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<h1>Trigonometric Approximation on the Hexagon</h1>
<p class="authors">
<span class="author">Ali Guven\(^\ast \)</span>
</p>
<p class="date">November 20, 2020; accepted: December 21, 2020; published online: February 20, 2021.</p>
</div>
<div class="abstract"><p> The degree of trigonometric approximation of continuous functions, which are periodic with respect to the hexagon lattice, is estimated in uniform and Hölder norms. Approximating trigonometric polynomials are matrix means of hexagonal Fourier series. </p>
<p><b class="bf">MSC.</b> 41A25, 41A63, 42B08 </p>
<p><b class="bf">Keywords.</b> Hexagonal Fourier series, Hölder class, matrix mean, regular hexagon. </p>
<p>Dedicated to the memory of Dr. Figen Kiraz. </p>
</div>
<p>\(^\ast \)Department of Mathematics, Balikesir University, 10145 Balikesir, Turkey, e-mail: <span class="tt">guvennali@gmail.com</span>. </p>
<h1 id="a0000000002">1 Introduction</h1>
<p>Approximation problems of functions of several variables defined on cubes of the Euclidean space are usually studied by assuming that the functions are periodic in each of their variables (see, for example <span class="cite">
	[
	<a href="#timan" >20</a>
	, 
	§
	§
	
	5.3,
	
	6.3
	]
</span>, <span class="cite">
	[
	<a href="#zygmund" >23</a>
	, 
	vol.
	
	II,
	
	ch.
	
	XVII
	]
</span>, <span class="cite">
	[
	<a href="#zhi" >22</a>
	, 
	part
	
	2
	]
</span>, <span class="cite">
	[
	<a href="#mr1" >15</a>
	]
</span>, <span class="cite">
	[
	<a href="#mr2" >16</a>
	]
</span> and <span class="cite">
	[
	<a href="#mshi" >17</a>
	]
</span>). But, we need other definitions of periodicity to study approximation problems on non-tensor product domains, for example on hexagonal domains of \(\mathbb {R}^{2}\). The periodicity defined by lattices is the most useful one. </p>
<p>In the Euclidean plane \(\mathbb {R}^{2},\) besides the standard lattice \(\mathbb {Z}^{2}\) and the rectangular domain \([ -\frac{1}{2},\frac{1}{2}) ^{2},\) the simplest lattice is the hexagon lattice and the simplest spectral set is the regular hexagon. The hexagon lattice has importance, since it offers the densest packing of the plane with unit circles. In this section we give basic information about hexagonal lattice and hexagonal Fourier series. More detailed information can be found in <span class="cite">
	[
	<a href="#li sun xu" >11</a>
	]
</span> and <span class="cite">
	[
	<a href="#xu" >21</a>
	]
</span>. </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="a0000000003">
  \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="a0000000004">
  \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 satisfies \(t_{1}+t_{2}+t_{3}=0\). As it is pointed out in <span class="cite">
	[
	<a href="#xu" >21</a>
	]
</span>, using homogeneous coordinates reveals symmetry in various formulas. If we set</p>
<div class="displaymath" id="a0000000005">
  \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}, \end{equation*}
</div>
<p>the hexagon \(\Omega _{H}\) becomes </p>
<div class="displaymath" id="a0000000006">
  \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>which is the intersection of the plane \(t_{1}+t_{2}+t_{3}=0\) with the cube \(\left[ -1,1\right] ^{3}.\) </p>
<p>We use bold letters \(\mathbf{t}\) for homogeneous coordinates and we set </p>
<div class="displaymath" id="a0000000007">
  \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\}  ,\text{ }\mathbb {Z}_{H}^{3}:=\mathbb {Z}^{3}\cap \mathbb {R}_{H}^{3}. \end{equation*}
</div>
<p>A function \(f:\mathbb {R}^{2}\rightarrow \mathbb {C}\) is called \(H\)-periodic if </p>
<div class="displaymath" id="a0000000008">
  \begin{equation*}  f\left( x+Hk\right) =f\left( x\right) \end{equation*}
</div>
<p>for all \(k\in \mathbb {Z}^{2}\) and \(x\in \mathbb {R}^{2}.\) If we define \(\mathbf{t}\equiv \mathbf{s}\) \(\left( \operatorname {mod}3\right) \) as </p>
<div class="displaymath" id="a0000000009">
  \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>for \(\mathbf{t}=\left( t_{1},t_{2},t_{3}\right) ,\) \(\mathbf{s}=\left( s_{1},s_{2},s_{3}\right) \mathbf{\in }\) \(\mathbb {R}_{H}^{3},\) it follows that the function \(f\) is \(H\)-periodic if and only if \(f\left( \mathbf{t}\right) =f\left( \mathbf{t}+\mathbf{s}\right) \) whenever \(\mathbf{s}\equiv \mathbf{0}\) \(\left( \operatorname {mod}3\right) ,\) and</p>
<div class="displaymath" id="a0000000010">
  \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 \(H\)-periodic integrable function \(f\) <span class="cite">
	[
	<a href="#xu" >21</a>
	]
</span>\(.\) </p>
<p>\(L^{2}\left( \Omega \right) \) becomes a Hilbert space with respect to the inner product</p>
<div class="displaymath" id="a0000000011">
  \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 \(\Omega .\) The functions </p>
<div class="displaymath" id="a0000000012">
  \begin{equation*}  \phi _{\mathbf{j}}\left( \mathbf{t}\right) :=e^{\frac{2\pi i}{3}\left\langle \mathbf{j},\mathbf{t}\right\rangle }\qquad ( \mathbf{t}\in \mathbb {R}_{H}^{3}) , \end{equation*}
</div>
<p>where \(\left\langle \mathbf{j},\mathbf{t}\right\rangle \) is the usual Euclidean inner product of \(\mathbf{j}\) and \(\mathbf{t},\) are \(H\)-periodic, and by a theorem of B. Fuglede the set </p>
<div class="displaymath" id="a0000000013">
  \begin{equation*}  \left\{  \phi _{\mathbf{j}}:\mathbf{j}\in \mathbb {Z}_{H}^{3}\right\}  \end{equation*}
</div>
<p>becomes an orthonormal basis of \(L^{2}\left( \Omega \right) \) <span class="cite">
	[
	<a href="#fuglede" >3</a>
	]
</span> (see also <span class="cite">
	[
	<a href="#li sun xu" >11</a>
	]
</span>)\(.\) </p>
<p>For every natural number \(n,\) we define a subset of \(\mathbb {Z}_{H}^{3}\) by </p>
<div class="displaymath" id="a0000000014">
  \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>The subspace </p>
<div class="displaymath" id="a0000000015">
  \begin{equation*}  \mathcal{H}_{n}:=\text{span}\left\{  \phi _{\mathbf{j}}:\mathbf{j}\in \mathbb {H}_{n}\right\}  \qquad \left( n\in \mathbb {N}\right) \end{equation*}
</div>
<p>has dimension \(\# \mathbb {H}_{n}=3n^{2}+3n+1,\) and its members are called hexagonal trigonometric polynomials of degree \(n\). </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">1</span>
</p>
</div>
<p>where </p>
<div class="displaymath" id="a0000000016">
  \begin{equation*}  \widehat{f}_{\mathbf{j}}=\tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }f\left( \mathbf{t}\right) \overline{\phi _{\mathbf{j}}\left( \mathbf{t}\right) }d\mathbf{t}\qquad ( \mathbf{j}\in \mathbb {Z}_{H}^{3}) . \end{equation*}
</div>
<p>The \(n\)th hexagonal partial sum of the series \((\ref{fourier})\) is defined by </p>
<div class="displaymath" id="a0000000017">
  \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) \qquad \left( n\in \mathbb {N}\right) . \end{equation*}
</div>
<p>It is clear that </p>
<div class="displaymath" id="a0000000018">
  \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{u}\right) D_{n}\left( \mathbf{u}\right) d\mathbf{u}, \end{equation*}
</div>
<p>where \(D_{n}\) is the Dirichlet kernel, defined by </p>
<div class="displaymath" id="a0000000019">
