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<title>Global Smoothness and Approximation <br />by Generalized Discrete Singular Operators: Global Smoothness and Approximation <br />by Generalized Discrete Singular Operators</title>
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<h1>Global Smoothness and Approximation <br />by Generalized Discrete Singular Operators</h1>
<p class="authors">
<span class="author">George A. Anastassiou\(^\ast \) Merve Kester\(^\ast \)</span>
</p>
<p class="date">September 23, 2014.<br />Published online: January 23, 2015.</p>
</div>
<p>\(^\ast \)Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, U.S.A., e-mail: <span class="tt">ganastss@memphis.edu, mkester@memphis.edu</span>. </p>

<div class="abstract"><p> In this article we continue with the study of generalized discrete singular operators over the real line regarding their simultaneous global smoothness preservation property with respect to \(L_{p}\) norm for \(1\leq p\leq \infty ,\) by involving higher order moduli of smoothness. Additionally we study their simultaneous approximation to the unit operator with rates involving the modulus of smoothness. The Jackson type inequalities that produced in this article are almost sharp, containing neat constants, and they reflect the high order of differentiability of involved function. </p>
<p><b class="bf">MSC.</b> 26A15, 26D15, 41A17, 41A25, 41A28, 41A35, 41A80 </p>
<p><b class="bf">Keywords.</b> Simultaneous global smoothness, simultaneous approximation with rates, generalized discrete singular operators, modulus of smoothness. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>This article is motivated mainly by <span class="cite">
	[
	<a href="#Anastass" >3</a>
	]
</span>, <span class="cite">
	[
	<a href="#Anastass-Mezei" >6</a>
	]
</span>, Chapter 18, and <span class="cite">
	[
	<a href="#Favard" >8</a>
	]
</span>, where J. Favard in 1944 introduced the discrete version of Gauss-Weierstrass operator</p>
<div class="equation" id="I1">
<p>
  <div class="equation_content">
    \begin{equation}  \left( F_{n}f\right) (x)=\tfrac {1}{\sqrt{\pi n}}\sum \limits _{\nu =-\infty }^{\infty }f\left( \tfrac {\nu }{n}\right) \exp \left( -n\left( \tfrac {\nu }{n}-x\right) ^{2}\right) , \label{I1} \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p>\(n\in \mathbb {N} ,\) which has the property that \(\left( F_{n}f\right) (x)\) converges to \(f(x)\) pointwise for each \(x\in \mathbb {R} ,\) and uniformly on any compact subinterval of \(\mathbb {R} ,\) for each continuous function \(f\) \(\left( f\in C(\mathbb {R} )\right) \) that fulfills \(\left\vert f(t)\right\vert \leq Ae^{Bt^{2}},\) \(t\in \mathbb {R} ,\) where \(A,\) \(B\) are positive constants. </p>
<p>We are also greatly motivated by <span class="cite">
	[
	<a href="#Abel" >1</a>
	]
</span> and <span class="cite">
	[
	<a href="#Abel-Butzer" >2</a>
	]
</span>. </p>
<p>Furthermore, we are inspired by <span class="cite">
	[
	<a href="#Anastass-Kester1" >4</a>
	]
</span> and <span class="cite">
	[
	<a href="#Anastass-Kester2" >5</a>
	]
</span> where the authors studied pointwise, uniform, and \(L_{p}\), \(p\geq 1,\) approximation properties of generalized discrete singular operators of Picard, Gauss-Weierstrass, and Poisson-Cauchy type and their non-unitary analogs. </p>
<p>In this article, we study the discrete operators mentioned above regarding their global smoothness preservation properties, additionally we study their simultaneous global smoothness and approximation properties in \(L_{p}\) norm for \(1\leq p\leq \infty .\) </p>
<h1 id="a0000000003">2 Background</h1>
<p>In <span class="cite">
	[
	<a href="#Anastass-Mezei" >6</a>
	]
</span>, the authors studied the global smoothness preservation properties and differentiability, also approximations, of smooth general singular integral operators \(\Theta _{r,\xi }(f;x),\) defined as follows. </p>
<p>Let \(\xi {\gt}0\) and \(\mu _{\xi }\) be Borel probability measures on \(\mathbb {R}.\) For \(r\in \mathbb {N}\) and \(n\in \mathbb {Z}_{+}\) they defined</p>
<div class="equation" id="b21">
<p>
  <div class="equation_content">
    \begin{equation}  \alpha _{j}=\left\{  \begin{array}{ll} \left( -1\right) ^{r-j}\binom {r}{j}j^{-n}, &  j=1,\ldots ,r, \\ 1-\sum \limits _{i=1}^{r}\left( -1\right) ^{r-i}\binom {r}{i}i^{-n}, &  j=0,\end{array}\right. \label{b21} \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p>that is \(\sum \limits _{j=0}^{r}\alpha _{j}=1.\) Let \(f\colon \mathbb {R}\rightarrow \mathbb {R}\) be Borel measurable, they defined for \(x\in \mathbb {R},\)</p>
<div class="equation" id="b22">
<p>
  <div class="equation_content">
    \begin{equation}  \Theta _{r,\xi }(f;x):=\int _{-\infty }^{\infty }\Big( \textstyle \sum \limits _{j=0}^{r}\alpha _{j}f(x+jt)\Big) d\mu _{\xi }(t). \label{b22} \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p>They supposed \(\Theta _{r,\xi }(f;x)\in \mathbb {R},\) \(\forall x\in \mathbb {R}.\) </p>
<p>Let \(f\in C(\mathbb {R}),\) for \(m\in \mathbb {N}\) the \(m\)-th modulus of smoothness for \(1\leq p\leq \infty ,\) is given by</p>
<div class="equation" id="b23">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(f,h)_{p}:=\sup _{0\leq t\leq h}\Vert \Delta _{t}^{m}f(x)\Vert _{p,x}, \label{b23} \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p>where </p>
<div class="equation" id="b24">
<p>
  <div class="equation_content">
    \begin{equation}  \Delta _{t}^{m}f(x):=\sum _{j=0}^{m}(-1)^{m-j}{\tbinom {m}{j}}f(x+jt), \label{b24} \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p>see also <span class="cite">
	[
	<a href="#Devore-Lorentz" >7</a>
	, 
	p. 44
	]
</span>. </p>
<p>Denote </p>
<div class="equation" id="b25">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(f,h)_{\infty }=\omega _{m}(f,h). \label{b25} \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p>Notice that</p>
<div class="equation" id="b25*">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(cf,h)_{p}=\left\vert c\right\vert \omega _{m}(f,h)_{p} \label{b25*} \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p>where \(c\) is a real constant. </p>
<p>They gave the main global smoothness preservation result in <span class="cite">
	[
	<a href="#Anastass-Mezei" >6</a>
	]
</span> as follows: </p>
<p><div class="theorem_thmwrapper " id="bt1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p><i class="itshape">Let </i>\(h{\gt}0,\)<i class="itshape"> </i>\(f\in C(\mathbb {R})\)<i class="itshape">.</i> </p>
<p>\(\mathbf{i)}\) <i class="itshape">Suppose </i>\(\Theta _{r,\xi }(f;x)\in \mathbb {R},\) \(\xi {\gt}0,\) \(\forall x\in \mathbb {R}\) and \(\omega _{m}(f,h){\lt}\infty .\)<i class="itshape"> Then</i></p>
<div class="equation" id="b1">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(\Theta _{r,\xi }f,h)\leq \Big( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\Big) \omega _{m}(f,h). \label{b1} \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p>\(\mathbf{ii)}\) <i class="itshape">Suppose </i>\(f\in \left( C(\mathbb {R})\cap L_{p}(\mathbb {R})\right) \)<i class="itshape">,</i> \(p\geq 1\)<i class="itshape">. Then</i> </p>
<div class="equation" id="b3">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(\Theta _{r,\xi }f,h)_{p}\leq \Big( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\Big) \omega _{m}(f,h)_{p}. \label{b3} \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>

  </div>
</div> </p>
<p>Next, in <span class="cite">
	[
	<a href="#Anastass-Mezei" >6</a>
	]
</span>, the authors discussed about the derivatives of \(\Theta _{r,\xi }\left( f;x\right) \) and their impact to simultaneous global smoothness preservation and convergence of these operators. </p>
<p>In <span class="cite">
	[
	<a href="#Anastass-Mezei" >6</a>
	]
</span>, they obtained also the next differentiation result </p>
<p><div class="theorem_thmwrapper " id="bt2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2</span>
  </div>
  <div class="theorem_thmcontent">
  <p><i class="itshape">Let \(f\in C^{n-1}(\mathbb {R})\), such that \(f^{(n)}\) exists, \(n,r\in \mathbb {N}\). Furthermore assume that for each </i>\(x\in \mathbb {R}\) <i class="itshape">the function</i> <i class="itshape">\(f^{(i)}(x+jt)\in L_{1}(\mathbb {R},\mu _{\xi })\) as a function of </i>\(\mathit{t,}\)<i class="itshape"> for all \(i=0,1,\ldots ,n-1;\) </i>\(j=1,\ldots ,r.\)<i class="itshape"> Suppose that there exist \(g_{i,j}\geq 0\), \(i=1,\ldots ,n;\) \(j=1,\ldots ,r,\) with \(g_{i,j}\in L_{1}(\mathbb {R},\mu _{\xi })\) such that for each \(x\in \mathbb {R}\) we have</i> </p>
<div class="equation" id="b4">
<p>
  <div class="equation_content">
    \begin{equation}  |f^{(i)}(x+jt)|\leq g_{i,j}(t), \label{b4} \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p><i class="itshape">for </i>\(\mu _{\xi }\)<i class="itshape">-almost all \(t\in \mathbb {R}\), all \(i=1,\ldots ,n;\) </i>\(j=1,2,\ldots ,r.\)<i class="itshape"> Then \(f^{(i)}(x+jt)\) defines a </i>\(\mu _{\xi }\)<i class="itshape">-integrable function with respect to \(t\) for each \(x\in \mathbb {R}\), all \(i=1,\ldots ,n;\) \(j=1,\ldots ,r\), and</i> </p>
<div class="equation" id="b5">
<p>
  <div class="equation_content">
    \begin{equation}  \left( \Theta _{r,\xi }\left( f;x\right) \right) ^{(i)}=\Theta _{r,\xi }\big( f^{(i)};x\big) , \label{b5} \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p><i class="itshape">for all \(x\in \mathbb {R}\), all</i> \(i=1,\ldots ,n\). </p>

