{3\left( n+1\right) }\underset {k=1}{\overset {n}{\sum }}\left( \underset {k}{\overset {k+1}{\int }}\omega _{H}\big( f,\tfrac {1}{t}\big) dt\right) \\ & \leq & \tfrac {p_{n}}{n+1}\underset {k=1}{\overset {n}{\sum }}\omega _{H}\big( f,\tfrac {1}{k}\big) \leq \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \end{eqnarray*}
</div>
<p>For \(j=2,3,\) by (<a href="#sin1">20</a>) and (<a href="#sin2">21</a>),</p>
<div class="displaymath" id="a0000000060">
  \begin{eqnarray*} & & \! \! \! \! \! \! \! \! \underset {\Gamma _{2}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k,j}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\leq \\ & \leq & \underset {\frac{1}{n+1}}{\overset {1}{\int }}\overset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\underset {0}{\int }}\omega _{H}\left( f,u_{1}\right) \left( \underset {k=1}{\overset {n}{\sum }}p_{n-k}\left\vert D_{k,j}^{\ast }\left( u_{1},u_{2}\right) \right\vert \right) du_{2}du_{1} \\ & \ll & \underset {\frac{1}{n+1}}{\overset {1}{\int }}\overset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\underset {0}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}\left( \underset {k=1}{\overset {n}{\sum }}kp_{n-k}\right) du_{2}du_{1} \leq \end{eqnarray*}
</div>
<div class="displaymath" id="a0000000061">
  \begin{eqnarray*} & \leq & nP_{n}\underset {\frac{1}{n+1}}{\overset {1}{\int }}\overset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\underset {0}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}du_{2}du_{1}\leq p_{n}\underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}du_{1} \\ & =& p_{n}\underset {1}{\overset {n+1}{\int }}\tfrac {\omega _{H}\left( f,1/t\right) }{t}dt=\underset {k=1}{\overset {n}{\sum }}\left( \underset {k}{\overset {k+1}{\int }}\tfrac {\omega _{H}\left( f,1/t\right) }{t}dt\right) \\ & \leq & p_{n}\underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}\omega _{H}\big( f,\tfrac {1}{k}\big) =\underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}p_{n}\omega _{H}\big( f,\tfrac {1}{k}\big) \leq \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \end{eqnarray*}
</div>
<p>Hence we get </p>
<div class="equation" id="C">
<p>
  <div class="equation_content">
    \begin{equation}  \underset {\Gamma _{2}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\ll \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \label{C} \end{equation}
  </div>
  <span class="equation_label">23</span>
</p>
</div>
<p>To estimate the integrals \(I_{n,3}^{\ast }\) and</p>
<div class="displaymath" id="a0000000062">
  \begin{equation*}  \underset {\Gamma _{2}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \end{equation*}
</div>
<p>we shall use the expression \(\left( \ref{expression2}\right) \) of \(D_{k}^{\ast }\left( u_{1},u_{2}\right) .\) </p>
<p>Lemma 5. 11 of <span class="cite">
	[
	<a href="#mcfadden" >12</a>
	]
</span> yields</p>
<div class="displaymath" id="a0000000063">
  \begin{equation*}  \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}\cos \left( \left( 2k+1\right) u_{2}\pi \right) \right\vert \ll P\big( \tfrac {1}{2\pi u_{2}}\big) \end{equation*}
</div>
<p>and </p>
<div class="displaymath" id="a0000000064">
  \begin{equation*}  \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}\cos \left( \left( 2k+1\right) \tfrac {u_{1}-u_{2}}{2}\pi \right) \right\vert \ll P\left( \tfrac {1}{\left( u_{1}-u_{2}\right) \pi }\right) \end{equation*}
</div>
<p>for \(\left( u_{1},u_{2}\right) \in \Gamma _{2}^{\prime \prime }\cup \Gamma _{3}\), where \(P\left( t\right) :=P_{\left[ t\right] }.\) By Lemmas 5. 11 and 5. 10 of <span class="cite">