  \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>It is known that 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) \qquad \left( n\geq 1\right) , \label{dirichlet} \end{equation}
  </div>
  <span class="equation_label">2</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">3</span>
</p>
</div>
<p>for \(\mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \mathbb {R}_{H}^{3}\) <span class="cite">
	[
	<a href="#li sun xu" >11</a>
	]
</span>. </p>
<p>The degree of approximation of \(H\)-periodic continuous functions by Cesàro, Riesz and Nörlund means of their hexagonal Fourier series was investigated by us in <span class="cite">
	[
	<a href="#guvenjca" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#guvenmia" >5</a>
	]
</span>, <span class="cite">
	[
	<a href="#guvenzaa" >6</a>
	]
</span>, <span class="cite">
	[
	<a href="#guvenata" >7</a>
	]
</span> and <span class="cite">
	[
	<a href="#guvenx" >8</a>
	]
</span>. In this paper, we studied the degree of approximation by matrix means of hexagonal Fourier and we obtained generalizations of previous results. </p>
<h1 id="a0000000020">2 Main Results</h1>
<p>Let \(C_{H}( \overline{\Omega }) \) be the Banach space of complex valued \(H\)-periodic continuous functions defined on \(\mathbb {R}_{H}^{3},\) whose norm is the uniform norm:</p>
<div class="displaymath" id="a0000000021">
  \begin{equation*}  \left\Vert f\right\Vert _{C_{H}\left( \overline{\Omega }\right) }:=\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 \(f\in C_{H}( \overline{\Omega }) \) is defined by </p>
<div class="displaymath" id="a0000000022">
  \begin{equation*}  \omega _{H}\left( f,\delta \right) :=\underset {0{\lt}\left\Vert \mathbf{t}\right\Vert \leq \delta }{\sup }\left\Vert f-f\left( \cdot +\mathbf{t}\right) \right\Vert _{C_{H}( \overline{\Omega }) }, \end{equation*}
</div>
<p>where</p>
<div class="displaymath" id="a0000000023">
  \begin{equation*}  \left\Vert \mathbf{t}\right\Vert :=\max \left\{  \left\vert t_{1}\right\vert ,\left\vert t_{2}\right\vert ,\left\vert t_{3}\right\vert \right\}  \end{equation*}
</div>
<p>for \(\mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \mathbb {R}_{H}^{3}.\) \(\omega _{H}\left( f,\cdot \right) \) is a nonnegative and nondecreasing 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">4</span>
</p>
</div>
<p>for \(\lambda {\gt}0\) <span class="cite">
	[
	<a href="#xu" >21</a>
	]
</span>\(.\) </p>
<p>A function \(f\in C_{H}( \overline{\Omega }) \) is said to belong to the Hölder space \(H^{\alpha }(\overline{\Omega }) \) \(\left( 0{\lt}\alpha \leq 1\right) \) if </p>
<div class="displaymath" id="a0000000024">
  \begin{equation*}  \Lambda ^{\alpha }\left( f\right) :=\underset {\mathbf{t}\neq \mathbf{s}}{\sup }\tfrac {\left\vert f\left( \mathbf{t}\right) -f\left( \mathbf{s}\right) \right\vert }{\left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\alpha }}{\lt}\infty . \end{equation*}
</div>
<p>\(H^{\alpha }( \overline{\Omega }) \) becomes a Banach space with respect to the Hölder norm </p>
<div class="displaymath" id="a0000000025">
  \begin{equation*}  \left\Vert f\right\Vert _{H^{\alpha }\left( \overline{\Omega }\right) }:=\left\Vert f\right\Vert _{C_{H}\left( \overline{\Omega }\right) }+\Lambda ^{\alpha }\left( f\right) . \end{equation*}
</div>
<p>Let \(A=\left( a_{n,k}\right) \) \(\left( n,k=0,1,\ldots \right) \) be a lower triangular infinite matrix of real numbers. The \(A\)-transform of the sequence \(\left( S_{n}\left( f\right) \right) \) of partial sums the series (<a href="#fourier">1</a>) is defined by </p>
<div class="displaymath" id="a0000000026">
  \begin{equation*}  T_{n}^{\left( A\right) }\left( f\right) \left( \mathbf{t}\right) :=\overset {n}{\underset {k=0}{\sum }}a_{n,k}S_{k}\left( f\right) \left( \mathbf{t}\right) \qquad \left( n\in \mathbb {N}\right) . \end{equation*}
</div>
<p>We shall assume that the lower triangular matrix \(A=\left( a_{n,k}\right) \) satisfies the conditions </p>
<div class="equation" id="m1">
<p>
  <div class="equation_content">
    \begin{equation}  a_{n,k}\geq 0\qquad \left( n=0,1,\ldots ,0\leq k\leq n\right) , \label{m1} \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<div class="equation" id="m2">
<p>
  <div class="equation_content">
    \begin{equation}  a_{n,k+1}\geq a_{n,k}\qquad \left( n=0,1,\ldots ,0\leq k\leq n-1\right) , \label{m2} \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p>and </p>
<div class="equation" id="m3">
<p>
  <div class="equation_content">
    \begin{equation}  \underset {k=0}{\overset {n}{\sum }}a_{n,k}=1\qquad \left( n=0,1,\ldots \right) . \label{m3} \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p>Also we use the notations </p>
<div class="displaymath" id="a0000000027">
  \begin{equation*}  A_{n,k}^{\ast }:=\underset {\nu =k}{\overset {n}{\sum }}a_{n,\nu }\text{ }\left( 0\leq k\leq n\right) ,\text{ }A_{n}^{\ast }\left( u\right) :=A_{n,n-\left[ u\right] }^{\ast }\text{ , }a_{n}^{\ast }\left( u\right) :=a_{n,n-\left[ u\right] }\text{ }\left( u{\gt}0\right) , \end{equation*}
</div>
<p>where \(\left[ u\right] \) is the integer part of \(u.\) </p>
<p>Hereafter, the relation \(x\lesssim y\) will mean that there exists an absolute constant \(c{\gt}0\) such that \(x\leq cy\) holds for quantities \(x\) and \(y\). </p>
<p><div class="theorem_thmwrapper " id="th:1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(f\in C_{H}( \overline{\Omega }) \) and let \(A=\left( a_{n,k}\right) \) \(\left( n,k=0,1,\ldots \right) \) be a lower triangular infinite matrix of real numbers which satisfies \(\left( \ref{m1}\right) \), \(\left( \ref{m2}\right) \) and \(\left( \ref{m3}\right) \). Then the estimate </p>
<div class="equation" id="uniform">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\lesssim \log \left( n+1\right) \left\{  \omega _{H}\left( f,a_{n,n}\right) +\underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac1k\right) }{k}A_{n,n-k}^{\ast }\right\}  \label{uniform} \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p>holds\(.\) </p>
<p><div class="corollary_thmwrapper " id="cor:2">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">2</span>
  </div>
  <div class="corollary_thmcontent">
  <p> <i class="itshape">If </i>\(f\in H^{\alpha }( \overline{\Omega }) \) \(\left( 0{\lt}\alpha \leq 1\right) \) and \(A=\left( a_{n,k}\right) \) \(\left( n,k=0,1,\ldots \right) \)<i class="itshape"> as in <a href="#th:1">theorem 1</a>, then </i></p>
<div class="equation" id="uniformx">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\lesssim \log \left( n+1\right) \left\{  a_{n,n}^{\alpha }+\underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k^{1+\alpha }}\right\}  \text{.} \label{uniformx} \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>

  </div>
</div> </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="th:3">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(0\leq \beta {\lt}\alpha \leq 1\), \(f\in H^{\alpha }( \overline{\Omega }) \) and let \(A=\left( a_{n,k}\right) \) \(\left( n,k=0,1,\ldots \right) \) be a lower triangular infinite matrix of real numbers which satisfies \(\left( \ref{m1}\right) \), \(\left( \ref{m2}\right) \) and \(\left( \ref{m3}\right) \). Then </p>