  </div>
</div> </p>
<p>On the other hand, in <span class="cite">
	[
	<a href="#Anastass-Kester1" >4</a>
	]
</span>, the authors defined important special cases of \(\Theta _{r,\xi }\) operators for discrete probability measures \(\mu _{\xi }\) as follows: </p>
<p>Let \(f\in C^{n}(\mathbb {R} )\), \(n\in \mathbb {Z} ^{+},\) \(0{\lt}\xi \leq 1\), \(x\in \mathbb {R} \). </p>
<p>\(i)\) When</p>
<div class="equation" id="b26">
<p>
  <div class="equation_content">
    \begin{equation}  \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}\text{,} \label{b26} \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p>they defined the generalized discrete Picard operators as</p>
<div class="equation" id="b27">
<p>
  <div class="equation_content">
    \begin{equation}  P_{r,\xi }^{\ast }\left( f;x\right) :=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( \sum \limits _{j=0}^{r}\alpha _{j}f(x+j\nu )\right) e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}\text{.} \label{b27} \end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<p>\(ii)\) When</p>
<div class="equation" id="b28">
<p>
  <div class="equation_content">
    \begin{equation}  \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{^{\frac{-\nu ^{2}}{_{\xi }}}}}\text{,} \label{b28} \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<p>they defined the generalized discrete Gauss-Weierstrass operators as</p>
<div class="equation" id="b29">
<p>
  <div class="equation_content">
    \begin{equation}  W_{r,\xi }^{\ast }\left( f;x\right) :=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( \sum \limits _{j=0}^{r}\alpha _{j}f(x+j\nu )\right) e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{_{\xi }}}}\text{.} \label{b29} \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p>\(iii)\) Let \(\alpha \in \mathbb {N} \), and \(\beta {\gt}\tfrac {1}{\alpha }.\) When</p>
<div class="equation" id="b30">
<p>
  <div class="equation_content">
    \begin{equation}  \mu _{\xi }(\nu )=\tfrac {\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}, \label{b30} \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p>they defined the generalized discrete Poisson-Cauchy operators as</p>
<div class="equation" id="b31">
<p>
  <div class="equation_content">
    \begin{equation}  Q_{r,\xi }^{\ast }\left( f;x\right) :=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\Big( \sum \limits _{j=0}^{r}\alpha _{j}f(x+j\nu )\Big) \left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}\text{.} \label{b31} \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p>They observed that for \(c\) constant they have</p>
<div class="equation" id="b32">
<p>
  <div class="equation_content">
    \begin{equation}  P_{r,\xi }^{\ast }\left( c;x\right) =W_{r,\xi }^{\ast }\left( c;x\right) =Q_{r,\xi }^{\ast }\left( c;x\right) =c\text{.} \label{b32} \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p>They assumed that the operators \(P_{r,\xi }^{\ast }\left( f;x\right) \), \(W_{r,\xi }^{\ast }\left( f;x\right) \), and \(Q_{r,\xi }^{\ast }\left( f;x\right) \in \mathbb {R} ,\) for \(x\in \mathbb {R} .\) This is the case when \(\left\Vert f\right\Vert _{\infty ,\mathbb {R} }{\lt}\infty .\) </p>
<p>\(iv)\) When </p>
<div class="equation" id="b33">
<p>
  <div class="equation_content">
    \begin{equation}  \mu _{\xi }(\nu ):=\mu _{\xi ,P}(\nu ):=\tfrac {e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{1+2\xi e^{-\frac{1}{\xi }}}, \label{b33} \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p>they defined the generalized discrete non-unitary Picard operators as</p>
<div class="equation" id="b34">
<p>
  <div class="equation_content">
    \begin{equation}  P_{r,\xi }\left( f;x\right) :=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( \sum \limits _{j=0}^{r}\alpha _{j}f(x+j\nu )\right) e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{1+2\xi e^{-\frac{1}{\xi }}}. \label{b34} \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<p>Here \(\mu _{\xi ,P}(\nu )\) has mass</p>
<div class="equation" id="b35">
<p>
  <div class="equation_content">
    \begin{equation}  m_{\xi ,P}:=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{1+2\xi e^{-\frac{1}{\xi }}}. \label{b35} \end{equation}
  </div>
  <span class="equation_label">21</span>
</p>
</div>
<p>They observed that</p>
<div class="equation" id="b36">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {\mu _{\xi ,P}(\nu )}{m_{\xi ,P}}=\tfrac {e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}, \label{b36} \end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p>which is the probability measure \(\left( \ref{b26}\right) \) defining the operators \(P_{r,\xi }^{\ast }.\) </p>
<p>\(v)\) When </p>
<div class="equation" id="b37">
<p>
  <div class="equation_content">
    \begin{equation}  \mu _{\xi }(\nu ):=\mu _{\xi ,W}(\nu ):=\tfrac {e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}, \label{b37} \end{equation}
  </div>
  <span class="equation_label">23</span>
</p>
</div>
<p>with \({\rm erf}(x)=\tfrac {2}{\sqrt{\pi }}\int _{0}^{x}e^{-t^{2}}dt,\) \({\rm erf}(\infty )=1,\) they defined the generalized discrete non-unitary Gauss-Weierstrass operators as</p>
<div class="equation" id="b38">
<p>
  <div class="equation_content">
    \begin{equation}  W_{r,\xi }\left( f;x\right) :=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( \sum \limits _{j=0}^{r}\alpha _{j}f(x+j\nu )\right) e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}. \label{b38} \end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<p>Here \(\mu _{\xi ,W}(\nu )\) has mass</p>
<div class="equation" id="b39">
<p>
  <div class="equation_content">
    \begin{equation}  m_{\xi ,W}:=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}. \label{b39} \end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p>They observed that</p>
<div class="equation" id="b40">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {\mu _{\xi ,W}(\nu )}{m_{\xi ,W}}=\tfrac {e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{_{\xi }}}}, \label{b40} \end{equation}
  </div>
  <span class="equation_label">26</span>
</p>
</div>
<p>which is the probability measure \(\left( \ref{b28}\right) \) defining the operators \(W_{r,\xi }^{\ast }.\) </p>
<p>The authors observed that \(P_{r,\xi }\left( f;x\right) \), \(W_{r,\xi }\left( f;x\right) \in \mathbb {R} ,\) for \(x\in \mathbb {R} \). </p>
<p>We notice that</p>
<div class="equation" id="b40.1">
<p>
  <div class="equation_content">
    \begin{equation}  P_{r,\xi }\left( f;x\right) =\lambda _{1}\left( \xi \right) P_{r,\xi }^{\ast }\left( f;x\right) , \label{b40.1} \end{equation}
  </div>
  <span class="equation_label">27</span>
</p>
</div>
<p>where</p>
<div class="equation" id="b40.2">
<p>
  <div class="equation_content">
    \begin{equation}  \lambda _{1}\left( \xi \right) =\tfrac {\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{1+2\xi e^{-\frac{1}{\xi }}}, \label{b40.2} \end{equation}
  </div>
  <span class="equation_label">28</span>
</p>
</div>
<p>and</p>
<div class="equation" id="b40.3">
<p>
  <div class="equation_content">
    \begin{equation}  W_{r,\xi }\left( f;x\right) =\lambda _{2}\left( \xi \right) W_{r,\xi }^{\ast }\left( f;x\right) , \label{b40.3} \end{equation}
  </div>
  <span class="equation_label">29</span>
</p>
</div>
<p>where</p>
<div class="equation" id="b40.4">
<p>
  <div class="equation_content">
    \begin{equation}  \lambda _{2}\left( \xi \right) =\tfrac {\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}. \label{b40.4} \end{equation}
  </div>
  <span class="equation_label">30</span>
</p>
</div>
<p>In <span class="cite">
	[
	<a href="#Anastass-Kester1" >4</a>
	]
</span>, for \(k=1,...,n\), the authors defined the ratios of sums </p>
<div class="equation" id="b41">
<p>
  <div class="equation_content">
    \begin{equation}  c_{k,\xi }^{\ast }:=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\nu ^{k}e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}, \label{b41} \end{equation}
  </div>
  <span class="equation_label">31</span>
</p>
</div>
<div class="equation" id="b42">
<p>
  <div class="equation_content">
    \begin{equation}  p_{k,\xi }^{\ast }:=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\nu ^{k}e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{_{\xi }}}}, \label{b42} \end{equation}
  </div>
  <span class="equation_label">32</span>
</p>
</div>
<p>and for \(\alpha \in \mathbb {N} ,\) \(\beta {\gt}\tfrac {n+r+1}{2\alpha },\) they  introduced</p>
<div class="equation" id="b43">
<p>
  <div class="equation_content">
    \begin{equation}  q_{k,\xi }^{\ast }:=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\nu ^{k}\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}. \label{b43} \end{equation}
  </div>
  <span class="equation_label">33</span>
</p>
</div>
<p>Furthermore, they proved that these ratios of sums \(c_{k,\xi }^{\ast },\) \(p_{k,\xi }^{\ast }\), and \(q_{k,\xi }^{\ast }\) are finite for all \(\xi \in \left( 0,1\right] \). </p>
<p>In <span class="cite">
	[
	<a href="#Anastass-Kester1" >4</a>
	]
</span>, the authors also proved</p>
<div class="equation" id="b44">
<p>
  <div class="equation_content">
    \begin{equation}  m_{\xi ,P}=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }e^{-\frac{\left\vert \nu \right\vert }{\xi }}}{1+2\xi e^{-\frac{1}{\xi }}}\rightarrow 1,\quad \text{ as }\xi \rightarrow 0^{+} \label{b44} \end{equation}
  </div>
  <span class="equation_label">34</span>
</p>
</div>
<p>and</p>
<div class="equation" id="b45">
<p>
  <div class="equation_content">
    \begin{equation}  m_{\xi ,W}=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }e^{-\frac{\nu ^{2}}{\xi }}}{1+\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) }\rightarrow 1,\quad \text{ as }\xi \rightarrow 0^{+}. \label{b45} \end{equation}
  </div>
  <span class="equation_label">35</span>
</p>
</div>
<p>The authors introduced also</p>
<div class="equation" id="b54">
<p>
  <div class="equation_content">
    \begin{equation}  \delta _{k}:=\sum _{j=1}^{r}\alpha _{j}j^{k},\quad k=1,\ldots ,n\in \mathbb {N}. \label{b54} \end{equation}
  </div>
  <span class="equation_label">36</span>
</p>
</div>
<p>Additionally, in <span class="cite">
	[
	<a href="#Anastass-Kester1" >4</a>
	]
</span>, the authors defined the following error quantities: </p>
<div class="displaymath" id="b46">
  \begin{align}  E_{0,P}(f,x) & :=P_{r,\xi }(f;x)-f(x) \label{b46} \\ & =\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\Big( \sum \limits _{j=0}^{r}\alpha _{j}f(x+j\nu )\Big) e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{1+2\xi e^{-\frac{1}{\xi }}}-f(x), \nonumber \end{align}
</div>
<div class="displaymath" id="b47">
  \begin{align}  E_{0,W}(f,x) & :=W_{r,\xi }(f;x)-f(x) \label{b47} \\ & =\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\Big( \sum \limits _{j=0}^{r}\alpha _{j}f(x+j\nu )\Big) e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}-f(x). \nonumber \end{align}
</div>
<p>Furthermore, they introduced the errors \((n\in \mathbb {N} )\):</p>
<div class="equation" id="b48">
<p>
  <div class="equation_content">
    \begin{equation}  E_{n,P}(f,x):=P_{r,\xi }(f;x)-f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{(k)}(x)}{k!}\delta _{k}\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\nu ^{k}e^{-\frac{\left\vert \nu \right\vert }{\xi }}}{1+2\xi e^{-\frac{1}{\xi }}} \label{b48} \end{equation}
  </div>
  <span class="equation_label">39</span>
</p>
</div>
<p>and</p>
<div class="equation" id="b49">
<p>
  <div class="equation_content">
    \begin{equation}  E_{n,W}(f,x):=W_{r,\xi }(f;x)-f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{(k)}(x)}{k!}\delta _{k}\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\nu ^{k}e^{-\frac{\nu ^{2}}{\xi }}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}. \label{b49} \end{equation}
  </div>
  <span class="equation_label">40</span>
</p>
</div>
<p>Next, they obtained the inequalities</p>
<div class="equation" id="b50">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert E_{0,P}(f,x)\right\vert \leq m_{\xi ,P}\left\vert P_{r,\xi }^{\ast }(f;x)-f(x)\right\vert +\left\vert f(x)\right\vert \left\vert m_{\xi ,P}-1\right\vert , \label{b50} \end{equation}
  </div>
  <span class="equation_label">41</span>
</p>
</div>
<div class="equation" id="b51">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert E_{0,W}(f,x)\right\vert \leq m_{\xi ,W}\left\vert W_{r,\xi }^{\ast }(f;x)-f(x)\right\vert +\left\vert f(x)\right\vert \left\vert m_{\xi ,W}-1\right\vert , \label{b51} \end{equation}
  </div>
  <span class="equation_label">42</span>
</p>
</div>
<p>and</p>
<div class="displaymath" id="b52">
  \begin{align} & \left\vert E_{n,P}(f,x)\right\vert \leq \label{b52} \\ & \leq m_{\xi ,P}\bigg\vert P_{r,\xi }^{\ast }(f;x)-f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{(k)}(x)}{k!}\delta _{k}c_{k,\xi }^{\ast }\bigg\vert +\left\vert f(x)\right\vert \left\vert m_{\xi ,P}-1\right\vert , \nonumber \end{align}
</div>
<p>with</p>
<div class="displaymath" id="b53">
  \begin{align} & \left\vert E_{n,W}(f,x)\right\vert \leq \label{b53} \\ & \leq m_{\xi ,W}\bigg\vert W_{r,\xi }^{\ast }(f;x)-f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{(k)}(x)}{k!}\delta _{k}p_{k,\xi }^{\ast }\bigg\vert +\left\vert f(x)\right\vert \left\vert m_{\xi ,W}-1\right\vert . \nonumber \end{align}
</div>
<p>In <span class="cite">
	[
	<a href="#Anastass-Kester1" >4</a>
	]
</span>, they first gave the following simultaneous approximation results for unitary operators. They showed </p>
<p><div class="theorem_thmwrapper " id="bt3">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\in C^{n}(\mathbb {R})\) with \(f^{(n)}\in C_{u}(\mathbb {R} )\) (uniformly continuous functions on \(\mathbb {R} \))\(.\) </p>
<p>\(\mathbf{i)}\) For \(n\in \mathbb {N} ,\)</p>
<div class="displaymath" id="b6">
  \begin{align} & \bigg\Vert P_{r,\xi }^{\ast }\left( f;x\right) -f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{\left( k\right) }(x)}{k!}\delta _{k}c_{k,\xi }^{\ast }\bigg\Vert _{\infty ,x}\leq \label{b6} \\ & \leq \tfrac {\omega _{r}(f^{(n)},\xi )}{n!}\Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }|\nu |^{n}\left( 1+\frac{|\nu |}{\xi }\right) ^{r}e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}\Bigg) , \nonumber \end{align}
</div>
<p>and</p>
<div class="displaymath" id="b7">
  \begin{align} & \bigg\Vert W_{r,\xi }^{\ast }\left( f;x\right) -f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{\left( k\right) }(x)}{k!}\delta _{k}p_{k,\xi }^{\ast }\bigg\Vert _{\infty ,x}\leq \label{b7} \\ & \leq \tfrac {\omega _{r}(f^{(n)},\xi )}{n!}\Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }|\nu |^{n}\left( 1+\frac{|\nu |}{\xi }\right) ^{r}e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{_{\xi }}}}\Bigg) . \nonumber \end{align}
</div>
<p>\(\mathbf{ii)}\) For \(n=0,\)</p>
<div class="equation" id="b6*">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert P_{r,\xi }^{\ast }\left( f;x\right) -f(x)\right\Vert _{\infty ,x}\leq \omega _{r}(f,\xi )\Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( 1+\frac{|\nu |}{\xi }\right) ^{r}e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}\Bigg) , \label{b6*} \end{equation}
  </div>
  <span class="equation_label">47</span>
</p>
</div>
<p>and</p>
<div class="equation" id="b7*">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert W_{r,\xi }^{\ast }\left( f;x\right) -f(x)\right\Vert _{\infty ,x}\leq \omega _{r}(f,\xi )\Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( 1+\frac{|\nu |}{\xi }\right) ^{r}e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{_{\xi }}}}\Bigg) . \label{b7*} \end{equation}
  </div>
  <span class="equation_label">48</span>
</p>
</div>
<p>In the above inequalities <span class="rm">(<a href="#b6">45</a>)–(<a href="#b7*">48</a>)</span>, the ratios of sums in their right hand sides \(\left( \text{R.H.S.}\right) \) are uniformly bounded with respect to \(\xi \in \left( 0,1\right] .\) </p>