	[
	<a href="#mcfadden" >12</a>
	]
</span>, the fact </p>
<div class="displaymath" id="a0000000065">
  \begin{equation*}  \sin \tfrac {u_{1}\pi }{2}\leq \tfrac {2}{\sqrt{3}}\sin \left( \tfrac {u_{1}+u_{2}}{2}\pi \right) , \end{equation*}
</div>
<p>and (<a href="#sin2">21</a>), we get </p>
<div class="displaymath" id="a0000000066">
  \begin{equation*}  \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}\cos \left( \left( 2k+1\right) \tfrac {u_{1}+u_{2}}{2}\pi \right) \right\vert \ll P\big( \tfrac {1}{u_{1}\pi }\big) \end{equation*}
</div>
<p>for \(\left( u_{1},u_{2}\right) \in \Gamma _{2}^{\prime \prime }\cup \Gamma _{3}.\) Hence by considering these inequalities and (<a href="#sin2">21</a>) we obtain   </p>
<div class="equation" id="hk1">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert \ll \tfrac {1}{u_{1}^{2}}P\big( \tfrac {1}{2\pi u_{2}}\big) \label{hk1} \end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<p>and </p>
<div class="equation" id="hk23">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}H_{k,j}\left( u_{1},u_{2}\right) \right\vert \ll \tfrac {1}{u_{1}u_{2}}P\big( \tfrac {3}{2\pi u_{1}}\big) \text{ }\left( j=2,3\right) \label{hk23} \end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p>for \(\left( u_{1},u_{2}\right) \in \Gamma _{2}^{\prime \prime }\cup \Gamma _{3}.\) </p>
<p>By (<a href="#sin2">21</a>) we obtain</p>
<div class="displaymath" id="a0000000067">
  \begin{eqnarray*} & & \! \! \! \! \! \! \! \! \! \! \underset {\Gamma _{2}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\leq \\ & \leq & \underset {\frac{1}{n+1}}{\overset {1}{\int }}\overset {\frac{1}{3\left( n+1\right) }}{\underset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\int }}\omega _{H}\left( f,u_{1}\right) \left( \underset {k=1}{\overset {n}{\sum }}p_{n-k}\left\vert H_{k,1}\left( u_{1},u_{2}\right) \right\vert \right) du_{2}du_{1} \\ & \leq & P_{n}\underset {\frac{1}{n+1}}{\overset {1}{\int }}\overset {\frac{1}{3\left( n+1\right) }}{\underset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{2}du_{1}\leq \tfrac {P_{n}}{n+1}\underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{1} \\ & =& \tfrac {P_{n}}{n+1}\underset {1}{\overset {n+1}{\int }}\omega _{H}\big( f,\tfrac {1}{t}\big) dt\leq \tfrac {P_{n}}{n+1}\underset {k=1}{\overset {n}{\sum }}\omega _{H}\big( f,\tfrac {1}{k}\big) \\ & \leq & \underset {k=1}{\overset {n}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \end{eqnarray*}
</div>
<p> For \(j=2,3\) by (<a href="#hk23">25</a>) we get </p>
<div class="displaymath" id="a0000000068">
  \begin{eqnarray*} & & \! \! \! \! \! \! \! \! \! \! \underset {\Gamma _{2}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}H_{k,j}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\ll \\ & \ll & \underset {\frac{1}{n+1}}{\overset {1}{\int }}\overset {\frac{1}{3\left( n+1\right) }}{\underset {\frac{p_{n}}{3\left( n+1\right) P_{n}}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}u_{2}}P\left( \tfrac {3}{2\pi u_{1}}\right) du_{2}du_{1} \\ & =& \log \left( \tfrac {P_{n}}{p_{n}}\right) \underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}P\left( \tfrac {3}{2\pi u_{1}}\right) du_{1} \\ & =& \log \left( \tfrac {P_{n}}{p_{n}}\right) \underset {\frac{3}{2\pi }}{\overset {\frac{3}{2\pi }\left( n+1\right) }{\int }}\omega _{H}\left( f,\tfrac {3}{2\pi t}\right) \tfrac {P\left( t\right) }{t}dt \\ & =& \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\left( \underset {\frac{3}{2\pi }k}{\overset {\frac{3}{2\pi }\left( k+1\right) }{\int }}\omega _{H}\left( f,\tfrac {3}{2\pi t}\right) \tfrac {P\left( t\right) }{t}dt\right) \leq \end{eqnarray*}