<div class="equation" id="holderbeta">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{H^{\beta }\left( \overline{\Omega }\right) }\lesssim \log \left( n+1\right) \left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k}\right) ^{\frac\beta \alpha }\left\{  a_{n,n}^{\alpha -\beta }\! +\! \left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k^{1+\alpha }}\right) ^{1-\frac\beta \alpha }\right\}  . \label{holderbeta} \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>

  </div>
</div> </p>
<p>Analogues of these results were obtained in <span class="cite">
	[
	<a href="#khs" >10</a>
	]
</span>, <span class="cite">
	[
	<a href="#mohapatra sahney" >13</a>
	]
</span>, <span class="cite">
	[
	<a href="#mohapatra chandra" >14</a>
	]
</span> and <span class="cite">
	[
	<a href="#chandra88" >2</a>
	]
</span> for matrix means of trigonometric Fourier series of \(2\pi \)-periodic continuous functions. </p>
<h1 id="a0000000028">3 Proofs of Main Results</h1>
<p><div class="proof_wrapper" id="a0000000029">
  <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="#th:1">theorem 1</a>] It is clear that </p>
<div class="displaymath" id="a0000000030">
  \begin{eqnarray*}  \left\vert f\left( \mathbf{t}\right) -T_{n}^{\left( A\right) }\left( f\right) \left( \mathbf{t}\right) \right\vert & \leq & \tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }\left\vert f\left( \mathbf{t}\right) -f\left( \mathbf{t}-\mathbf{u}\right) \right\vert \left\vert \overset {n}{\underset {k=0}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u} \\ & \lesssim & \tfrac {1}{\left\vert \Omega \right\vert }\underset {\Omega }{\int }\omega _{H}\left( f,\left\Vert \mathbf{u}\right\Vert \right) \left\vert \overset {n}{\underset {k=0}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}. \end{eqnarray*}
</div>
<p>If we set \(\Theta _{-1}\left( \mathbf{u}\right) :=0,\) by \(\left( \ref{dirichlet}\right) \) we get </p>
<div class="displaymath" id="a0000000031">
  \begin{align*} & \underset {\Omega }{\int }\omega _{H}\left( f,\left\Vert \mathbf{u}\right\Vert \right) \left\vert \overset {n}{\underset {k=0}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}= \\ & =\underset {\Omega }{\int }\omega _{H}\left( f,\left\Vert \mathbf{u}\right\Vert \right) \left\vert \overset {n}{\underset {k=0}{\sum }}a_{n,k}\Big( \Theta _{k}\left( \mathbf{u}\right) \! -\! \Theta _{k-1}\left( \mathbf{u}\right) \Big) \right\vert \!  d\mathbf{u}. \end{align*}
</div>
<p>The function </p>
<div class="displaymath" id="a0000000032">
  \begin{equation*}  \mathbf{t}\rightarrow \omega _{H}\left( f,\left\Vert \mathbf{t}\right\Vert \right) \left\vert \overset {n}{\underset {k=0}{\sum }}a_{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 \(\mathbf{t}=\left( t_{1},t_{2},t_{3}\right) \in \Omega .\) Hence it is sufficient to estimate the integral over the triangle</p>
<div class="displaymath" id="a0000000033">
  \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 \((\ref{thetan})\), we obtain </p>
<div class="displaymath" id="a0000000034">
  \begin{align*} & \underset {\Delta }{\int }\omega _{H}\left( f,\left\Vert \mathbf{t}\right\Vert \right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{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 \underset {k=0}{\overset {n}{\sum }}a_{n,k}\bigg( \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{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}}\bigg) \bigg\vert d\mathbf{t}. \end{align*}
</div>
<p>If we use the change of variables</p>
<div class="equation" id="change1">
<p>
  <div class="equation_content">
    \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}, \label{change1} \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p>the integral becomes </p>
<div class="displaymath" id="a0000000035">
  \begin{align*}  3\underset {\widetilde{\Delta }}{\int }\omega _{H}\left( f,s_{1}+s_{2}\right) \bigg\vert \underset {k=0}{\overset {n}{\sum }}a_{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 \(\widetilde{\Delta }\) is the image of \(\Delta \) in the plane, that is</p>
<div class="displaymath" id="a0000000036">
  \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 \(s_{2},\) estimating the integral over the triangle </p>
<div class="displaymath" id="a0000000037">
  \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>which is the half of \(\widetilde{\Delta },\) will be sufficient\(.\) The change of variables</p>
<div class="equation" id="change2">
<p>
  <div class="equation_content">
    \begin{equation}  s_{1}:=\tfrac {u_{1}-u_{2}}{2},\text{ }s_{2}:=\tfrac {u_{1}+u_{2}}{2} \label{change2} \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p>transforms the triangle \(\Delta ^{\ast }\) to the triangle</p>
<div class="displaymath" id="a0000000038">
  \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>Thus we have to estimate the integral</p>
<div class="displaymath" id="a0000000039">
  \begin{equation*}  I_{n}:=\underset {\Gamma }{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{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="a0000000040">
  \begin{eqnarray*}  D_{k}^{\ast }\left( u_{1},u_{2}\right) & =& \tfrac {\sin \left( \left( k+1\right) \left( u_{2}\right) \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( \left( u_{2}\right) \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( k\left( u_{2}\right) \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( \left( u_{2}\right) \pi \right) \sin \left( \frac{u_{1}+u_{2}}{2}\pi \right) \sin \left( \frac{u_{1}-u_{2}}{2}\pi \right) }. \end{eqnarray*}
</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">13</span>
</p>
</div>
<p>where </p>
<div class="displaymath" id="a0000000041">
  \begin{align*}  D_{k,1}^{\ast }\left( u_{1},u_{2}\right) & :=2\cos \left( ( k+\tfrac {1}{2}) 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( ( k+\tfrac {1}{2}) \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="a0000000042">
  \begin{eqnarray*}  D_{k,3}^{\ast }\left( u_{1},u_{2}\right) :=2\cos \left( ( k+\tfrac {1}{2}) \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{eqnarray*}
</div>
<p>We partition the triangle \(\Gamma \) as \(\Gamma =\Gamma _{1}\cup \Gamma _{2}\cup \Gamma _{3},\) where </p>
<div class="displaymath" id="a0000000043">
  \begin{align*}  \Gamma _{1} & :=\left\{  \left( u_{1},u_{2}\right) \in \Gamma :u_{1}\leq a_{n,n}\right\}  , \\ \Gamma _{2} & :=\left\{  \left( u_{1},u_{2}\right) \in \Gamma :u_{1}\geq a_{n,n},\text{ }u_{2}\leq \tfrac {a_{n,n}}{3}\right\}  , \\ \Gamma _{3} & :=\left\{  \left( u_{1},u_{2}\right) \in \Gamma :u_{1}\geq a_{n,n},\text{ }u_{2}\geq \tfrac {a_{n,n}}{3}\right\}  . \end{align*}
</div>
<p>Hence \(I_{n}=I_{n,1}+I_{n,2}+I_{n,3},\) where </p>
<div class="displaymath" id="a0000000044">
  \begin{equation*}  I_{n,j}:=\underset {\Gamma _{j}}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\qquad \left( j=1,2,3\right) . \end{equation*}
</div>
<p>We need the inequalities </p>
<div class="equation" id="sin1">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert \frac{\sin nt}{\sin t}\right\vert \leq n,\qquad \left( n\in \mathbb {N}\right) , \label{sin1} \end{equation}
  </div>