  </div>
</div> </p>
<p>In <span class="cite">
	[
	<a href="#Anastass-Kester1" >4</a>
	]
</span>, they had also </p>
<p><div class="theorem_thmwrapper " id="bt5">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\in C^{n}(\mathbb {R})\) with \(f^{(n)}\in C_{u}(\mathbb {R} ),\) \(n\in \mathbb {\mathbb {N} }\), and \(\beta {\gt}\tfrac {n+r+1}{2\alpha }.\) </p>
<p>\(\mathbf{i)}\) For \(n\in \mathbb {N} ,\)</p>
<div class="displaymath" id="b8">
  \begin{align} & \bigg\Vert Q_{r,\xi }^{\ast }\left( f;x\right) -f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{\left( k\right) }(x)}{k!}\delta _{k}q_{k,\xi }^{\ast }\bigg\Vert _{\infty ,x}\leq \label{b8} \\ & \leq \tfrac {\omega _{r}(f^{(n)},\xi )}{n!}\Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }|\nu |^{n}\left( 1+\frac{|\nu |}{\xi }\right) ^{r}\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}\Bigg) . \nonumber \end{align}
</div>
<p>\(\mathbf{ii)}\) For \(n=0,\)</p>
<div class="equation" id="b8*">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert Q_{r,\xi }^{\ast }\left( f;x\right) -f(x)\right\Vert _{\infty ,x}\leq \omega _{r}(f,\xi )\Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( 1+\frac{|\nu |}{\xi }\right) ^{r}\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}\Bigg) . \label{b8*} \end{equation}
  </div>
  <span class="equation_label">50</span>
</p>
</div>
<p>In the above inequalities <span class="rm">(<a href="#b8">49</a>)–(<a href="#b8*">50</a>)</span>, the ratios of sums in their R.H.S. are uniformly bounded with respect to \(\xi \in \left( 0,1\right] .\) </p>

  </div>
</div> </p>
<p>Next, they stated their results in <span class="cite">
	[
	<a href="#Anastass-Kester1" >4</a>
	]
</span> for the errors \(E_{0,P},\) \(E_{0,W},\) \(E_{n,P},\) and \(E_{n,W}.\) They had </p>
<p><div class="corollary_thmwrapper " id="bc1">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">5</span>
  </div>
  <div class="corollary_thmcontent">
  <p>Let \(f\in C_{u}(\mathbb {R} ).\) Then </p>
<p>\(\mathbf{i)}\)</p>
<div class="equation" id="b55">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert E_{0,P}(f,x)\right\vert \leq \Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\omega _{r}(f,\left\vert \nu \right\vert )e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{1+2\xi e^{-\frac{1}{\xi }}}\Bigg) +\left\vert f(x)\right\vert \cdot \left\vert m_{\xi ,P}-1\right\vert , \label{b55} \end{equation}
  </div>
  <span class="equation_label">51</span>
</p>
</div>
<p>\(\mathbf{ii)}\)</p>
<div class="equation" id="b56">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert E_{0,W}(f,x)\right\vert \leq \Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\omega _{r}(f,\left\vert \nu \right\vert )e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\Bigg) +\left\vert f(x)\right\vert \cdot \left\vert m_{\xi ,W}-1\right\vert . \label{b56} \end{equation}
  </div>
  <span class="equation_label">52</span>
</p>
</div>

  </div>
</div> </p>
<p>In <span class="cite">
	[
	<a href="#Anastass-Kester1" >4</a>
	]
</span>, for \(E_{n,P}\) and \(E_{n,W}\), the authors presented </p>
<p><div class="theorem_thmwrapper " id="bt12">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">6</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\in C^{n}(\mathbb {R} )\) with \(f^{(n)}\in C_{u}(\mathbb {R} )\), \(n\in \mathbb {N} ,\)and \(\left\Vert f\right\Vert _{\infty ,\mathbb {R} }{\lt}\infty .\) Then </p>
<p>\(\mathbf{i)}\)</p>
<div class="displaymath" id="b57">
  \begin{align} & \left\Vert E_{n,P}(f,x)\right\Vert _{\infty ,x}\leq \label{b57} \\ & \leq \tfrac {\omega _{r}(f^{(n)},\xi )}{n!}\Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }|\nu |^{n}\left( 1+\frac{|\nu |}{\xi }\right) ^{r}e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{1+2\xi e^{-\frac{1}{\xi }}}\Bigg) +\left\Vert f\right\Vert _{\infty ,\mathbb {R} }\left\vert m_{\xi ,P}-1\right\vert , \nonumber \end{align}
</div>
<p>\(\mathbf{ii)}\)</p>
<div class="displaymath" id="b58">
  \begin{align} & \left\Vert E_{n,W}(f,x)\right\Vert _{\infty ,x}\leq \label{b58} \\ & \leq \tfrac {\omega _{r}(f^{(n)},\xi )}{n!}\Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }|\nu |^{n}\left( 1+\frac{|\nu |}{\xi }\right) ^{r}e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\Bigg) +\left\Vert f\right\Vert _{\infty ,\mathbb {R} }\left\vert m_{\xi ,W}-1\right\vert . \nonumber \end{align}
</div>
<p>In the above inequalities <span class="rm">(<a href="#b57">53</a>)–(<a href="#b58">54</a>)</span>, the ratios of sums in their R.H.S. are uniformly bounded with respect to \(\xi \in \left( 0,1\right] .\) </p>

  </div>
</div> </p>
<p>In <span class="cite">
	[
	<a href="#Anastass-Kester2" >5</a>
	]
</span>, the authors represented simultaneous \(L_{p}\) approximation results. They started with </p>
<p><div class="theorem_thmwrapper " id="bt6">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">7</span>
  </div>
  <div class="theorem_thmcontent">
  <p>\(\mathbf{i)}\) <i class="itshape">Let </i>\(f\in C^{n}(\mathbb {R}),\)<i class="itshape"> with </i>\(f^{(n)}\in L_{p}\left( \mathbb {R}\right) ,\)<i class="itshape"> </i>\(n\in \mathbb {N},\)<i class="itshape"> </i>\(p,q{\gt}1\)<span class="rm">:</span> \(\tfrac {1}{p}+\tfrac {1}{q}=1\), and <i class="itshape">rest as above in this section</i>\(.\)Then</p>
<div class="displaymath" id="b9">
  \begin{align} & \bigg\Vert P_{r,\xi }^{\ast }\left( f;x\right) -f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{\left( k\right) }(x)}{k!}\delta _{k}c_{k,\xi }^{\ast }\bigg\Vert _{p}\leq \label{b9} \\ & \leq \tfrac {1}{((n-1)!)(q(n-1)+1)^{\frac{1}{q}}\left( rp+1\right) ^{\frac{1}{p}}}\left( M_{p,\xi }^{\ast }\right) ^{\frac{1}{p}}\xi ^{\frac{1}{p}}\omega _{r}(f^{(n)},\xi )_{p} \nonumber \end{align}
</div>
<p>where</p>
<div class="equation" id="b9*">
<p>
  <div class="equation_content">
    \begin{equation}  M_{p,\xi }^{\ast }:=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( \left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp+1}-1\right) \left\vert \nu \right\vert ^{np-1}e^{\frac{-\left\vert \nu \right\vert }{\xi }}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{\xi }}} \label{b9*} \end{equation}
  </div>
  <span class="equation_label">56</span>
</p>
</div>
<p>which is uniformly bounded for all \(\xi \in \left( 0,1\right] .\) </p>
<p>Additionally, as \(\xi \rightarrow 0^{+}\) we obtain that \(R.H.S.\) of \((\ref{b9})\) goes to zero. </p>
<p>\(\mathbf{ii)}\) When \(p=1,\) let <i class="itshape"> </i>\(f\in C^{n}(\mathbb {R})\), \(f^{(n)}\in L_{1}(\mathbb {R}),\) and \(n\in \mathbb {N} -\{ 1\} .\) Then</p>
<div class="displaymath" id="b10">
  \begin{align} & \bigg\Vert P_{r,\xi }^{\ast }\left( f;x\right) -f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{\left( k\right) }(x)}{k!}\delta _{k}c_{k,\xi }^{\ast }\bigg\Vert _{1}\leq \label{b10} \\ & \leq \tfrac {1}{(n-1)!\left( r+1\right) }M_{1,\xi }^{\ast }\xi \omega _{r}(f^{(n)},\xi )_{1} \nonumber \end{align}
</div>
<p>holds where \(M_{1,\xi }^{\ast }\) is defined as in \(\left( \ref{b9*}\right) \). Hence, as \(\xi \rightarrow 0^{+},\) we obtain that \(R.H.S.\) of \((\ref{b10})\) goes to zero. </p>
<p>\(\mathbf{iii)}\) When \(n=0\), <i class="itshape">let </i>\(f\in \left( C\left( \mathbb {R} \right) \cap L_{p}(\mathbb {R})\right) ,\) <i class="itshape"> </i>\(p,q{\gt}1\)<i class="itshape"> such that </i>\(\tfrac {1}{p}+\tfrac {1}{q}=1\)<i class="itshape"> and the rest as above in this section. Then</i></p>
<div class="equation" id="b11">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert P_{r,\xi }^{\ast }\left( f;x\right) -f(x)\right\Vert _{p}\leq \left( \bar{M}_{p,\xi }^{\ast }\right) ^{1/p}\omega _{r}(f,\xi )_{p} \label{b11} \end{equation}
  </div>
  <span class="equation_label">58</span>
</p>
</div>
<p>where</p>
<div class="equation" id="b11*">
<p>
  <div class="equation_content">
    \begin{equation}  \bar{M}_{p,\xi }^{\ast }:=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp}e^{\frac{-\left\vert \nu \right\vert }{\xi }}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{\xi }}} \label{b11*} \end{equation}
  </div>
  <span class="equation_label">59</span>
</p>
</div>
<p> which is uniformly bounded for all \(\xi \in \left( 0,1\right] .\) </p>
<p>Hence, as \(\xi \rightarrow 0^{+},\) we obtain that \(P_{r,\xi }^{\ast }\rightarrow \) unit operator \(I\) in the \(L_{p}\) norm for \(p{\gt}1\). </p>
<p>\(\mathbf{iv)}\)<b class="bfseries"> </b>When \(n=0\) and \(p=1,\) let \(f\in \left( C\left( \mathbb {R} \right) \cap L_{1}(\mathbb {R})\right) \) <i class="itshape">and the rest as above in this section</i>. Then the inequality</p>
<div class="equation" id="b12">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert P_{r,\xi }^{\ast }\left( f;x\right) -f(x)\right\Vert _{1}\leq \bar{M}_{1,\xi }^{\ast }\omega _{r}(f,\xi )_{1} \label{b12} \end{equation}
  </div>
  <span class="equation_label">60</span>
</p>
</div>
<p>holds where \(\bar{M}_{1,\xi }^{\ast }\) is defined as in \(\left( \ref{b11*}\right) \). Furthermore, we get \(P_{r,\xi }^{\ast }\rightarrow I\) in the \(L_{1}\) norm as \(\xi \rightarrow 0^{+}.\) </p>