</div>
<div class="displaymath" id="a0000000069">
  \begin{eqnarray*} & \leq & \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}P\left( \tfrac {3}{2\pi }\left( k+1\right) \right) \\ & \ll & \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \end{eqnarray*}
</div>
<p>Thus, (<a href="#C">23</a>) and this inequality give</p>
<div class="displaymath" id="a0000000070">
  \begin{equation*}  \underset {\Gamma _{2}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\ll \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) , \end{equation*}
</div>
<p>and hence</p>
<div class="equation" id="D">
<p>
  <div class="equation_content">
    \begin{equation}  I_{n,2}^{\ast }\ll \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {P_{k}\omega _{H}\left( f,1/k\right) }{k}. \label{D} \end{equation}
  </div>
  <span class="equation_label">26</span>
</p>
</div>
<p>By (<a href="#hk1">24</a>) and by the inequality</p>
<div class="displaymath" id="a0000000071">
  \begin{equation*}  \tfrac {\omega _{H}\left( f,\delta _{2}\right) }{\delta _{2}}\leq 2\tfrac {\omega _{H}\left( f,\delta _{1}\right) }{\delta _{1}}\text{ }\left( \delta _{1}{\lt}\delta _{2}\right) \end{equation*}
</div>
<p>which is easily obtained from (<a href="#modc">11</a>),</p>
<div class="displaymath" id="a0000000072">
  \begin{eqnarray*} & & \! \! \! \! \! \! \! \! \! \! \underset {\Gamma _{3}}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\ll \\ & \ll & \underset {\frac{1}{3\left( n+1\right) }}{\overset {\frac{1}{3}}{\int }}\underset {3u_{2}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}P\big( \tfrac {1}{2\pi u_{2}}\big) du_{1}du_{2} \\ & \ll & \underset {\frac{1}{3\left( n+1\right) }}{\overset {\frac{1}{3}}{\int }}\underset {3u_{2}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{1}u_{2}}P\big( \tfrac {1}{2\pi u_{2}}\big) du_{1}du_{2} \\ & =& \underset {\frac{1}{3\left( n+1\right) }}{\overset {\frac{1}{3}}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}P\big( \tfrac {1}{2\pi u_{2}}\big) \log \big( \tfrac {1}{3u_{2}}\big) du_{2} \\ & \leq & \log \left( n+1\right) \underset {\frac{1}{3\left( n+1\right) }}{\overset {\frac{1}{3}}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}P\big( \tfrac {1}{2\pi u_{2}}\big) du_{2} \\ & =& \log \left( n+1\right) \underset {\frac{3}{2\pi }}{\overset {\frac{3}{2\pi }\left( n+1\right) }{\int }}\omega _{H}\big( f,\tfrac {3}{2\pi t}\big) \tfrac {P\left( t\right) }{t}dt \\ & \ll & \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \end{eqnarray*}
</div>
<p>By (<a href="#hk23">25</a>), for \(j=2,3,\)</p>
<div class="displaymath" id="a0000000073">
  \begin{eqnarray*} & & \! \! \! \! \! \! \! \! \! \! \underset {\Gamma _{3}}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=1}{\overset {n}{\sum }}p_{n-k}H_{k,j}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\leq \\ & \leq & \underset {\frac{1}{n+1}}{\overset {1}{\int }}\underset {\frac{1}{3\left( n+1\right) }}{\overset {\frac{u_{1}}{3}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}u_{2}}P\big( \tfrac {3}{2\pi u_{1}}\big) du_{2}du_{1} \\ & =& \underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}P\big( \tfrac {3}{2\pi u_{1}}\big) \log \left( \left( n+1\right) u_{1}\right) du_{1} \\ & \leq & \log \left( n+1\right) \underset {\frac{1}{n+1}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}P\big( \tfrac {3}{2\pi u_{1}}\big) du_{1} \\ & \ll & \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \end{eqnarray*}
</div>
<p>Thus, </p>