  <span class="equation_label">14</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,\qquad \left( 0\leq t\leq \tfrac {\pi }{2}\right) \label{sin2} \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p>to estimate integrals \(I_{n,1},I_{n,2}\) and \(I_{n,3}.\) </p>
<p>We divide \(\Gamma _{1}\) into three parts to estimate \(I_{n,1}\) as follows:</p>
<div class="displaymath" id="a0000000045">
  \begin{align*}  \Gamma _{1}^{\prime } & :=\left\{  \left( u_{1},u_{2}\right) \in \Gamma _{1}:u_{1}\leq \tfrac {1}{n+1}\right\}  , \\ \Gamma _{1}^{\prime \prime } & :=\left\{  \left( u_{1},u_{2}\right) \in \Gamma _{1}:u_{1}\geq \tfrac {1}{n+1},u_{2}\leq \tfrac {1}{3\left( n+1\right) }\right\}  , \\ \Gamma _{1}^{\prime \prime \prime } & :=\left\{  \left( u_{1},u_{2}\right) \in \Gamma _{1}:u_{1}\geq \tfrac {1}{n+1},u_{2}\geq \tfrac {1}{3\left( n+1\right) }\right\}  . \end{align*}
</div>
<p>Hence we have </p>
<div class="displaymath" id="a0000000046">
  \begin{equation*}  I_{n,1}=\left( \underset {\Gamma _{1}^{\prime }}{\int }+\underset {\Gamma _{1}^{\prime \prime }}{\int }+\underset {\Gamma _{1}^{\prime \prime \prime }}{\int }\right) \omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}. \end{equation*}
</div>
<p>By (<a href="#sin1">14</a>),</p>
<div class="displaymath" id="a0000000047">
  \begin{align*} & \underset {\Gamma _{1}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim \underset {\Gamma _{1}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left( \underset {k=0}{\overset {n}{\sum }}\left( k+1\right) ^{2}a_{n,k}\right) du_{1}du_{2} \\ & \leq \left( n+1\right) ^{2}\overset {\frac{1}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {3u_{2}}{\overset {\frac{1}{n+1}}{\int }}\omega _{H}\left( f,u_{1}\right) du_{1}du_{2} \\ & \leq \omega _{H}\left( f,\tfrac {1}{n+1}\right) \leq \omega _{H}\left( f,a_{n,n}\right) . \end{align*}
</div>
<p>By \(\left( \ref{sin2}\right) ,\)</p>
<div class="displaymath" id="a0000000048">
  \begin{align*} & \underset {\Gamma _{1}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k,1}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim \overset {\frac{1}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {\frac{1}{n+1}}{\overset {a_{n,n}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{1}du_{2} \leq \omega _{H}\left( f,a_{n,n}\right) \overset {\frac{1}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {\frac{1}{n+1}}{\overset {a_{n,n}}{\int }}\tfrac {1}{u_{1}^{2}}du_{1}du_{2} \leq \omega _{H}\left( f,a_{n,n}\right) . \end{align*}
</div>
<p>\(\left( \ref{sin1}\right) \) and \(\left( \ref{sin2}\right) \) gives for \(j=1,2, \)</p>
<div class="displaymath" id="a0000000049">
  \begin{eqnarray*} & & \underset {\Gamma _{1}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k,j}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\lesssim \\ & \lesssim & \left( n+1\right) \overset {\frac{1}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {\frac{1}{n+1}}{\overset {a_{n,n}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}du_{1}du_{2}\leq \left( n+1\right) \omega _{H}\left( f,a_{n,n}\right) \overset {\frac{1}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {\frac{1}{n+1}}{\overset {a_{n,n}}{\int }}\tfrac {1}{u_{1}}du_{1}du_{2} \\ & \leq & \log \left( n+1\right) \omega _{H}\left( f,a_{n,n}\right) . \end{eqnarray*}
</div>
<p>Since </p>
<div class="displaymath" id="a0000000050">
  \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">16</span>
</p>
</div>
<p>where</p>
<div class="displaymath" id="a0000000051">
  \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) }, \end{align*}
</div>
<div class="displaymath" id="a0000000052">
  \begin{align*}  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 the method used in <span class="cite">
	[
	<a href="#mcfadden" >12</a>
	, 
	p.
	
	179
	]
</span> we get </p>
<div class="displaymath" id="Ant1">
  \begin{align} \label{Ant1} \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}\cos \left( 2k+1\right) t\right\vert \lesssim A_{n}^{\ast }\left( \tfrac {1}{t}\right) +\tfrac {1}{\sin t }a_{n}^{\ast }\left( \tfrac {1}{t}\right) \quad \text{ }\left( 0{\lt}t{\lt}\pi \right) \end{align}
</div>
<p> and </p>
<div class="displaymath" id="Ant2">
  \begin{align} \label{Ant2} \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}\cos \left( 2k+1\right) t\right\vert \lesssim A_{n}^{\ast }\left( \tfrac {1}{t}\right) \qquad \left( 0{\lt}t\leq \tfrac {\pi }{2}\right). \end{align}
</div>
<p>By aim of (<a href="#Ant2">18</a>) and \(\left( \ref{sin2}\right) \) we obtain</p>
<div class="equation" id="hk1">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert \lesssim \tfrac {1}{u_{1}^{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) \label{hk1} \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p>and </p>
<div class="equation" id="hk3">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,3}\left( u_{1},u_{2}\right) \right\vert \lesssim \tfrac {1}{u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) , \label{hk3} \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<p>where both of for \(u_{1}\) and \(u_{2}\) are away from the origin. Also, for such \(u_{1}\) and \(u_{2},\) it follows from \(\left( \ref{Ant1}\right) ,\) \(\left( \ref{sin2}\right) \) and from the fact </p>
<div class="displaymath" id="a0000000053">
  \begin{equation*}  \sin \left( \tfrac {u_{1}\pi }{2}\right) \lesssim \sin \left( \tfrac {\left( u_{1}+u_{2}\right) \pi }{2}\right) \end{equation*}
</div>
<p>that </p>
<div class="equation" id="hk2">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,2}\left( u_{1},u_{2}\right) \right\vert \lesssim \tfrac {1}{u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) . \label{hk2} \end{equation}
  </div>
  <span class="equation_label">21</span>
</p>
</div>
<p>Using \(\left( \ref{hk1}\right) \) and the inequality </p>
<div class="equation" id="modc2">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\omega _{H}\left( f,\delta _{2}\right) }{\delta _{2}}\leq 2\frac{\omega _{H}\left( f,\delta _{1}\right) }{\delta _{1}}\qquad \left( \delta _{1}<\delta _{2}\right) , \label{modc2} \end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p>which is obtained from \(\left( \ref{modc}\right) ,\) and considering that the function \(A_{n}^{\ast }\) is nondecreasing yield</p>
<div class="displaymath" id="a0000000054">
  \begin{align*} & \underset {\Gamma _{1}^{\prime \prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\lesssim \end{align*}
</div>
<div class="displaymath" id="a0000000055">