  </div>
</div> </p>
<p>Next, the authors presented their quantitative results for the Gauss-Weierstrass operators, see <span class="cite">
	[
	<a href="#Anastass-Kester2" >5</a>
	]
</span>. They started with </p>
<p><div class="theorem_thmwrapper " id="bt8">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">8</span>
  </div>
  <div class="theorem_thmcontent">
  <p>\(\mathbf{i)}\) <i class="itshape">Let </i>\(f\in C^{n}(\mathbb {R}),\)<i class="itshape"> with </i>\(f^{(n)}\in L_{p}\left( \mathbb {R}\right) ,\)<i class="itshape"> </i>\(n\in \mathbb {N},\)<i class="itshape"> </i>\(p,q{\gt}1\)<span class="rm">:</span> \(\tfrac {1}{p}+\tfrac {1}{q}=1\), and the rest as above in this section. Then</p>
<div class="displaymath" id="b13">
  \begin{align} & \bigg\Vert W_{r,\xi }^{\ast }\left( f;x\right) -f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{\left( k\right) }(x)}{k!}\delta _{k}p_{k,\xi }^{\ast }\bigg\Vert _{p}\leq \label{b13} \\ & \leq \tfrac {1}{((n-1)!)(q(n-1)+1)^{\frac{1}{q}}\left( rp+1\right) ^{\frac{1}{p}}}\left( N_{p,\xi }^{\ast }\right) ^{\frac{1}{p}}\xi ^{\frac{1}{p}}\omega _{r}(f^{(n)},\xi )_{p} \nonumber \end{align}
</div>
<p>where</p>
<div class="equation" id="b13*">
<p>
  <div class="equation_content">
    \begin{equation}  N_{p,\xi }^{\ast }:=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( \left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp+1}-1\right) \left\vert \nu \right\vert ^{np-1}e^{\frac{-\nu ^{2}}{\xi }}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{\xi }}} \label{b13*} \end{equation}
  </div>
  <span class="equation_label">62</span>
</p>
</div>
<p>which is uniformly bounded for all \(\xi \in \left( 0,1\right] .\) </p>
<p>Additionally, as \(\xi \rightarrow 0^{+}\) we obtain that \(R.H.S.\) of \((\ref{b13})\) goes to zero. </p>
<p>\(\mathbf{ii)}\) For \(p=1\), let <i class="itshape"> </i>\(f\in C^{n}(\mathbb {R})\), \(f^{(n)}\in L_{1}(\mathbb {R}),\) and \(n\in \mathbb {N} -\{ 1\} .\) Then</p>
<div class="displaymath" id="b14">
  \begin{align} & \bigg\Vert W_{r,\xi }^{\ast }\left( f;x\right) -f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{\left( k\right) }(x)}{k!}\delta _{k}p_{k,\xi }^{\ast }\bigg\Vert _{1}\leq \label{b14} \\ & \leq \tfrac {1}{(n-1)!\left( r+1\right) }N_{1,\xi }^{\ast }\xi \omega _{r}(f^{(n)},\xi )_{1} \nonumber \end{align}
</div>
<p>holds where \(N_{1,\xi }^{\ast }\) is defined as in \(\left( \ref{b13*}\right) \). Hence, as \(\xi \rightarrow 0^{+},\) we obtain that \(R.H.S.\) of \((\ref{b14})\) goes to zero. </p>
<p>\(\mathbf{iii)}\) For \(n=0\),<i class="itshape"> let </i>\(f\in \left( C\left( \mathbb {R} \right) \cap L_{p}(\mathbb {R})\right) ,\) \(p,q{\gt}1\)<i class="itshape"> such that </i>\(\tfrac {1}{p}+\tfrac {1}{q}=1\)<i class="itshape"> and the rest as above in this section. Then</i></p>
<div class="equation" id="b15">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert W_{r,\xi }^{\ast }\left( f;x\right) -f(x)\right\Vert _{p}\leq \left( \bar{N}_{p,\xi }^{\ast }\right) ^{1/p}\omega _{r}(f,\xi )_{p} \label{b15} \end{equation}
  </div>
  <span class="equation_label">64</span>
</p>
</div>
<p>where</p>
<div class="equation" id="b15*">
<p>
  <div class="equation_content">
    \begin{equation}  \bar{N}_{p,\xi }^{\ast }:=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp}e^{\frac{-\nu ^{2}}{\xi }}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{\xi }}} \label{b15*} \end{equation}
  </div>
  <span class="equation_label">65</span>
</p>
</div>
<p>which is uniformly bounded for all \(\xi \in \left( 0,1\right] .\) </p>
<p>Hence, as \(\xi \rightarrow 0^{+},\) we obtain that \(W_{r,\xi }^{\ast }\rightarrow \) unit operator \(I\) in the \(L_{p}\) norm for \(p{\gt}1\). </p>
<p>\(\mathbf{iv)}\) For \(n=0\) and \(p=1\), let \(f\in \left( C\left( \mathbb {R} \right) \cap L_{1}(\mathbb {R})\right) \) <i class="itshape">and the rest as above in this section. Then </i>the inequality</p>
<div class="equation" id="b16">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert W_{r,\xi }^{\ast }\left( f;x\right) -f(x)\right\Vert _{1}\leq \bar{N}_{1,\xi }^{\ast }\omega _{r}(f,\xi )_{1} \label{b16} \end{equation}
  </div>
  <span class="equation_label">66</span>
</p>
</div>
<p>holds where \(\bar{N}_{1,\xi }^{\ast }\) is defined as in \(\left( \ref{b15*}\right) \). Furthermore, we get \(W_{r,\xi }^{\ast }\rightarrow I\) in the \(L_{1}\) norm as \(\xi \rightarrow 0^{+}.\) </p>

  </div>
</div> </p>
<p>For the Poisson-Cauchy operators, in <span class="cite">
	[
	<a href="#Anastass-Kester2" >5</a>
	]
</span>, the authors showed </p>
<p><div class="theorem_thmwrapper " id="bt10">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">9</span>
  </div>
  <div class="theorem_thmcontent">
  <p>\(\mathbf{i)}\) <i class="itshape">Let </i>\(f\in C^{n}(\mathbb {R}),\)<i class="itshape"> with </i>\(f^{(n)}\in L_{p}\left( \mathbb {R}\right) ,\)<i class="itshape"> </i>\(n\in \mathbb {N},\)<i class="itshape"> </i>\(p,q{\gt}1\)<span class="rm">:</span> \(\tfrac {1}{p}+\tfrac {1}{q}=1\), \(\beta {\gt}\tfrac {p(r+n)+1}{2\alpha },\alpha \in \mathbb {N} ,\) and the rest as above in this section. Then</p>
<div class="displaymath" id="b17">
  \begin{align} & \bigg\Vert Q_{r,\xi }^{\ast }\left( f;x\right) -f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{\left( k\right) }(x)}{k!}\delta _{k}q_{k,\xi }^{\ast }\bigg\Vert _{p}\leq \label{b17} \\ & \leq \tfrac {1}{((n-1)!)(q(n-1)+1)^{\frac{1}{q}}\left( rp+1\right) ^{\frac{1}{p}}}\left( S_{p,\xi }^{\ast }\right) ^{\frac{1}{p}}\xi ^{\frac{1}{p}}\omega _{r}(f^{(n)},\xi )_{p} \nonumber \end{align}
</div>
<p>where</p>
<div class="equation" id="b17*">
<p>
  <div class="equation_content">
    \begin{equation}  S_{p,\xi }^{\ast }:=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( \left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp+1}-1\right) \left\vert \nu \right\vert ^{np-1}\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }} \label{b17*} \end{equation}
  </div>
  <span class="equation_label">68</span>
</p>
</div>
<p>is uniformly bounded for all \(\xi \in \left( 0,1\right] .\) </p>
<p>Additionally, as \(\xi \rightarrow 0^{+},\) we obtain that \(R.H.S.\) of \((\ref{b17})\) goes to zero. </p>
<p>\(\mathbf{ii)}\) When \(p=1\), let <i class="itshape"> </i>\(f\in C^{n}(\mathbb {R})\), \(f^{(n)}\in L_{1}(\mathbb {R}),\) \(\beta {\gt}\tfrac {r+n+1}{2\alpha },\) and \(n\in \mathbb {N} -\{ 1\} .\) Then</p>
<div class="displaymath" id="b18">
  \begin{align} & \bigg\Vert Q_{r,\xi }^{\ast }\left( f;x\right) -f(x)-\sum \limits _{k=1}^{n}\tfrac {f^{\left( k\right) }(x)}{k!}\delta _{k}q_{k,\xi }^{\ast }\bigg\Vert _{1}\leq \label{b18} \\ & \leq \tfrac {1}{(n-1)!\left( r+1\right) }S_{1,\xi }^{\ast }\xi \omega _{r}(f^{(n)},\xi )_{1} \nonumber \end{align}
</div>
<p>holds where \(S_{1,\xi }^{\ast }\) is defined as in \(\left( \ref{b17*}\right) \). Hence, as \(\xi \rightarrow 0^{+},\) we obtain that \(R.H.S.\) of \((\ref{b18})\) goes to zero. </p>
<p>\(\mathbf{iii)}\)When \(n=0\), <i class="itshape">let </i>\(f\in \left( C\left( \mathbb {R} \right) \cap L_{p}(\mathbb {R})\right) ,\) <i class="itshape"> </i>\(p,q{\gt}1\)<i class="itshape"> such that </i>\(\tfrac {1}{p}+\tfrac {1}{q}=1,\) \(\beta {\gt}\tfrac {p\left( r+2\right) +1}{2\alpha },\)<i class="itshape"> and the rest as above in this section. Then</i></p>
<div class="equation" id="b19">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert Q_{r,\xi }^{\ast }\left( f;x\right) -f(x)\right\Vert _{p}\leq \left( \bar{S}_{p,\xi }^{\ast }\right) ^{1/p}\omega _{r}(f,\xi )_{p} \label{b19} \end{equation}
  </div>
  <span class="equation_label">70</span>
</p>
</div>
<p>where</p>
<div class="equation" id="b19*">
<p>
  <div class="equation_content">
    \begin{equation}  \bar{S}_{p,\xi }^{\ast }:=\tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp}\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }} \label{b19*} \end{equation}
  </div>
  <span class="equation_label">71</span>
</p>
</div>
<p>which is uniformly bounded for all \(\xi \in \left( 0,1\right] .\) </p>
<p>Hence, as \(\xi \rightarrow 0^{+},\) we obtain that \(Q_{r,\xi }^{\ast }\rightarrow \) unit operator \(I\) in the \(L_{p}\) norm for \(p{\gt}1\). </p>
<p>\(\mathbf{iv)}\) When \(n=0\) and \(p=1\), let \(f\in \left( C\left( \mathbb {R} \right) \cap L_{1}(\mathbb {R})\right) ,\) \(\beta {\gt}\tfrac {r+3}{2\alpha }\) and the rest as above in this section. The inequality</p>
<div class="equation" id="b20">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert Q_{r,\xi }^{\ast }\left( f;x\right) -f(x)\right\Vert _{1}\leq \bar{S}_{1,\xi }^{\ast }\omega _{r}(f,\xi )_{1} \label{b20} \end{equation}
  </div>
  <span class="equation_label">72</span>
</p>
</div>
<p>holds where \(\bar{S}_{1,\xi }^{\ast }\) is defined as in \(\left( \ref{b19*}\right) \). Furthermore, we get \(Q_{r,\xi }^{\ast }\rightarrow I\) in the \(L_{1}\) norm as \(\xi \rightarrow 0^{+}.\) </p>