<div class="equation" id="E">
<p>
  <div class="equation_content">
    \begin{equation}  I_{n,3}^{\ast }\ll \log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {1}{k}P_{k}\omega _{H}\big( f,\tfrac {1}{k}\big) . \label{E} \end{equation}
  </div>
  <span class="equation_label">27</span>
</p>
</div>
<p>Combining (<a href="#fminusNn">16</a>), (<a href="#A">19</a>), (<a href="#B2">22</a>), (<a href="#D">26</a>) and (<a href="#E">27</a>) give (<a href="#Th1">15</a>). <div class="proof_wrapper" id="a0000000074">
  <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="a0000000075">4 Conclusions</h1>
<p>\(\textsc{Conclusion 1}. \) For \(f\in H^{\alpha }\big( \overline{\Omega }\big) \) \(\left( 0{\lt}\alpha \leq 1\right) ,\) Theorem 1 yields the following analogue of (<a href="#SG">5</a>): </p>
<div class="displaymath" id="a0000000076">
  \begin{equation*}  \left\Vert f-N_{n}\left( p;f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\ll \tfrac {1}{P_{n}}\log \left( \tfrac {P_{n}}{p_{n}}\right) \overset {n}{\underset {k=1}{\sum }}\tfrac {P_{k}}{k^{1+\alpha }}. \end{equation*}
</div>
<p>Note that this estimate was obtained directly in <span class="cite">
	[
	<a href="#guven" >7</a>
	]
</span>. </p>
<p>\(\textsc{Conclusion 2}. \) In the case \(p_{n}=1,\) \(\left( n=0,1,...\right) ,\) (<a href="#Th1">15</a>) reduces to </p>
<div class="displaymath" id="a0000000077">
  \begin{equation*}  \left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\ll \tfrac {\log n}{n+1}\overset {n}{\underset {k=1}{\sum }}\omega _{H}\big( f,\tfrac {1}{k}\big) , \end{equation*}
</div>
<p>which is the analogue of (<a href="#S">3</a>) for hexagonal Fourier series. </p>
<p>\(\textsc{Conclusion 3}. \) In the case \(p_{n}=1,\) \(\left( n=0,1,...\right) \) and \(f\in H^{\alpha }\big( \overline{\Omega }\big) \) \(\left( 0{\lt}\alpha \leq 1\right) ,\) (<a href="#Th1">15</a>) gives</p>
<div class="displaymath" id="a0000000078">
  \begin{equation*}  \left\Vert f-\sigma _{n}\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\ll \left\{  \begin{array}{cc} \frac{\log n}{n^{\alpha }}, &  0{\lt}\alpha {\lt}1 \\[2mm] \frac{\left( \log n\right) ^{2}}{n}, &  \alpha =1.\end{array}\right. \end{equation*}
</div>
<p>This estimate yields the same approximation order with (<a href="#G">12</a>) in the case \(\alpha =1.\) </p>
<p><div class="acknowledgement_thmwrapper " id="a0000000079">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgements
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The author wants to thank the referee for her/his valuable suggestions. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="alexits">1</a></dt>
  <dd><p><i class="sc">G. Alexits</i>, <i class="it">Convergence Problems of Orthogonal Series</i>, Pergamon Press, New York-Oxford-Paris, 1961. </p>
</dd>
  <dt><a name="bernstein">2</a></dt>
  <dd><p><i class="sc">S.N. Bernstein</i>, <i class="it">Sur l’ordre de la meilleure approximation des fonctions continues par les polynômes de degré donné</i>, Mem. Cl. Sci. Acad. Roy. Belg., <b class="bf">4</b> (1912), pp.&#160;1–103. </p>
</dd>
  <dt><a name="butzer nessel">3</a></dt>
  <dd><p><i class="sc">P.L. Butzer, R.J. Nessel</i>, <i class="it">Fourier analysis and Approximation</i>, Academic Press, New York-London, 1971. </p>
</dd>
  <dt><a name="devore lorentz">4</a></dt>
  <dd><p><i class="sc">R.A. DeVore, G.G. Lorentz</i>, <i class="it">Constructive Approximation</i>, Springer-Verlag, Berlin, 1993. </p>
</dd>