  \begin{align*} & \lesssim \overset {\frac{a_{n,n}}{3}}{\underset {\frac{1}{3\left( n+1\right) }}{\int }}\underset {3u_{2}}{\overset {a_{n,n}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{1}du_{2} \\ & \leq 2\overset {\frac{a_{n,n}}{3}}{\underset {\frac{1}{3\left( n+1\right) }}{\int }}\underset {3u_{2}}{\overset {a_{n,n}}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{3u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{1}du_{2} \\ &  = \tfrac {2}{3}\overset {\frac{a_{n,n}}{3}}{\underset {\frac{1}{3\left( n+1\right) }}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}\log \left( \tfrac {a_{n,n}}{3u_{2}}\right) A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{2} \\ & \leq \log \left( \left( n+1\right) a_{n,n}\right) \overset {\frac{a_{n,n}}{3}}{\underset {\frac{1}{3\left( n+1\right) }}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{2} \\ &  \leq \log \left( n+1\right) \overset {\frac{a_{n,n}}{3}}{\underset {\frac{1}{3\left( n+1\right) }}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{2} =\log \left( n+1\right) \overset {\frac{3}{\pi }\left( n+1\right) }{\underset {\frac{3}{\pi }}{\int }}\tfrac {\omega _{H}( f,\frac{3}{\pi t}) }{t}A_{n}^{\ast }\left( t\right) dt \\ & =\log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\left( \overset {\frac{3}{\pi }\left( k+1\right) }{\underset {\frac{3}{\pi }k}{\int }}\tfrac {\omega _{H}\left( f,\frac{3}{\pi t}\right) }{t}A_{n}^{\ast }\left( t\right) dt\right) \\ & \leq \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}( f,\frac{1}{k}) }{k}A_{n}^{\ast }\left( k+1\right) . \end{align*}
</div>
<p>For \(j=2,3,\) by \(\left( \ref{hk3}\right) \) and (<a href="#hk2">21</a>)</p>
<div class="displaymath" id="a0000000056">
  \begin{align*} & \underset {\Gamma _{1}^{\prime \prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,j}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim \overset {a_{n,n}}{\underset {\frac{1}{n+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}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{2}du_{1} \\ & =\overset {a_{n,n}}{\underset {\frac{1}{n+1}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}\log \left( \left( n+1\right) u_{1}\right) A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1} \\ & \leq \log \left( \left( n+1\right) a_{n,n}\right) \overset {a_{n,n}}{\underset {\frac{1}{n+1}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1}= \end{align*}
</div>
<div class="displaymath" id="a0000000057">
  \begin{align*} & =\log \left( \left( n+1\right) a_{n,n}\right) \overset {\frac{3}{\pi }\left( n+1\right) }{\underset {\frac{3}{\pi }}{\int }}\tfrac {\omega _{H}\left( f,\frac{3}{\pi t}\right) }{t}A_{n}^{\ast }\left( t\right) dt \\ & \leq \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) . \end{align*}
</div>
<p>Thus we get the estimate </p>
<div class="equation" id="In1">
<p>
  <div class="equation_content">
    \begin{equation}  I_{n,1}\lesssim \log \left( n+1\right) \left\{  \omega _{H}\left( f,a_{n,n}\right) +\underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) \right\}  . \label{In1} \end{equation}
  </div>
  <span class="equation_label">23</span>
</p>
</div>
<p>We divide \(\Gamma _{2}\) into two domains as </p>
<div class="displaymath" id="a0000000058">
  \begin{align*}  \Gamma _{2}^{\prime } & :=\left\{  \left( u_{1},u_{2}\right) \in \Gamma _{2}:u_{2}\leq \tfrac {a_{n,n}}{3\left( n+1\right) }\right\}  , \\ \Gamma _{2}^{\prime \prime } & :=\left\{  \left( u_{1},u_{2}\right) \in \Gamma _{2}:u_{2}\geq \tfrac {a_{n,n}}{3\left( n+1\right) }\right\}  \end{align*}
</div>
<p>to estimate \(I_{n,2}.\) By \(\left( \ref{sin2}\right) \) and \(\left( \ref{modc2}\right) ,\)</p>
<div class="displaymath" id="a0000000059">
  \begin{eqnarray*} & & \underset {\Gamma _{2}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k,1}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim & \overset {\frac{a_{n,n}}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{1}du_{2}\leq 2\tfrac {\omega _{H}\left( f,a_{n,n}\right) }{a_{n,n}}\overset {\frac{a_{n,n}}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}\tfrac {1}{u_{1}}du_{1}du_{2} \\ & =& 2\tfrac {\omega _{H}\left( f,a_{n,n}\right) }{a_{n,n}}\log \left( \tfrac {1}{a_{n,n}}\right) \tfrac {a_{n,n}}{3\left( n+1\right) }\leq \log \left( n+1\right) \omega _{H}\left( f,a_{n,n}\right) . \end{eqnarray*}
</div>
<p>\(\left( \ref{sin1}\right) ,\) \(\left( \ref{sin2}\right) \) and \(\left( \ref{modc2}\right) \) yield</p>
<div class="displaymath" id="a0000000060">
  \begin{eqnarray*} & & \underset {\Gamma _{2}^{\prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k,j}^{\ast }\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim & \left( n+1\right) \overset {\frac{a_{n,n}}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}du_{1}du_{2}\leq 2\left( n+1\right) \tfrac {\omega _{H}\left( f,a_{n,n}\right) }{a_{n,n}}\overset {\frac{a_{n,n}}{3\left( n+1\right) }}{\underset {0}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}du_{1}du_{2} \\ & \leq & \omega _{H}\left( f,a_{n,n}\right) \end{eqnarray*}
</div>
<p>for \(j=2,3.\) Since \(\left( \ref{sin2}\right) \) implies \(\left\vert H_{k,1}\left( u_{1},u_{2}\right) \right\vert \lesssim \frac{1}{u_{1}^{2}}\) for \(\left( u_{1},u_{2}\right) \in \Gamma _{2}^{\prime \prime },\) we get </p>
<div class="displaymath" id="a0000000061">
  \begin{align*} & \underset {\Gamma _{2}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2}\lesssim \end{align*}
</div>
<div class="displaymath" id="a0000000062">
  \begin{align*} & \lesssim \overset {\frac{a_{n,n}}{3}}{\underset {\frac{a_{n,n}}{3\left( n+1\right) }}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}du_{1}du_{2}\leq 2\tfrac {\omega _{H}\left( f,a_{n,n}\right) }{a_{n,n}}\overset {\frac{a_{n,n}}{3}}{\underset {\frac{a_{n,n}}{3\left( n+1\right) }}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}\tfrac {1}{u_{1}}du_{1}du_{2} \\ & =2\tfrac {\omega _{H}\left( f,a_{n,n}\right) }{a_{n,n}}\log \left( \tfrac {1}{a_{n,n}}\right) \tfrac {a_{n,n}}{3}\left( 1-\tfrac {1}{n+1}\right) \leq \log \left( n+1\right) \omega _{H}\left( f,a_{n,n}\right) . \end{align*}
</div>
<p>By considering \(\left( \ref{hk3}\right) \) and \(\left( \ref{hk2}\right) \) we obtain</p>
<div class="displaymath" id="a0000000063">
  \begin{eqnarray*} & & \underset {\Gamma _{2}^{\prime \prime }}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,j}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim & \overset {\frac{a_{n,n}}{3}}{\underset {\frac{a_{n,n}}{3\left( n+1\right) }}{\int }}\underset {a_{n,n}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1}du_{2}=\log \left( n+1\right) \underset {a_{n,n}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}A_{n}^{\ast }\left( \tfrac {3}{\pi 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}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1}\leq \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) \end{eqnarray*}
</div>
<p>for \(j=2,3.\) Thus we obtain </p>
<div class="equation" id="In2">
<p>
  <div class="equation_content">
    \begin{equation}  I_{n,2}\lesssim \log \left( n+1\right) \left\{  \omega _{H}\left( f,a_{n,n}\right) +\underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) \right\}  . \label{In2} \end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<p>If we use \(\left( \ref{hk1}\right) \) and \(\left( \ref{modc2}\right) ,\)</p>
<div class="displaymath" id="a0000000064">
  \begin{align*} & \underset {\Gamma _{3}}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,1}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim \overset {\frac{1}{3}}{\underset {\frac{a_{n,n}}{3}}{\int }}\underset {3u_{2}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}^{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{1}du_{2} \\ & \leq \tfrac {2}{3}\overset {\frac{1}{3}}{\underset {\frac{a_{n,n}}{3}}{\int }}\underset {3u_{2}}{\overset {1}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{1}du_{2} \\ & =\tfrac {2}{3}\overset {\frac{1}{3}}{\underset {\frac{a_{n,n}}{3}}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}\log \left( \tfrac {1}{3u_{2}}\right) A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{2} \\ & \leq \tfrac {2}{3}\log \left( \tfrac {1}{a_{n,n}}\right) \overset {\frac{1}{3}}{\underset {\frac{a_{n,n}}{3}}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{2} \leq \end{align*}