  </div>
</div> </p>
<p>Next in <span class="cite">
	[
	<a href="#Anastass-Kester2" >5</a>
	]
</span>, they stated their results for the errors \(E_{0,P},\) \(E_{0,W},\) \(E_{n,P},\) and \(E_{n,W}\) as follows </p>
<p><div class="theorem_thmwrapper " id="bt13">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">10</span>
  </div>
  <div class="theorem_thmcontent">
  <p>\(\mathbf{i)}\) Let \(p,q{\gt}1\) such that \(\tfrac {1}{p}+\tfrac {1}{q}=1\), \(n\in \mathbb {N} \) such that \(np\neq 1\), \(f\in L_{p}(\mathbb {R} ),\) and the rest as above in this section. Then</p>
<div class="displaymath" id="b59">
  \begin{align}  \left\Vert E_{n,P}(f,x)\right\Vert _{p}& \leq \label{b59} \tfrac {\xi ^{\frac{1}{p}}\omega _{r}(f^{(n)},\xi )_{p}\left( \sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{\xi }}\right) ^{\frac{1}{q}}}{((n-1)!)(q(n-1)+1)^{\frac{1}{q}}\left( rp+1\right) ^{\frac{1}{p}}}\cdot \\ & \quad \cdot \left[ \tfrac {\left( \sum \limits _{\nu =-\infty }^{\infty }\left( \left( 1\! +\! \frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp+1}\! -\! 1\right) \left\vert \nu \right\vert ^{np-1}e^{\frac{-\left\vert \nu \right\vert }{\xi }}\right) }{1+2\xi e^{-\frac{1}{\xi }}}^{\frac{1}{p}}\right]\! +\! \left\Vert f(x)\right\Vert _{p}\left\vert m_{\xi ,P}\! -\! 1\right\vert \nonumber \end{align}
</div>
<p>holds. Additionally, as \(\xi \rightarrow 0^{+},\) we obtain that \(R.H.S.\) of \((\ref{b59})\) goes to zero. </p>
<p>\(\mathbf{ii)}\) When \(p=1,\) let <i class="itshape"> </i>\(f\in C^{n}(\mathbb {R}),f\in L_{1}(\mathbb {R} )\), \(f^{(n)}\in L_{1}(\mathbb {R}),\) and \(n\in \mathbb {N} -\{ 1\} .\) Then</p>
<div class="displaymath" id="b60">
  \begin{align}  \left\Vert E_{n,P}(f,x)\right\Vert _{1}&  \label{b60} \leq \tfrac {\xi \omega _{r}(f^{(n)},\xi )_{1}}{(n-1)!\left( r+1\right) }\cdot \\ & \quad \cdot \left[ \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( \left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{r+1}-1\right) \left\vert \nu \right\vert ^{n-1}e^{\frac{-\left\vert \nu \right\vert }{\xi }}}{1+2\xi e^{-\frac{1}{\xi }}}\right] +\left\Vert f(x)\right\Vert _{1}\left\vert m_{\xi ,P}-1\right\vert \nonumber \end{align}
</div>
<p>holds. Additionally, as \(\xi \rightarrow 0^{+},\) we obtain that \(R.H.S.\) of \((\ref{b60})\) goes to zero. </p>
<p>\(\mathbf{iii)}\) When \(n=0,\) <i class="itshape">let </i>\(p,q{\gt}1\)<i class="itshape"> such that </i>\(\tfrac {1}{p}+\tfrac {1}{q}=1,\) \(f\in L_{p}(\mathbb {R} )\),<i class="itshape"> and the rest as above in this section. Then</i></p>
<div class="displaymath" id="b61">
  \begin{align}  \left\Vert E_{0,P}(f,x)\right\Vert _{p} & \leq \omega _{r}(f,\xi )_{p}\left( \sum \limits _{\nu =-\infty }^{\infty }e^{\tfrac {-\left\vert \nu \right\vert }{\xi }}\right) ^{\frac{1}{q}} \label{b61} \cdot \\ & \quad \cdot \left[ \tfrac {\left( \sum \limits _{\nu =-\infty }^{\infty }\left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp}e^{\frac{-\left\vert \nu \right\vert }{\xi }}\right) }{1+2\xi e^{-\frac{1}{\xi }}}^{1/p}\right] +\left\Vert f(x)\right\Vert _{p}\left\vert m_{\xi ,P}-1\right\vert \nonumber \end{align}
</div>
<p>holds. Hence, as \(\xi \rightarrow 0^{+},\) we obtain that \(R.H.S.\) of \((\ref{b61})\) goes to zero. </p>
<p>\(\mathbf{iv)}\) When \(n=0\) and \(p=1,\) the inequality</p>
<div class="displaymath" id="b62">
  \begin{align}  \left\Vert E_{0,P}(f,x)\right\Vert _{1} & \leq \left( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( 1+\tfrac {\left\vert \nu \right\vert }{\xi }\right) ^{r}e^{-\frac{\left\vert \nu \right\vert }{\xi }}}{1+2\xi e^{-\frac{1}{\xi }}}\right) \omega _{r}(f,\xi )_{1} \label{b62} +\left\Vert f(x)\right\Vert _{1}\left\vert m_{\xi ,P}-1\right\vert \end{align}
</div>
<p>holds. Hence, as \(\xi \rightarrow 0^{+},\) we obtain that \(R.H.S.\) of \((\ref{b62})\) goes to zero. </p>

  </div>
</div> </p>
<p>Next in <span class="cite">
	[
	<a href="#Anastass-Kester2" >5</a>
	]
</span>, the authors gave quantitative results for \(E_{n,W}(f,x)\) </p>
<p><div class="theorem_thmwrapper " id="bt15">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">11</span>
  </div>
  <div class="theorem_thmcontent">
  <p>\(\mathbf{i)}\) Let \(p,q{\gt}1\) such that \(\tfrac {1}{p}+\tfrac {1}{q}=1\), \(n\in \mathbb {N} \) such that \(np\neq 1\), \(f\in L_{p}(\mathbb {R} ),\) and the rest as above in this section. Then</p>
<div class="displaymath" id="b63">
  \begin{align}  \left\Vert E_{n,W}(f,x)\right\Vert _{p}&  \label{b63} \leq \frac{\xi ^{\tfrac {1}{p}}\omega _{r}(f^{(n)},\xi )_{p}\left( \sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{\xi }}\right) ^{\frac{1}{q}}}{((n-1)!)(q(n-1)+1)^{\frac{1}{q}}\left( rp+1\right) ^{\frac{1}{p}}}\cdot \\ & \quad \cdot \left[ \tfrac {\left( \sum \limits _{\nu =-\infty }^{\infty }\left( \left( 1\! +\! \frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp+1}\! -\! 1\right) \left\vert \nu \right\vert ^{np-1}e^{\frac{-\nu ^{2}}{\xi }}\right) }{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}^{\frac{1}{p}}\right]\!  +\! \left\Vert f(x)\right\Vert _{p}\left\vert m_{\xi ,W}\! -\! 1\right\vert \nonumber \end{align}
</div>
<p>holds. Additionally, as \(\xi \rightarrow 0^{+},\) we obtain that \(R.H.S.\) of \((\ref{b63})\) goes to zero. </p>
<p>\(\mathbf{ii)}\) For \(p=1,\) let <i class="itshape"> </i>\(f\in C^{n}(\mathbb {R}),f\in L_{1}(\mathbb {R} )\), \(f^{(n)}\in L_{1}(\mathbb {R}),\) and \(n\in \mathbb {N} -\{ 1\} .\) Then</p>
<div class="displaymath" id="b64">
  \begin{align}  \left\Vert E_{n,W}(f,x)\right\Vert _{1} \label{b64} & \leq \tfrac {\xi \omega _{r}(f^{(n)},\xi )_{1}}{(n-1)!\left( r+1\right) }\cdot \\ & \quad \cdot \left[ \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( \left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{r+1}-1\right) \left\vert \nu \right\vert ^{n-1}e^{\frac{-\nu ^{2}}{\xi }}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\right] +\left\Vert f(x)\right\Vert _{1}\left\vert m_{\xi ,W}-1\right\vert \nonumber \end{align}
</div>
<p>holds. Additionally, as \(\xi \rightarrow 0^{+},\) we obtain that \(R.H.S.\) of \((\ref{b64})\) goes to zero. </p>
<p>\(\mathbf{iii)}\) For \(n=0,\) <i class="itshape">let </i>\(p,q{\gt}1\)<i class="itshape"> such that </i>\(\tfrac {1}{p}+\tfrac {1}{q}=1,\) \(f\in L_{p}(\mathbb {R} )\),<i class="itshape"> and the rest as above in this section. Then</i></p>
<div class="displaymath" id="b65">
  \begin{align}  \left\Vert E_{0,W}(f,x)\right\Vert _{p} & \leq \omega _{r}(f,\xi )_{p}\left( \sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{\xi }}\right) ^{\frac{1}{q}} \label{b65} \cdot \left[ \tfrac {\left( \sum \limits _{\nu =-\infty }^{\infty }\left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp}e^{\frac{-\nu ^{2}}{\xi }}\right) ^{\frac{1}{p}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\right] \\ & \quad +\left\Vert f(x)\right\Vert _{p}\left\vert m_{\xi ,W}-1\right\vert \nonumber \end{align}
</div>
<p>holds. Hence, as \(\xi \rightarrow 0^{+},\) we obtain that \(R.H.S.\) of \((\ref{b65})\) goes to zero. </p>
<p>\(\mathbf{iv)}\) For \(n=0\) and \(p=1,\) the inequality</p>
<div class="displaymath" id="b66">
  \begin{align}  \left\Vert E_{0,W}(f,x)\right\Vert _{1} & \leq \left( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{r}e^{\frac{-\nu ^{2}}{\xi }}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\right) \omega _{r}(f,\xi )_{1} \label{b66} +\left\Vert f(x)\right\Vert _{1}\left\vert m_{\xi ,W}-1\right\vert \end{align}
</div>
<p>holds. Hence, as \(\xi \rightarrow 0^{+},\) we obtain that \(R.H.S.\) of \((\ref{b66})\) goes to zero. </p>