  <dt><a name="fuglede">5</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/0022-1236(74)90072-x"> <i class="sc">B. Fuglede</i>, <i class="it">Commuting self-adjoint partial differential operators and a group theoretic problem</i>, J. Funct. Anal., <b class="bf">16</b> (1974), pp.&#160;101–121. <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="guven2013">6</a></dt>
  <dd><p><i class="sc">A. Guven</i>, <i class="it">Approximation by \(\left( C,1\right) \) and Abel-Poisson means of Fourier series on hexagonal domains</i>, Math. Inequal. Appl., <b class="bf">16</b> (2013), pp.&#160;175–191. </p>
</dd>
  <dt><a name="guven">7</a></dt>
  <dd><p><i class="sc">A. Guven</i>, <i class="it">Approximation by Nörlund means of hexagonal Fourier series</i>, Anal. Theory Appl., <b class="bf">33</b> (2017), pp.&#160;384–400. </p>
</dd>
  <dt><a name="hardy">8</a></dt>
  <dd><p><i class="sc">G.H. Hardy</i>, <i class="it">Divergent Series</i>, Clarendon Press, Oxford, 1949. </p>
</dd>
  <dt><a name="holland s t">9</a></dt>
  <dd><p><i class="sc">A.S.B. Holland, B.N. Sahney, J. Tzimbalario</i>, <i class="it">On degree of approximation of a class of functions by means of Fourier series</i>, Acta Sci. Math., <b class="bf">38</b> (1976), pp.&#160;69–72. </p>
</dd>
  <dt><a name="holland">10</a></dt>
  <dd><p><a href ="https://doi.org/10.1137/1023064 "> <i class="sc">A.S.B. Holland</i>, <i class="it">A survey of degree of approximation of continuous functions</i>, SIAM Rev., <b class="bf">23</b> (1981), pp.&#160;344–379. <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="li sun xu">11</a></dt>
  <dd><p><a href ="https:doi.org/10.1137/060671851"> <i class="sc">H. Li, J. Sun, Y. Xu</i>, <i class="it">Discrete Fourier analysis, cubature and interpolation on a hexagon and a triangle</i>, SIAM J. Numer. Anal., <b class="bf">46</b> (2008), pp.&#160;1653–1681. <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="mcfadden">12</a></dt>
  <dd><p><a href ="https://doi.org/10.1215/s0012-7094-42-00913-x "> <i class="sc">L. Mcfadden</i>, <i class="it">Absolute Nörlund summability</i>, Duke Math. J., <b class="bf">9</b> (1942), pp.&#160;168–207. <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="sahney goel">13</a></dt>
  <dd><p><i class="sc">B.N. Sahney, D.S. Goel</i>, <i class="it">On the degree of approximation of continuous functions</i>, Ranchii Univ. Math. J., <b class="bf">4</b> (1973), pp.&#160;50–53. </p>
</dd>
  <dt><a name="stechkin">14</a></dt>
  <dd><p><i class="sc">S.B. Stechkin</i>, <i class="it">The approximation of periodic functions by Fejér sums</i>, Trudy Mat. Inst. Steklov., <b class="bf">62</b> (1961), pp.&#160;48–60 (in Russian). </p>
</dd>
  <dt><a name="sun">15</a></dt>
  <dd><p><i class="sc">J. Sun</i>, <i class="it">Multivariate Fourier series over a class of non tensor-product partition domains</i>, J. Comput. Math., <b class="bf">21</b> (2003), pp.&#160;53–62. </p>
</dd>
  <dt><a name="timan">16</a></dt>
  <dd><p><i class="sc">A.F. Timan</i>, <i class="it">Theory of Approximation of Functions of a Real Variable</i>, Pergamon Press, New York, 1963. </p>
</dd>
  <dt><a name="xu">17</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s00365-008-9034-y"> <i class="sc">Y. Xu</i>, <i class="it">Fourier series and approximation on hexagonal and triangular domains</i>, Constr. Approx., <b class="bf">31</b> (2010), pp.&#160;115–138. <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">18</a></dt>
  <dd><p><i class="sc">A. Zygmund</i>, <i class="it">Trigonometric Series</i>, 2nd ed., vol. I, Cambridge Univ. Press, New York, 1959. </p>
</dd>
</dl>


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