</div>
<div class="displaymath" id="a0000000065">
  \begin{align*} & \leq \log \left( n+1\right) \overset {\frac{1}{3}}{\underset {\frac{1}{3\left( n+1\right) }}{\int }}\tfrac {\omega _{H}\left( f,3u_{2}\right) }{u_{2}}A_{n}^{\ast }\left( \tfrac {1}{\pi u_{2}}\right) du_{2} \\ & =\log \left( n+1\right) \overset {\frac{3}{\pi }\left( n+1\right) }{\underset {\frac{3}{\pi }}{\int }}\tfrac {\omega _{H}\left( f,\frac{3}{\pi t}\right) }{t}A_{n}^{\ast }\left( t\right) dt \\ & \leq \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) . \end{align*}
</div>
<p>By \(\left( \ref{hk3}\right) \) and \(\left( \ref{hk2}\right) \) we get</p>
<div class="displaymath" id="a0000000066">
  \begin{eqnarray*} & & \underset {\Gamma _{3}}{\int }\omega _{H}\left( f,u_{1}\right) \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}H_{k,j}\left( u_{1},u_{2}\right) \right\vert du_{1}du_{2} \lesssim \\ & \lesssim & \overset {1}{\underset {a_{n,n}}{\int }}\underset {\frac{a_{n,n}}{3}}{\overset {\frac{u_{1}}{3}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}u_{2}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{2}du_{1}=\overset {1}{\underset {a_{n,n}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}\log \left( \tfrac {u_{1}}{a_{n,n}}\right) A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1} \\ & \leq & \log \left( \tfrac {1}{a_{n,n}}\right) \overset {1}{\underset {\frac{1}{n+1}}{\int }}\tfrac {\omega _{H}\left( f,u_{1}\right) }{u_{1}}A_{n}^{\ast }\left( \tfrac {3}{\pi u_{1}}\right) du_{1}\leq \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) \end{eqnarray*}
</div>
<p>for \(j=2,3.\) Thus, we have </p>
<div class="equation" id="In3">
<p>
  <div class="equation_content">
    \begin{equation}  I_{n,3}\lesssim \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {\omega _{H}\left( f,\frac{1}{k}\right) }{k}A_{n}^{\ast }\left( k+1\right) . \label{In3} \end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p>Inequalities \(\left( \ref{In1}\right) \), \(\left( \ref{In2}\right) \), \(\left( \ref{In3}\right) \) and the fact \(A_{n}^{\ast }\left( k+1\right) \lesssim A_{n,n-k}^{\ast }\) imply \(\left( \ref{uniform}\right) \). <div class="proof_wrapper" id="a0000000067">
  <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="proof_wrapper" id="a0000000068">
  <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="#th:3">theorem 3</a>] The method used in proof of <a href="#th:1">theorem 1</a> gives </p>
<div class="equation" id="d1">
<p>
  <div class="equation_content">
    \begin{equation}  \underset {\Omega }{\int }\left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u\lesssim }\log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k} \label{d1} \end{equation}
  </div>
  <span class="equation_label">26</span>
</p>
</div>
<p>and </p>
<div class="equation" id="d2">
<p>
  <div class="equation_content">
    \begin{equation}  \underset {\Omega }{\int }\left\Vert \mathbf{u}\right\Vert ^{\alpha }\left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u\lesssim }\log \left( n+1\right) \left\{  a_{n,n}^{\alpha }+\underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,k}}{k^{1+\alpha }}\right\}  \qquad \left( 0<\alpha \leq 1\right) . \label{d2} \end{equation}
  </div>
  <span class="equation_label">27</span>
</p>
</div>
<p>If we set \(e_{n}\left( \mathbf{t}\right) :=f\left( \mathbf{t}\right) -T_{n}^{\left( A\right) }\left( f\right) \left( \mathbf{t}\right) ,\) we have </p>
<div class="equation" id="en">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{H^{\beta }\left( \overline{\Omega }\right) }=\left\Vert f-T_{n}^{\left( A\right) }\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }+\Lambda ^{\beta }\left( e_{n}\right) . \label{en} \end{equation}
  </div>
  <span class="equation_label">28</span>
</p>
</div>
<p>Since </p>
<div class="displaymath" id="a0000000069">
  \begin{equation*}  \left\vert e_{n}\left( \mathbf{t}\right) -e_{n}\left( \mathbf{s}\right) \right\vert \leq \tfrac {1}{\left\vert \Omega \right\vert }\! \underset {\Omega }{\int }\! \Big\vert f\left( \mathbf{t}\right) -f\left( \mathbf{t}\! -\! \mathbf{u}\right) -f\left( \mathbf{s}\right) +f\left( \mathbf{s}\! -\! \mathbf{u}\right) \Big\vert \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}, \end{equation*}
</div>
<p>we have to estimate the integral </p>
<div class="displaymath" id="a0000000070">
  \begin{equation*}  J_{n}:=\underset {\Omega }{\int }\Big\vert f\left( \mathbf{t}\right) -f\left( \mathbf{t}-\mathbf{u}\right) -f\left( \mathbf{s}\right) +f\left( \mathbf{s}-\mathbf{u}\right) \Big\vert \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}. \end{equation*}
</div>
<p>Since \(f\in H^{\alpha }( \overline{\Omega }) \) we have </p>
<div class="displaymath" id="a0000000071">
  \begin{equation*}  \Big\vert f\left( \mathbf{t}\right) -f\left( \mathbf{t}-\mathbf{u}\right) -f\left( \mathbf{s}\right) +f\left( \mathbf{s}-\mathbf{u}\right) \Big\vert \lesssim \left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\alpha }, \end{equation*}
</div>
<p>hence by \(\left( \ref{d1}\right) \) we get</p>
<div class="displaymath" id="a0000000072">
  \begin{eqnarray*}  \left( J_{n}\right) ^{\frac{\beta }{\alpha }} & =& \left( \underset {\Omega }{\int }\Big\vert f\left( \mathbf{t}\right) -f\left( \mathbf{t}-\mathbf{u}\right) -f\left( \mathbf{s}\right) +f\left( \mathbf{s}-\mathbf{u}\right) \Big\vert \left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}\right) ^{\frac{\beta }{\alpha }} \\ & \lesssim & \left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\beta }\left( \underset {\Omega }{\int }\left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}\right) ^{\frac{\beta }{\alpha }} \\ & \lesssim & \left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\beta }\left( \log \left( n+1\right) \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k}\right) ^{\frac{\beta }{\alpha }}. \end{eqnarray*}
</div>
<p>We also have </p>
<div class="displaymath" id="a0000000073">
  \begin{equation*}  \left\vert f\left( \mathbf{t}\right) -f\left( \mathbf{t}-\mathbf{u}\right) -f\left( \mathbf{s}\right) +f\left( \mathbf{s}-\mathbf{u}\right) \right\vert \lesssim \left\Vert \mathbf{u}\right\Vert ^{\alpha } \end{equation*}
</div>
<p>for \(f\in H^{\alpha }( \overline{\Omega }) .\) Thus by \(\left( \ref{d2}\right) \) we obtain </p>
<div class="displaymath" id="a0000000074">
  \begin{align*}  \left( J_{n}\right) ^{1-\frac{\beta }{\alpha }} & \lesssim \left( \underset {\Omega }{\int }\left\Vert \mathbf{u}\right\Vert ^{\alpha }\left\vert \underset {k=0}{\overset {n}{\sum }}a_{n,k}D_{k}\left( \mathbf{u}\right) \right\vert d\mathbf{u}\right) ^{1-\frac{\beta }{\alpha }} \\ & \lesssim \left( \log \left( n+1\right) \left\{  a_{n,n}^{\alpha }+\underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k^{1+\alpha }}\right\}  \right) ^{1-\frac{\beta }{\alpha }}. \end{align*}