  </div>
</div> </p>
<h1 id="a0000000004">3 Main Results</h1>
<p>We start with global smoothness preservation properties of the operators \(P_{r,\xi }^{\ast },\) \(W_{r,\xi }^{\ast },\) and \(\Theta _{r,\xi }^{\ast }.\) </p>
<p><div class="theorem_thmwrapper " id="mt1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">12</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(h{\gt}0\) and \(0{\lt}\xi \leq 1.\) </p>
<p>\(\mathbf{i)}\) Suppose \(f\in C\left( \mathbb {R} \right) ,\) and \(P_{r,\xi }^{\ast }\left( f;x\right) ,\) \(W_{r,\xi }^{\ast }\left( f;x\right) ,\) \(Q_{r,\xi }^{\ast }\left( f;x\right) \in \mathbb {R} \) for all \(x\in \mathbb {R} ,\) \(\omega _{m}(f,h){\lt}\infty .\) Then</p>
<div class="equation" id="m1">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(P_{r,\xi }^{\ast }f,h)\leq \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h), \label{m1} \end{equation}
  </div>
  <span class="equation_label">81</span>
</p>
</div>
<div class="equation" id="m2">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(W_{r,\xi }^{\ast }f,h)\leq \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h), \label{m2} \end{equation}
  </div>
  <span class="equation_label">82</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m3">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(Q_{r,\xi }^{\ast }f,h)\leq \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h). \label{m3} \end{equation}
  </div>
  <span class="equation_label">83</span>
</p>
</div>
<p>\(\mathbf{ii)}\) Suppose \(f\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R} \right) \right) ,\) \(p\geq 1.\) Then</p>
<div class="equation" id="m4">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(P_{r,\xi }^{\ast }f,h)_{p}\leq \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h)_{p}, \label{m4} \end{equation}
  </div>
  <span class="equation_label">84</span>
</p>
</div>
<div class="equation" id="m5">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(W_{r,\xi }^{\ast }f,h)_{p}\leq \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h)_{p}, \label{m5} \end{equation}
  </div>
  <span class="equation_label">85</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m6">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(Q_{r,\xi }^{\ast }f,h)_{p}\leq \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h)_{p}. \label{m6} \end{equation}
  </div>
  <span class="equation_label">86</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000005">
  <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 Theorem \(\ref{bt1}.\) <div class="proof_wrapper" id="a0000000006">
  <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>For \(r=1,\) we get \(\alpha _{0}=0\) and \(\alpha _{1}=1.\) Hence, we obtain</p>
<div class="equation" id="m7">
<p>
  <div class="equation_content">
    \begin{equation}  P_{1,\xi }^{\ast }\left( f;x\right) =P_{\xi }^{\ast }\left( f;x\right) =\tfrac {\sum \limits _{\nu =-\infty }^{\infty }f\left( x+\nu \right) e^{-\frac{\left\vert \nu \right\vert }{\xi }}}{\sum \limits _{\nu =-\infty }^{\infty }e^{-\frac{\left\vert \nu \right\vert }{\xi }}}, \label{m7} \end{equation}
  </div>
  <span class="equation_label">87</span>
</p>
</div>
<div class="equation" id="m8">
<p>
  <div class="equation_content">
    \begin{equation}  W_{1,\xi }^{\ast }\left( f;x\right) =W_{\xi }^{\ast }\left( f;x\right) =\tfrac {\sum \limits _{\nu =-\infty }^{\infty }f\left( x+\nu \right) e^{-\frac{\nu ^{2}}{\xi }}}{\sum \limits _{\nu =-\infty }^{\infty }e^{-\frac{\nu ^{2}}{\xi }}}, \label{m8} \end{equation}
  </div>
  <span class="equation_label">88</span>
</p>
</div>
<p>and for \(\beta {\gt}\tfrac {1}{\alpha },\) \(\alpha \in \mathbb {N} \)</p>
<div class="equation" id="m9">
<p>
  <div class="equation_content">
    \begin{equation}  Q_{1,\xi }^{\ast }\left( f;x\right) =Q_{\xi }^{\ast }\left( f;x\right) =\tfrac {\sum \limits _{\nu =-\infty }^{\infty }f\left( x+\nu \right) \left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}. \label{m9} \end{equation}
  </div>
  <span class="equation_label">89</span>
</p>
</div>
<p>Therefore, by <i class="itshape">Theorem</i> \(\ref{mt1},\) we have </p>
<p><div class="theorem_thmwrapper " id="mt2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">13</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(h{\gt}0\) and \(0{\lt}\xi \leq 1.\) </p>
<p>\(\mathbf{i)}\) Suppose \(f\in C\left( \mathbb {R} \right) ,\) and \(P_{\xi }^{\ast }\left( f;x\right) ,\) \(W_{\xi }^{\ast }\left( f;x\right) ,\) \(Q_{\xi }^{\ast }\left( f;x\right) \in \mathbb {R} \) for all \(x\in \mathbb {R} ,\) \(\omega _{m}(f,h){\lt}\infty .\) Then</p>
<div class="equation" id="m10">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(P_{\xi }^{\ast }f,h)\leq \omega _{m}(f,h), \label{m10} \end{equation}
  </div>
  <span class="equation_label">90</span>
</p>
</div>
<div class="equation" id="m11">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(W_{\xi }^{\ast }f,h)\leq \omega _{m}(f,h), \label{m11} \end{equation}
  </div>
  <span class="equation_label">91</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m12">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(Q_{\xi }^{\ast }f,h)\leq \omega _{m}(f,h). \label{m12} \end{equation}
  </div>
  <span class="equation_label">92</span>
</p>
</div>
<p>Inequalities <span class="rm">(<a href="#m10">90</a>), (<a href="#m11">91</a>),</span> and <span class="rm">(<a href="#m12">92</a>)</span> are sharp, that is attained by \(f\left( x\right) =g\left( x\right) =x^{m}.\) </p>
<p>\(\mathbf{ii)}\) Suppose \(f\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R} \right) \right) ,\) \(p\geq 1.\) Then</p>
<div class="equation" id="m13">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(P_{\xi }^{\ast }f,h)_{p}\leq \omega _{m}(f,h)_{p}, \label{m13} \end{equation}
  </div>
  <span class="equation_label">93</span>
</p>
</div>
<div class="equation" id="m14">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(W_{\xi }^{\ast }f,h)_{p}\leq \omega _{m}(f,h)_{p}, \label{m14} \end{equation}
  </div>
  <span class="equation_label">94</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m15">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(Q_{\xi }^{\ast }f,h)_{p}\leq \omega _{m}(f,h)_{p}. \label{m15} \end{equation}
  </div>
  <span class="equation_label">95</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000007">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> It suffices to show the attainability of the inequalities \(\left( \ref{m10}\right) ,\) \(\left( \ref{m11}\right) ,\) and \(\left( \ref{m12}\right)\). We notice that</p>
<div class="equation" id="m16">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(g,h)=\omega _{m}(x^{m},h)=m!h^{m}. \label{m16} \end{equation}
  </div>
  <span class="equation_label">96</span>
</p>
</div>
<p>On the other hand, we have</p>
<div class="displaymath" id="m17">
  \begin{eqnarray}  \Delta _{t}^{m}(P_{\xi }^{\ast }g)\left( x\right) & =& \textstyle \sum \limits _{j=0}^{m}(-1)^{m-j}{\tbinom {m}{j}}P_{\xi }^{\ast }g(x+jt) \label{m17} \\ & =& \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left[ \sum \limits _{j=0}^{m}(-1)^{m-j}{\binom {m}{j}}(\left( x+\nu \right) +jt)^{m}\right] e^{-\frac{\left\vert \nu \right\vert }{\xi }}}{\sum \limits _{\nu =-\infty }^{\infty }e^{-\frac{\left\vert \nu \right\vert }{\xi }}} \nonumber \\ & =& \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( \Delta _{t}^{m}(x+\nu )^{m}\right) e^{-\frac{\left\vert \nu \right\vert }{\xi }}}{\sum \limits _{\nu =-\infty }^{\infty }e^{-\frac{\left\vert \nu \right\vert }{\xi }}} \nonumber \\ & =& m!t^{m}. \nonumber \end{eqnarray}
</div>
<p>Thus, we get</p>
<div class="equation" id="m18">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(g,h)=\omega _{m}(P_{\xi }^{\ast }g,h). \label{m18} \end{equation}
  </div>
  <span class="equation_label">98</span>
</p>
</div>
<p>Similarly, we obtain</p>
<div class="equation" id="m19">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(g,h)=\omega _{m}(W_{\xi }^{\ast }g,h), \label{m19} \end{equation}
  </div>
  <span class="equation_label">99</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m20">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(g,h)=\omega _{m}(Q_{\xi }^{\ast }g,h). \label{m20} \end{equation}
  </div>
  <span class="equation_label">100</span>
</p>
</div>
<p> <div class="proof_wrapper" id="a0000000008">
  <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>Next, we present the following theorem for the non-unitary operators \(P_{r,\xi }\) and \(W_{r,\xi }\) </p>
<p><div class="theorem_thmwrapper " id="mt1n">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">14</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(h{\gt}0\) and \(0{\lt}\xi \leq 1.\) </p>
<p>\(\mathbf{i)}\) Suppose that \(f\in C\left( \mathbb {R} \right) ,\) and \(P_{r,\xi }^{\ast }\left( f;x\right) ,\) \(W_{r,\xi }^{\ast }\left( f;x\right) \in \mathbb {R} \) for all \(x\in \mathbb {R} ,\) \(\omega _{m}(f,h){\lt}\infty .\) Then</p>
<div class="equation" id="m20.1">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(P_{r,\xi }f,h)\leq \bigg( \tfrac {1+2e^{\frac{-1}{\xi }}\left( \xi +1\right) }{1+2\xi e^{\frac{-1}{\xi }}}\bigg) \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h), \label{m20.1} \end{equation}
  </div>
  <span class="equation_label">101</span>
</p>
</div>
<div class="equation" id="m20.2">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(W_{r,\xi }f,h)\leq \Bigg( 1+\tfrac {2e^{\tfrac {-1}{\xi }}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\Bigg) \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h). \label{m20.2} \end{equation}
  </div>
  <span class="equation_label">102</span>
</p>
</div>
<p>\(\mathbf{ii)}\) Suppose \(f\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R} \right) \right) ,\) \(p\geq 1.\) Then</p>
<div class="equation" id="m20.3">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(P_{r,\xi }f,h)_{p}\leq \bigg( \tfrac {1+2e^{\frac{-1}{\xi }}\left( \xi +1\right) }{1+2\xi e^{\frac{-1}{\xi }}}\bigg) \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h)_{p}, \label{m20.3} \end{equation}
  </div>
  <span class="equation_label">103</span>
</p>
</div>
<div class="equation" id="m20.4">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(W_{r,\xi }f,h)_{p}\leq \Bigg( 1+\tfrac {2e^{\frac{-1}{\xi }}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\Bigg) \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h)_{p}. \label{m20.4} \end{equation}
  </div>
  <span class="equation_label">104</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000009">
  <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 \(\left( \ref{b25*}\right) \), \(\left( \ref{b40.1}\right) ,\) \(\left( \ref{b40.3}\right) \) and Theorem \(\ref{mt1}\), we have</p>
<div class="equation" id="m20.5">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(P_{r,\xi }f,h)\leq \lambda _{1}\left( \xi \right) \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h), \label{m20.5} \end{equation}
  </div>
  <span class="equation_label">105</span>
</p>
</div>
<div class="equation" id="m20.5*">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(P_{r,\xi }f,h)_{p}\leq \lambda _{1}\left( \xi \right) \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h)_{p}, \label{m20.5*} \end{equation}
  </div>
  <span class="equation_label">106</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m20.6">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(W_{r,\xi }f,h)\leq \lambda _{2}\left( \xi \right) \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h), \label{m20.6} \end{equation}
  </div>
  <span class="equation_label">107</span>
</p>
</div>
<div class="equation" id="m20.6*">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(W_{r,\xi }f,h)_{p}\leq \lambda _{2}\left( \xi \right) \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f,h)_{p}. \label{m20.6*} \end{equation}
  </div>
  <span class="equation_label">108</span>
</p>
</div>
<p>Additionally, in <span class="cite">
	[
	<a href="#Anastass" >3</a>
	]
</span>, it was shown that</p>
<div class="equation" id="m20.7">
<p>
  <div class="equation_content">
    \begin{equation}  \lambda _{1}\left( \xi \right) \leq \tfrac {1+2e^{\frac{-1}{\xi }}\left( \xi +1\right) }{1+2\xi e^{\frac{-1}{\xi }}}, \label{m20.7} \end{equation}
  </div>
  <span class="equation_label">109</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m20.8">
<p>
  <div class="equation_content">
    \begin{equation}  \lambda _{2}\left( \xi \right) \leq 1+\tfrac {2e^{\frac{-1}{\xi }}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}. \label{m20.8} \end{equation}
  </div>
  <span class="equation_label">110</span>
</p>
</div>
<p>Thus, by <span class="rm">(<a href="#m20.5">105</a>)–(<a href="#m20.8">110</a>)</span>, we obtain the inequalities <span class="rm">(<a href="#m20.1">101</a>)–(<a href="#m20.4">104</a>)</span>. <div class="proof_wrapper" id="a0000000010">
  <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>Now, we give our results for the derivatives of the unitary operators <br />\( P_{r,\xi }^{\ast }\left( f;x\right) ,\) \(W_{r,\xi }^{\ast }\left( f;x\right) , \) and \(Q_{r,\xi }^{\ast }\left( f;x\right) \) mentioned above. First, we get </p>
<p><div class="theorem_thmwrapper " id="mt3">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">15</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\in C^{n-1}(\mathbb {R} )\), such that \(f^{(n)}\) exists, \(n,r\in \mathbb {N} ,\) \(0{\lt}\xi \leq 1.\) Additionally, suppose that for each \(x\in \mathbb {R} \) the function \(f^{(i)}(x+j\nu )\in L_{1}(\mathbb {R} ,\mu _{\xi })\) as a function of \(\nu ,\) for all \(i=0,1,\ldots ,n-1;\) \(j=1,\ldots ,r.\) Assume that there exist \(g_{i,j}\geq 0\), \(i=1,\ldots ,n;\) \(j=1,\ldots ,r,\) with \(g_{i,j}\in L_{1}(\mathbb {R} ,\mu _{\xi })\) such that for each \(x\in \mathbb {R} \) we have </p>
<div class="equation" id="m21">
<p>
  <div class="equation_content">
    \begin{equation}  |f^{(i)}(x+j\nu )|\leq g_{i,j}(\nu ), \label{m21} \end{equation}
  </div>
  <span class="equation_label">111</span>
</p>
</div>
<p>for \(\mu _{\xi }\)-almost all \(\nu \in \mathbb {R} \), all \(i=1,\ldots ,n;\) \(j=1,2,\ldots ,r.\) Then, \(f^{(i)}(x+j\nu )\) <i class="itshape">defines</i> a \(\mu _{\xi }\)-integrable function with respect to \(\nu \) for each \(x\in \mathbb {R} \), all \(i=1,\ldots ,n;\) \(j=1,\ldots ,r\). </p>
<p>\(\mathbf{i)}\) When</p>
<div class="displaymath" id="a0000000011">
  \[  \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}  \]
</div>
<p>we get</p>
<div class="equation" id="m22">
<p>
  <div class="equation_content">
    \begin{equation}  \left( P_{r,\xi }^{\ast }\left( f;x\right) \right) ^{(i)}=P_{r,\xi }^{\ast }\big( f^{(i)};x\big) , \label{m22} \end{equation}
  </div>
  <span class="equation_label">112</span>
</p>
</div>
<p>for all \(x\in \mathbb {R} \), and for all \(i=1,\ldots ,n\). </p>
<p>\(\mathbf{ii)}\) When</p>
<div class="displaymath" id="a0000000012">
  \[  \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{_{\xi }}}}  \]
</div>
<p>we have</p>
<div class="equation" id="m23">
<p>
  <div class="equation_content">
    \begin{equation}  \left( W_{r,\xi }^{\ast }\left( f;x\right) \right) ^{(i)}=W_{r,\xi }^{\ast }\big( f^{(i)};x\big) , \label{m23} \end{equation}
  </div>
  <span class="equation_label">113</span>
</p>
</div>
<p>for all \(x\in \mathbb {R} \), and for all \(i=1,\ldots ,n\). </p>
<p>\(\mathbf{iii)}\) Let \(\alpha \in \mathbb {N} \), and \(\beta {\gt}\tfrac {1}{\alpha }.\) When</p>
<div class="displaymath" id="a0000000013">
  \[  \mu _{\xi }(\nu )=\tfrac {\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}\text{,}  \]
</div>
<p>we obtain</p>
<div class="equation" id="m24">
<p>
  <div class="equation_content">
    \begin{equation}  \left( Q_{r,\xi }^{\ast }\left( f;x\right) \right) ^{(i)}=Q_{r,\xi }^{\ast }\big( f^{(i)};x\big) , \label{m24} \end{equation}
  </div>
  <span class="equation_label">114</span>
</p>
</div>
<p>for all \(x\in \mathbb {R} \), and for all \(i=1,\ldots ,n\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000014">
  <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 Theorem \(\ref{bt2}.\) <div class="proof_wrapper" id="a0000000015">
  <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>Next, we present our results for the derivatives of non-unitary operators \(P_{r,\xi }\left( f;x\right) \) and \(W_{r,\xi }\left( f;x\right) .\) </p>
<p><div class="proposition_thmwrapper " id="mp3n">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">16</span>
  </div>
  <div class="proposition_thmcontent">
  <p>Let the assumptions of the Theorem \(\ref{mt3}\) be valid. </p>
<p>\(\mathbf{i)}\) When</p>
<div class="displaymath" id="a0000000016">
  \[  \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{1+2\xi e^{\frac{-1}{\xi }}}  \]
</div>
<p>we get</p>
<div class="equation" id="m24.2">
<p>
  <div class="equation_content">
    \begin{equation}  \left( P_{r,\xi }\left( f;x\right) \right) ^{(i)}=P_{r,\xi }\left( f^{(i)};x\right) , \label{m24.2} \end{equation}
  </div>
  <span class="equation_label">115</span>
</p>
</div>
<p>for all \(x\in \mathbb {R} \), and for all \(i=1,\ldots ,n\). </p>
<p>\(\mathbf{ii)}\) When</p>
<div class="displaymath" id="a0000000017">
  \[  \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}  \]
</div>
<p>we have</p>
<div class="equation" id="m24.3">
<p>
  <div class="equation_content">
    \begin{equation}  \left( W_{r,\xi }\left( f;x\right) \right) ^{(i)}=W_{r,\xi }\big( f^{(i)};x\big) , \label{m24.3} \end{equation}
  </div>
  <span class="equation_label">116</span>
</p>
</div>
<p>for all \(x\in \mathbb {R} \), and for all \(i=1,\ldots ,n\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000018">
  <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 \(\left( \ref{b40.1}\right) \) and \(\left( \ref{b40.3}\right) \) we have</p>
<div class="equation" id="m24.4">
<p>
  <div class="equation_content">
    \begin{equation}  \left( P_{r,\xi }\left( f;x\right) \right) ^{\left( i\right) }=\lambda _{1}\left( \xi \right) \left( P_{r,\xi }^{\ast }\left( f;x\right) \right) ^{\left( i\right) } \label{m24.4} \end{equation}
  </div>
  <span class="equation_label">117</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m24.5">
<p>
  <div class="equation_content">
    \begin{equation}  \left( W_{r,\xi }\left( f;x\right) \right) ^{\left( i\right) }=\lambda _{2}\left( \xi \right) \left( W_{r,\xi }^{\ast }\left( f;x\right) \right) ^{\left( i\right) }. \label{m24.5} \end{equation}
  </div>
  <span class="equation_label">118</span>
</p>
</div>
<p>Thus by <i class="itshape">Theorem</i> \(\ref{mt3},\) we get</p>
<div class="displaymath" id="m24.6">
  \begin{eqnarray}  \left( P_{r,\xi }\left( f;x\right) \right) ^{\left( i\right) } & =& \lambda _{1}\left( \xi \right) P_{r,\xi }^{\ast }( f^{\left( i\right) };x) \label{m24.6} \\ & =& P_{r,\xi }( f^{\left( i\right) };x) \nonumber \end{eqnarray}
</div>
<p>and</p>
<div class="displaymath" id="m24.7">
  \begin{eqnarray}  \left( W_{r,\xi }\left( f;x\right) \right) ^{\left( i\right) } & =& \lambda _{2}\left( \xi \right) W_{r,\xi }^{\ast }( f^{\left( i\right) };x) \label{m24.7} \\ & =& W_{r,\xi }( f^{\left( i\right) };x) . \nonumber \end{eqnarray}
</div>
<p> <div class="proof_wrapper" id="a0000000019">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>We have the following application of the <i class="itshape">Theorem</i> \(\ref{mt3}\) for the case of \(r=1.\) </p>
<p><div class="proposition_thmwrapper " id="mp1">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">17</span>
  </div>
  <div class="proposition_thmcontent">
  <p>Let \(f\in C^{n-1}(\mathbb {R} )\), such that \(f^{(n)}\) exists, \(n\in \mathbb {N} ,\) \(0{\lt}\xi \leq 1.\) Additionally, suppose that for each \(x\in \mathbb {R} \) the function \(f^{(i)}(x+\nu )\in L_{1}(\mathbb {R} ,\mu _{\xi })\) as a function of \(\nu ,\) for all \(i=0,1,\ldots ,n-1.\) Assume that there exist \(g_{i}\geq 0\), \(i=1,\ldots ,n\) with \(g_{i}\in L_{1}(\mathbb {R} ,\mu _{\xi })\) such that for each \(x\in \mathbb {R} \) we have </p>
<div class="equation" id="m25">
<p>
  <div class="equation_content">
    \begin{equation}  |f^{(i)}(x+\nu )|\leq g_{i}(\nu ), \label{m25} \end{equation}
  </div>
  <span class="equation_label">121</span>
</p>
</div>
<p>for \(\mu _{\xi }\)-almost all \(\nu \in \mathbb {R} \), all \(i=1,\ldots ,n.\) Then, \(f^{(i)}(x+\nu )\) <i class="itshape">defines</i> a \(\mu _{\xi }\)-integrable function with respect to \(\nu \) for each \(x\in \mathbb {R} \), and for all \(i=1,\ldots ,n\). </p>
<p>\(\mathbf{i)}\) When</p>
<div class="displaymath" id="a0000000020">
  \[  \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}  \]
</div>
<p>we get</p>
<div class="equation" id="m26">
<p>
  <div class="equation_content">
    \begin{equation}  \left( P_{\xi }^{\ast }\left( f;x\right) \right) ^{(i)}=P_{\xi }^{\ast }( f^{(i)};x) , \label{m26} \end{equation}
  </div>
  <span class="equation_label">122</span>
</p>
</div>
<p>for all \(x\in \mathbb {R} \), and for all \(i=1,\ldots ,n\). </p>
<p>\(\mathbf{ii)}\) When</p>
<div class="displaymath" id="a0000000021">
  \[  \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{_{\xi }}}}  \]
</div>
<p>we have</p>
<div class="equation" id="m27">
<p>
  <div class="equation_content">
    \begin{equation}  \left( W_{\xi }^{\ast }\left( f;x\right) \right) ^{(i)}=W_{\xi }^{\ast }( f^{(i)};x) , \label{m27} \end{equation}
  </div>
  <span class="equation_label">123</span>
</p>
</div>
<p>for all \(x\in \mathbb {R} \), and for all \(i=1,\ldots ,n\). </p>
<p>\(\mathbf{iii)}\) Let \(\alpha \in \mathbb {N} \), and \(\beta {\gt}\tfrac {1}{\alpha }.\) When</p>
<div class="displaymath" id="a0000000022">
  \[  \mu _{\xi }(\nu )=\tfrac {\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}\text{,}  \]
</div>
<p>we obtain</p>
<div class="equation" id="m28">
<p>
  <div class="equation_content">
    \begin{equation}  \left( Q_{\xi }^{\ast }\left( f;x\right) \right) ^{(i)}=Q_{\xi }^{\ast }( f^{(i)};x) , \label{m28} \end{equation}
  </div>
  <span class="equation_label">124</span>
</p>
</div>
<p>for all \(x\in \mathbb {R} \), and for all \(i=1,\ldots ,n\). </p>

  </div>
</div> </p>
<p>We obtain </p>
<p><div class="theorem_thmwrapper " id="mt4">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">18</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(h{\gt}0\) and the assumptions of the Theorem \(\ref{mt3}\) be valid. </p>
<p>\(\mathbf{i)}\) Suppose that \(\omega _{m}(f^{(i)},h){\lt}\infty ,\) for all \(i=0,1,...,n.\) Then</p>
<div class="equation" id="m29">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(( P_{r,\xi }^{\ast }f) ^{(i)},h)\leq \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f^{(i)},h), \label{m29} \end{equation}
  </div>
  <span class="equation_label">125</span>
</p>
</div>
<div class="equation" id="m30">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(( W_{r,\xi }^{\ast }f) ^{(i)},h)\leq \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f^{(i)},h), \label{m30} \end{equation}
  </div>
  <span class="equation_label">126</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m31">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(( Q_{r,\xi }^{\ast }f) ^{(i)},h)\leq \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f^{(i)},h). \label{m31} \end{equation}
  </div>
  <span class="equation_label">127</span>
</p>
</div>
<p>\(\mathbf{ii)}\) Assume \(f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R} \right) \right) ,\) \(i=0,1,...,n,\) \(p\geq 1.\) Then</p>
<div class="equation" id="m32">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(( P_{r,\xi }^{\ast }f) ^{(i)},h)_{p}\leq \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f^{(i)},h)_{p}, \label{m32} \end{equation}
  </div>
  <span class="equation_label">128</span>
</p>
</div>
<div class="equation" id="m33">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(( W_{r,\xi }^{\ast }f) ^{(i)},h)_{p}\leq \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f^{(i)},h)_{p}, \label{m33} \end{equation}
  </div>
  <span class="equation_label">129</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m34">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(( Q_{r,\xi }^{\ast }f) ^{(i)},h)_{p}\leq \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f^{(i)},h)_{p}. \label{m34} \end{equation}
  </div>
  <span class="equation_label">130</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000023">
  <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 Theorems \(\ref{mt1}\), \(\ref{mt3}.\) <div class="proof_wrapper" id="a0000000024">
  <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>Next, we state our results for the non-unitary operators </p>
<p><div class="theorem_thmwrapper " id="mt4n">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">19</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(h{\gt}0\) and the assumptions of the Theorem \(\ref{mt3}\) be valid. </p>
<p>\(\mathbf{i)}\) Suppose that \(\omega _{m}(f^{(i)},h){\lt}\infty ,\) for all \(i=0,1,...,n.\) Then</p>
<div class="equation" id="m34.1">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(\left( P_{r,\xi }f\right) ^{(i)},h)\leq \Bigg( \tfrac {1+2e^{\frac{-1}{\xi }}\left( \xi +1\right) }{1+2\xi e^{\frac{-1}{\xi }}}\Bigg) \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f^{(i)},h), \label{m34.1} \end{equation}
  </div>
  <span class="equation_label">131</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m34.2">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(\left( W_{r,\xi }f\right) ^{(i)},h)\leq \Bigg( 1+\tfrac {2e^{\frac{-1}{\xi }}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\Bigg) \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f^{(i)},h). \label{m34.2} \end{equation}
  </div>
  <span class="equation_label">132</span>
</p>
</div>
<p>\(\mathbf{ii)}\) Assume \(f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R} \right) \right) ,\) \(i=0,1,...,n,\) \(p\geq 1.\) Then</p>
<div class="equation" id="m34.3">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(\left( P_{r,\xi }f\right) ^{(i)},h)_{p}\leq \Bigg( \tfrac {1+2e^{\frac{-1}{\xi }}\left( \xi +1\right) }{1+2\xi e^{\frac{-1}{\xi }}}\Bigg) \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f^{(i)},h)_{p}, \label{m34.3} \end{equation}
  </div>
  <span class="equation_label">133</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m34.4">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(\left( W_{r,\xi }f\right) ^{(i)},h)_{p}\leq \Bigg( 1+\tfrac {2e^{\frac{-1}{\xi }}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\Bigg) \bigg( \textstyle \sum \limits _{j=0}^{r}|\alpha _{j}|\bigg) \omega _{m}(f^{(i)},h)_{p}. \label{m34.4} \end{equation}
  </div>
  <span class="equation_label">134</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000025">
  <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 \(\left( \ref{b40.1}\right) ,\) \(\left( \ref{b40.3}\right) ,\) \(\left( \ref{m20.7}\right) ,\) \(\left( \ref{m20.8}\right) ,\) and Theorem \(\ref{mt4}.\) <div class="proof_wrapper" id="a0000000026">
  <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>For the case of \(r=1\) we have </p>
<p><div class="proposition_thmwrapper " id="mp2">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">20</span>
  </div>
  <div class="proposition_thmcontent">
  <p>Let \(h{\gt}0\) and the assumptions of the Proposition \(\ref{mp1}\) be valid. </p>
<p>\(\mathbf{i)}\) Assume that \(\omega _{m}(f^{(i)},h){\lt}\infty ,\) for all \(i=0,1,...,n.\) Then</p>
<div class="equation" id="m35">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(( P_{\xi }^{\ast }f) ^{(i)},h)\leq \omega _{m}(f^{(i)},h), \label{m35} \end{equation}
  </div>
  <span class="equation_label">135</span>
</p>
</div>
<div class="equation" id="m36">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(( W_{\xi }^{\ast }f) ^{(i)},h)\leq \omega _{m}(f^{(i)},h), \label{m36} \end{equation}
  </div>
  <span class="equation_label">136</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m37">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(( Q_{\xi }^{\ast }f) ^{(i)},h)\leq \omega _{m}(f^{(i)},h). \label{m37} \end{equation}
  </div>
  <span class="equation_label">137</span>
</p>
</div>
<p>\(\mathbf{ii)}\) Suppose that \(f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R} \right) \right) ,\) \(i=0,1,...,n,\) \(p\geq 1.\) Then</p>
<div class="equation" id="m38">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(( P_{\xi }^{\ast }f) ^{(i)},h)_{p}\leq \omega _{m}(f^{(i)},h)_{p}, \label{m38} \end{equation}
  </div>
  <span class="equation_label">138</span>
</p>
</div>
<div class="equation" id="m39">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(( W_{\xi }^{\ast }f) ^{(i)},h)_{p}\leq \omega _{m}(f^{(i)},h)_{p}, \label{m39} \end{equation}
  </div>
  <span class="equation_label">139</span>
</p>
</div>
<p>and</p>
<div class="equation" id="m40">
<p>
  <div class="equation_content">
    \begin{equation}  \omega _{m}(( Q_{\xi }^{\ast }f) ^{(i)},h)_{p}\leq \omega _{m}(f^{(i)},h)_{p}. \label{m40} \end{equation}
  </div>
  <span class="equation_label">140</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000027">
  <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 Theorem \(\ref{mt2}\) and Proposition \(\ref{mp1}.\) <div class="proof_wrapper" id="a0000000028">
  <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>Now, we demonstrate our simultaneous results for the operators \(P_{r,\xi }^{\ast },\) \(W_{r,\xi }^{\ast },\) and \(Q_{r,\xi }^{\ast }.\) We start with </p>
<p><div class="theorem_thmwrapper " id="mt5">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">21</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\in C^{n+\rho }(\mathbb {R}),\) \(n\in \mathbb {N} ,\) \(\rho \in \mathbb {Z}^{+}\) and \(f^{(n+i)}\in C_{u}(\mathbb {R} ),\) \(i=0,1,\ldots ,\rho ,\) and \(0{\lt}\xi \leq 1\). <i class="itshape">We consider the assumptions of Theorem </i>\(\ref{mt3}\)<i class="itshape"> valid for </i>\(n=\rho \)<i class="itshape"> there.</i> </p>
<div class="displaymath" id="m41">
  \begin{align} & \bigg\Vert \left( P_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)-\textstyle \sum \limits _{k=1}^{n}\tfrac {f^{(i+k)}(x)}{k!}\delta _{k}c_{k,\xi }^{\ast }\bigg\Vert _{\infty ,x}\leq \label{m41} \\ & \leq \tfrac {\omega _{r}\left( f^{(n+i)},\xi \right) }{n!}\Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left\vert \nu \right\vert ^{n}\left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{r}e^{-\frac{\nu ^{2}}{\xi }}}{\sum \limits _{\nu =-\infty }^{\infty }e^{-\frac{\nu ^{2}}{\xi }}}\Bigg) , \nonumber \end{align}
</div>
<div class="displaymath" id="m42">
  \begin{align} & \bigg\Vert \left( W_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)-\textstyle \sum \limits _{k=1}^{n}\tfrac {f^{(i+k)}(x)}{k!}\delta _{k}p_{k,\xi }^{\ast }\bigg\Vert _{\infty ,x}\leq \label{m42} \\ & \leq \tfrac {\omega _{r}\left( f^{(n+i)},\xi \right) }{n!}\Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left\vert \nu \right\vert ^{n}\left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{r}e^{-\frac{\nu ^{2}}{\xi }}}{\sum \limits _{\nu =-\infty }^{\infty }e^{-\frac{\nu ^{2}}{\xi }}}\Bigg) , \nonumber \end{align}
</div>
<p>and</p>
<div class="displaymath" id="m43">
  \begin{align} & \bigg\Vert \left( Q_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)-\textstyle \sum \limits _{k=1}^{n}\tfrac {f^{(i+k)}(x)}{k!}\delta _{k}q_{k,\xi }^{\ast }\bigg\Vert _{\infty ,x}\leq \label{m43} \\ & \leq \tfrac {\omega _{r}\left( f^{(n+i)},\xi \right) }{n!}\left( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left\vert \nu \right\vert ^{n}\left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{r}\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}\right) . \nonumber \end{align}
</div>
<p>where \(\beta {\gt}\tfrac {n+r+1}{2\alpha },\) \(\alpha \in \mathbb {N} .\) </p>