</div>
<p>Since </p>
<div class="displaymath" id="a0000000075">
  \begin{align*} & \left\vert e_{n}\left( \mathbf{t}\right) -e_{n}\left( \mathbf{s}\right) \right\vert \leq \\ & \leq J_{n}= \left( J_{n}\right) ^{\frac{\beta }{\alpha }}\left( J_{n}\right) ^{1-\frac{\beta }{\alpha }} \\ & \lesssim \left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\beta }\log \left( n+1\right) \left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k}\right) ^{\frac{\beta }{\alpha }}\left\{  a_{n,n}^{\alpha -\beta }+\left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k^{1+\alpha }}\right) ^{1-\frac{\beta }{\alpha }}\right\}  , \end{align*}
</div>
<p>we get</p>
<div class="displaymath" id="a0000000076">
  \begin{equation*}  \tfrac {\left\vert e_{n}\left( \mathbf{t}\right) -e_{n}\left( \mathbf{s}\right) \right\vert }{\left\Vert \mathbf{t}-\mathbf{s}\right\Vert ^{\beta }}\lesssim \log \left( n+1\right) \left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k}\right) ^{\frac{\beta }{\alpha }}\left\{  a_{n,n}^{\alpha -\beta }+\left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k^{1+\alpha }}\right) ^{1-\frac{\beta }{\alpha }}\right\}  \text{ }\left( \mathbf{t}\neq \mathbf{s}\right) , \end{equation*}
</div>
<p>which implies</p>
<div class="displaymath" id="a0000000077">
  \begin{equation*}  \Lambda ^{\beta }\left( e_{n}\right) \lesssim \log \left( n+1\right) \left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k}\right) ^{\frac{\beta }{\alpha }}\left\{  a_{n,n}^{\alpha -\beta }+\left( \underset {k=1}{\overset {n}{\sum }}\tfrac {A_{n,n-k}^{\ast }}{k^{1+\alpha }}\right) ^{1-\frac{\beta }{\alpha }}\right\}  . \end{equation*}
</div>
<p>The proof is completed by combining this last estimate, \(\left( \ref{uniformx}\right) \) and \(\left( \ref{en}\right) \). <div class="proof_wrapper" id="a0000000078">
  <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="a0000000079">4 Remarks</h1>
<p>The degree of approximation by \(\left( C,1\right) ,\) Riesz and Nörlund means of trigonometric Fourier series was investigated in <span class="cite">
	[
	<a href="#prossdorf" >18</a>
	]
</span>, <span class="cite">
	[
	<a href="#sahney goel" >19</a>
	]
</span>, <span class="cite">
	[
	<a href="#hst" >9</a>
	]
</span> and <span class="cite">
	[
	<a href="#chandra82" >1</a>
	]
</span>. In this section we conclude from ?? results about the degree of approximation of these means of hexagonal Fourier series. </p>
<p><div class="remark_thmwrapper " id="a0000000080">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">4</span>
  </div>
  <div class="remark_thmcontent">
  <p>Let \(p=\left( p_{k}\right) \) be a nondecreasing sequence of numbers positive real numbers. If we take </p>
<div class="displaymath" id="a0000000081">
  \begin{equation*}  a_{n,k}:= \begin{cases}  \frac{p_{k}}{P_{n}}, &  0\leq k\leq n \\ 0, &  k{\gt}n\end{cases}\end{equation*}
</div>
<p>where \(P_{n}:=\underset {k=0}{\overset {n}{\sum }}p_{k},\) then \(A=\left( a_{n,k}\right) \) satisfies \(\left( \ref{m1}\right) \), \(\left( \ref{m2}\right) \) and \(\left( \ref{m3}\right) \), and \(T_{n}^{\left( A\right) }\) becomes the Riesz mean </p>
<div class="displaymath" id="a0000000082">
  \begin{equation*}  R_{n}\left( p;f\right) =\tfrac {1}{P_{n}}\underset {k=0}{\overset {n}{\sum }}p_{k}S_{k}\left( f\right) . \end{equation*}
</div>
<p><a href="#th:1">theorem 1</a> gives </p>
<div class="equation" id="riesz1">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-R_{n}\left( p;f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\lesssim \log \left( \tfrac {P_{n}}{p_{0}}\right) \left\{  \omega _{H}\left( f,\tfrac {p_{n}}{P_{n}}\right) +\tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {Q_{n,k}\omega _{H}\left( f,1/k\right) }{k}\right\}  \label{riesz1} \end{equation}
  </div>
  <span class="equation_label">31</span>
</p>
</div>
<p>for \(f\in C_{H}( \overline{\Omega }) ,\) and <a href="#th:3">theorem 3</a> yields</p>
<div class="equation" id="riesz2">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-R_{n}\left( p;f\right) \right\Vert _{H^{\beta }\left( \overline{\Omega }\right) }\lesssim \log \left( \tfrac {P_{n}}{p_{0}}\right) \! \! \left( \!  \tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {Q_{n,k}}{k}\right) ^{\frac{\beta }{\alpha }}\! \! \left\{  \left(\!  \tfrac {p_{n}}{P_{n}}\right) ^{\alpha -\beta }\! \! \! +\! \! \left( \tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {Q_{n,k}}{k^{1+\alpha }}\right) ^{1-\frac{\beta }{\alpha }}\right\}  \label{riesz2} \end{equation}
  </div>
  <span class="equation_label">32</span>
</p>
</div>
<p>for \(f\in H^{\alpha }( \overline{\Omega }) ,\) \(\left( 0\leq \beta {\lt}\alpha \leq 1\right) ,\) where \(Q_{n,k}:=\underset {\nu =n-k}{\overset {n}{\sum }}p_{\nu }.\) <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000083">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">5</span>
  </div>
  <div class="remark_thmcontent">
  <p>Let \(\left( p_{k}\right) \) be a nonincreasing sequence of positive real numbers. In this case the matrix \(A=\left( a_{n,k}\right) \) with entries</p>
<div class="displaymath" id="a0000000084">
  \begin{equation*}  a_{n,k}= \begin{cases}  \frac{p_{n-k}}{P_{n}}, &  0\leq k\leq n , \\ 0, &  k{\gt}n, \end{cases}\end{equation*}
</div>
<p>satisfies \(\left( \ref{m1}\right)\), \(\left( \ref{m2}\right) \) and \(\left( \ref{m3}\right) ,\) and \(T_{n}^{\left( A\right) }\) becomes the Nörlund mean </p>
<div class="displaymath" id="a0000000085">
  \begin{equation*}  N_{n}\left( p;f\right) =\tfrac {1}{P_{n}}\underset {k=0}{\overset {n}{\sum }}p_{n-k}S_{k}\left( f\right) . \end{equation*}
</div>
<p>By <a href="#th:1">theorem 1</a> we conclude </p>
<div class="equation" id="norlund1">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-N_{n}\left( p;f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\lesssim \log \left( \tfrac {P_{n}}{p_{n}}\right) \left\{  \omega _{H}\left( f,\tfrac {p_{0}}{P_{n}}\right) +\tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {P_{k}\omega _{H}\left( f,1/k\right) }{k}\right\}  \label{norlund1} \end{equation}
  </div>
  <span class="equation_label">35</span>
</p>
</div>
<p>for \(f\in C_{H}\left( \overline{\Omega }\right) ,\) and by <a href="#th:3">theorem 3</a></p>
<div class="equation" id="norlund2">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f-N_{n}\left( p;f\right) \right\Vert _{H^{\beta }\left( \overline{\Omega }\right) }\lesssim \log \left( \tfrac {P_{n}}{p_{n}}\right) \! \! \left( \tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {P_{k}}{k}\right) ^{\frac{\beta }{\alpha }}\! \left\{  \left( \tfrac {p_{0}}{P_{n}}\right) ^{\alpha -\beta }\! \! \! +\! \! \left( \tfrac {1}{P_{n}}\underset {k=1}{\overset {n}{\sum }}\tfrac {P_{k}}{k^{1+\alpha }}\right) ^{1-\frac{\beta }{\alpha }}\right\}  \label{norlund2} \end{equation}
  </div>
  <span class="equation_label">36</span>
</p>
</div>
<p>for \(f\in H^{\alpha }( \overline{\Omega }) ,\) \(\left( 0\leq \beta {\lt}\alpha \leq 1\right) .\) <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000086">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">6</span>
  </div>
  <div class="remark_thmcontent">
  <p>f we take \(p_{k}=1\) \(\left( k=0,1,\ldots \right) ,\) \(R_{n}\left( p;f\right) \) and \(N_{n}\left( p;f\right) \) become \(\left( C,1\right) \) means \(S_{n}^{\left( 1\right) }\left( f\right) ,\) and both of \((\ref{riesz1})\) and \(\left( \ref{norlund1}\right) \) reduce to </p>
<div class="displaymath" id="a0000000087">
  \begin{equation*}  \left\Vert f-S_{n}^{\left( 1\right) }\left( f\right) \right\Vert _{C_{H}\left( \overline{\Omega }\right) }\lesssim \tfrac {\log \left( n+1\right) }{n+1}\underset {k=1}{\overset {n}{\sum }}\omega _{H}\left( f,\tfrac {1}{k}\right) \end{equation*}
</div>
<p>for \(f\in C_{H}( \overline{\Omega }) .\) Furthermore, \(\left( \ref{riesz2}\right) \) and \(\left( \ref{norlund2}\right) \) give the estimate</p>
<div class="displaymath" id="a0000000088">
  \begin{equation*}  \left\Vert f-S_{n}^{\left( 1\right) }\left( f\right) \right\Vert _{H^{\beta }\left( \overline{\Omega }\right) }\lesssim \left\{  \begin{array}{cc} \frac{\log \left( n+1\right) }{n^{\alpha -\beta }}, &  \alpha {\lt}1 \\[2mm] \frac{\left( \log \left( n+1\right) \right) ^{2-\beta }}{n^{1-\beta }}, &  \alpha =1\end{array}\right. \end{equation*}
</div>
<p>for \(\left( C,1\right) \) means of \(f\in H^{\alpha }( \overline{\Omega }) \) \(\left( 0\leq \beta {\lt}\alpha \leq 1\right) .\) <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
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</a> </p>
</dd>
  <dt><a name="chandra88">2</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/BF01951564"> <i class="sc">P. Chandra</i>, <i class="it">On the degree of approximation of a class of functions by means of Fourier series</i>, Acta Math. Hung., <b class="bf">52</b> (1988) nos. 3–4, pp.&#160;199–205. <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="fuglede">3</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) no. 1, 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>
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  <dd><p><a href ="http://doi.org/10.7153/jca-01-06"> <i class="sc">A. Guven</i>, <i class="it">Approximation by means of hexagonal Fourier series in Hölder norms</i>, J. Classical. Anal., <b class="bf">1</b> (2012) no. 1, pp.&#160;43–52. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
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  <dt><a name="guvenmia">5</a></dt>
  <dd><p><a href ="http://doi.org/10.7153/mia-16-13"> <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) no. 1, pp.&#160;175–191. <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="guvenzaa">6</a></dt>
  <dd><p><a href ="http://doi.org/10.4171/ZAA/1576"> <i class="sc">A. Guven</i>, <i class="it">Approximation by Riesz means of hexagonal Fourier series</i>, Z. Anal. Anwend., <b class="bf">36</b> (2017) no. 1, pp.&#160;1–16. <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="guvenata">7</a></dt>
  <dd><p><a href ="https://doi.org/10.4208/ata.2017.v33.n4.8"> <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) no. 4, pp.&#160;384–400. <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="guvenx">8</a></dt>
  <dd><p><a href ="https://ictp.acad.ro/jnaat/journal/article/view/1128"> <i class="sc">A. Guven</i>, <i class="it">Approximation of continuous functions on hexagonal domains</i>, J. Numer. Anal. Approx. Theory., <b class="bf">47</b> (2018) no. 1, pp.&#160;42–57. <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="hst">9</a></dt>
  <dd><p><i class="sc">A.S.B. Holland, B.N. Sahney</i>, <i class="sc">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) nos. 1-2, pp.&#160;69–72. </p>
</dd>
  <dt><a name="khs">10</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/BF01896187"> <i class="sc">P.D. Kathal, A.S.B. Holland</i>, <i class="sc">B.N. Sahney</i>, <i class="it">A class of continuous functions and their degree of approximation</i>, Acta Math. Acad. Sci. Hungar., <b class="bf">30</b> (1977) nos. 3–4, pp.&#160;227–231. <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</i>, <i class="sc">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) no. 4, 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 ="http://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) no. 1, 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="mohapatra sahney">13</a></dt>
  <dd><p><a href ="https://ictp.acad.ro/jnaat/journal/article/view/1981-vol10-no1-art9"> <i class="sc">R.N. Mohapatra</i>, <i class="sc">B.N. Sahney</i>, <i class="it">Approximation of continuous functions by their Fourier series</i>, Rev. Anal. Numér. Théor. Approx., <b class="bf">10</b> (1981) no. 1, pp.&#160;81–87. <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 chandra">14</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/BF01994063"> <i class="sc">R.N. Mohapatra</i>, <i class="sc">P. Chandra</i>, <i class="it">Degree of approximation of functions in the Hölder metric</i>, Acta Math. Hung., <b class="bf">41</b> (1983) nos. 1–2, pp.&#160;67–76. <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="mr1">15</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/0021-9045(87)90012-8"> <i class="sc">F. Moricz</i>, <i class="sc">B.E. Rhoades</i>, <i class="it">Approximation by Nörlund means of double Fourier series for Lipschitz functions</i>, J. Approx. Theory, <b class="bf">50</b> (1987) no. 4, pp.&#160;341–358. <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="mr2">16</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/BF01890571"> <i class="sc">F. Moricz</i>, <i class="sc">B.E. Rhoades</i>, <i class="it">Approximation by Nörlund means of double Fourier series to continuous functions in two variables</i>, Constr. Approx., <b class="bf">3</b> (1987) no. 1, pp.&#160;281–296. <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="mshi">17</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/0021-9045(87)90074-8"> <i class="sc">F. Moricz</i>, <i class="sc">X.L. Shi</i>, <i class="it">Approximation to continuous functions by Cesàro means of double Fourier series and conjugate series</i>, J. Approx. Theory, <b class="bf">49</b> (1987) no. 4, pp.&#160;346–377. <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="prossdorf">18</a></dt>
  <dd><p><a href ="https://doi.org/10.1002/mana.19750690102"> <i class="sc">S. Prössdorf</i>, <i class="it">Zur konvergenz der Fourierreihen hölderstetiger funktionen</i>, Math. Nachr., <b class="bf">69</b> (1975) no. 1, pp.&#160;7–14. <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">19</a></dt>
  <dd><p><i class="sc">B.N. Sahney</i>, <i class="sc">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="timan">20</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, 1963. </p>
</dd>
  <dt><a name="xu">21</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) no. 1, 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="zhi">22</a></dt>
  <dd><p><i class="sc">L. Zhizhiashvili</i>, <i class="it">Trigonometric Fourier Series and Their Conjugates</i>, Kluwer Academic Publishers, 1996. </p>
</dd>
  <dt><a name="zygmund">23</a></dt>
  <dd><p><i class="sc">A. Zygmund</i>, <i class="it">Trigonometric Series</i>, Cambridge Univ. Press, 2nd edition, 1959. </p>
</dd>
</dl>


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