  </div>
</div> </p>
<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> By Theorems \(\ref{bt3},\) \(\ref{bt5}.\) <div class="proof_wrapper" id="a0000000030">
  <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>Next we have </p>
<p><div class="theorem_thmwrapper " id="mt6">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">22</span>
  </div>
  <div class="theorem_thmcontent">
  <p><i class="itshape">Let </i>\(f\in C^{n+\rho }(\mathbb {R}),\) <i class="itshape">with</i> \(f^{(n+i)}\in L_{p}\left( \mathbb {R}\right) ,\) \(n\in \mathbb {N},\) \(i=0,1,\ldots ,\) \(\rho \in \mathbb {Z}^{+}.\) Let \(p,q{\gt}1:\tfrac {1}{p}+\tfrac {1}{q}=1. \) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\rm \ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there. Then</i> </p>
<p>\(\mathbf{i)}\)</p>
<div class="displaymath" id="m44">
  \begin{align} & \bigg\Vert \left( P_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)-\textstyle \sum \limits _{k=1}^{n}\tfrac {f^{(i+k)}(x)}{k!}\delta _{k}c_{k,\xi }^{\ast }\bigg\Vert _{p,x}\leq \label{m44} \\ & \leq \tfrac {1}{\left( \left( n-1\right) !\right) \left( q\left( n-1\right) +1\right) ^{\tfrac {1}{q}}\left( rp+1\right) ^{\tfrac {1}{p}}}\left( M_{p,\xi }^{\ast }\right) ^{\tfrac {1}{p}}\xi ^{\tfrac {1}{p}}\omega _{r}\left( f^{(n+i)},\xi \right) _{p}, \nonumber \end{align}
</div>
<p>\(\mathbf{ii)}\)</p>
<div class="displaymath" id="m45">
  \begin{align} & \bigg\Vert \left( W_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)-\textstyle \sum \limits _{k=1}^{n}\tfrac {f^{(i+k)}(x)}{k!}\delta _{k}p_{k,\xi }^{\ast }\bigg\Vert _{p,x}\leq \label{m45} \\ & \leq \tfrac {1}{\left( \left( n-1\right) !\right) \left( q\left( n-1\right) +1\right) ^{\tfrac {1}{q}}\left( rp+1\right) ^{\tfrac {1}{p}}}\left( N_{p,\xi }^{\ast }\right) ^{\tfrac {1}{p}}\xi ^{\tfrac {1}{p}}\omega _{r}\left( f^{(n+i)},\xi \right) _{p}, \nonumber \end{align}
</div>
<p>\(\mathbf{iii)}\) for \(\beta {\gt}\tfrac {p\left( n+r\right) +1}{2\alpha },\) \(\alpha \in \mathbb {N} \)</p>
<div class="displaymath" id="m46">
  \begin{align} & \bigg\Vert \left( Q_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)-\textstyle \sum \limits _{k=1}^{n}\tfrac {f^{(i+k)}(x)}{k!}\delta _{k}q_{k,\xi }^{\ast }\bigg\Vert _{p,x}\leq \label{m46} \\ & \leq \tfrac {1}{\left( \left( n-1\right) !\right) \left( q\left( n-1\right) +1\right) ^{\tfrac {1}{q}}\left( rp+1\right) ^{\tfrac {1}{p}}}\left( S_{p,\xi }^{\ast }\right) ^{\tfrac {1}{p}}\xi ^{\tfrac {1}{p}}\omega _{r}\left( f^{(n+i)},\xi \right) _{p}. \nonumber \end{align}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000031">
  <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 Theorems <a href="#bt6">7</a>-<a href="#bt10">9</a>. <div class="proof_wrapper" id="a0000000032">
  <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>Now, we give our results for the special case of \(n=0.\) </p>
<p><div class="proposition_thmwrapper " id="mp3">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">23</span>
  </div>
  <div class="proposition_thmcontent">
  <p><i class="itshape">Let </i> \(f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R}\right) \right) ,\) \(i=0,1,\ldots ,\rho \in \mathbb {Z}^{+};\) \(p,q{\gt}1\) such that \(\tfrac {1}{p}+\tfrac {1}{q}=1.\) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there. Then for all </i>\(i=0,1,\ldots ,\rho ,\) we have </p>
<p>\(\mathbf{i)}\)</p>
<div class="equation" id="m47">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert \left( P_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)\right\Vert _{p,x}\leq \left( \bar{M}_{p,\xi }^{\ast }\right) ^{\tfrac {1}{p}}\omega _{r}( f^{(i)},\xi ) _{p}, \label{m47} \end{equation}
  </div>
  <span class="equation_label">147</span>
</p>
</div>
<p>\(\mathbf{ii)}\)</p>
<div class="equation" id="m48">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert \left( W_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)\right\Vert _{p,x}\leq \left( \bar{N}_{p,\xi }^{\ast }\right) ^{\tfrac {1}{p}}\omega _{r}( f^{(i)},\xi ) _{p}, \label{m48} \end{equation}
  </div>
  <span class="equation_label">148</span>
</p>
</div>
<p>\(\mathbf{iii)}\) for \(\beta {\gt}\tfrac {p\left( r+2\right) +1}{2\alpha },\) \(\alpha \in \mathbb {N} \)</p>
<div class="equation" id="m49">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert \left( Q_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)\right\Vert _{p,x}\leq \left( \bar{S}_{p,\xi }^{\ast }\right) ^{\tfrac {1}{p}}\omega _{r}( f^{(i)},\xi ) _{p}. \label{m49} \end{equation}
  </div>
  <span class="equation_label">149</span>
</p>
</div>

  </div>
</div> </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 Theorems <a href="#bt6">7</a>-<a href="#bt10">9</a>. <div class="proof_wrapper" id="a0000000034">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>For the special case of \(p=1,\) we obtain </p>
<p><div class="theorem_thmwrapper " id="mt7">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">24</span>
  </div>
  <div class="theorem_thmcontent">
  <p><i class="itshape">Let </i>\(f\in C^{n+\rho }(\mathbb {R}),\) <i class="itshape">with</i> \(f^{(n+i)}\in L_{1}\left( \mathbb {R}\right) ,\) \(n\in \mathbb {N-}\left\{  1\right\}  ,\) \(i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.\) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there. Then for all </i> \(i=0,1,\ldots ,\rho \), we have</p>
<p>\(\mathbf{i)}\)</p>
<div class="displaymath" id="m50">
  \begin{align} & \bigg\Vert \left( P_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)-\textstyle \sum \limits _{k=1}^{n}\tfrac {f^{(i+k)}(x)}{k!}\delta _{k}c_{k,\xi }^{\ast }\bigg\Vert _{1,x}\leq \label{m50} \\ & \leq \tfrac {1}{\left( n-1\right) !\left( r+1\right) }M_{1,\xi }^{\ast }\xi \omega _{r}( f^{(n+i)},\xi ) _{1}, \nonumber \end{align}
</div>
<p>\(\mathbf{ii)}\)</p>
<div class="displaymath" id="m51">
  \begin{align} & \bigg\Vert \left( W_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)-\textstyle \sum \limits _{k=1}^{n}\tfrac {f^{(i+k)}(x)}{k!}\delta _{k}p_{k,\xi }^{\ast }\bigg\Vert _{1,x}\leq \label{m51} \\ & \leq \tfrac {1}{\left( n-1\right) !\left( r+1\right) }N_{1,\xi }^{\ast }\xi \omega _{r}( f^{(n+i)},\xi ) _{1}, \nonumber \end{align}
</div>
<p>\(\mathbf{iii)}\) for \(\beta {\gt}\tfrac {n+r+1}{2\alpha },\) \(\alpha \in \mathbb {N} \)</p>
<div class="displaymath" id="m52">
  \begin{align} & \bigg\Vert \left( Q_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)-\textstyle \sum \limits _{k=1}^{n}\tfrac {f^{(i+k)}(x)}{k!}\delta _{k}q_{k,\xi }^{\ast }\bigg\Vert _{1,x}\leq \label{m52} \\ & \leq \tfrac {1}{\left( n-1\right) !\left( r+1\right) }S_{1,\xi }^{\ast }\xi \omega _{r}( f^{(n+i)},\xi ) _{1}. \nonumber \end{align}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000035">
  <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 Theorems <span class="rm"><a href="#bt6">7</a>–<a href="#bt10">9</a></span>. <div class="proof_wrapper" id="a0000000036">
  <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>For \(p=1\) and \(n=0,\) we give </p>
<p><div class="proposition_thmwrapper " id="mp4">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">25</span>
  </div>
  <div class="proposition_thmcontent">
  <p><i class="itshape">Let </i> \(f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{1}\left( \mathbb {R}\right) \right) ,\) \(i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.\) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\rm \ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there. Then for all </i>\(i=0,1,\ldots ,\rho ,\) we have </p>
<p>\(\mathbf{i)}\)</p>
<div class="equation" id="m53">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert \left( P_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)\right\Vert _{1,x}\leq \bar{M}_{1,\xi }^{\ast }\omega _{r}( f^{(i)},\xi ) _{1}, \label{m53} \end{equation}
  </div>
  <span class="equation_label">153</span>
</p>
</div>
<p>\(\mathbf{ii)}\)</p>
<div class="equation" id="m54">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert \left( W_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)\right\Vert _{1,x}\leq \bar{N}_{1,\xi }^{\ast }\omega _{r}( f^{(i)},\xi ) _{1}, \label{m54} \end{equation}
  </div>
  <span class="equation_label">154</span>
</p>
</div>
<p>\(\mathbf{iii)}\) for \(\beta {\gt}\tfrac {r+3}{2\alpha },\) \(\alpha \in \mathbb {N} \)</p>
<div class="equation" id="m55">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert \left( Q_{r,\xi }^{\ast }(f;x)\right) ^{(i)}-f^{(i)}(x)\right\Vert _{1,x}\leq \bar{S}_{1,\xi }^{\ast }\omega _{r}( f^{(i)},\xi ) _{1}. \label{m55} \end{equation}
  </div>
  <span class="equation_label">155</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000037">
  <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 Theorems <span class="rm"><a href="#bt6">7</a>–<a href="#bt10">9</a></span>. <div class="proof_wrapper" id="a0000000038">
  <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>Next, we state our simultaneous approximation results for the errors \(E_{0,P},\) \(E_{0,W},\) \(E_{n,P},\) and \(E_{n,W}.\) We obtain </p>
<p><div class="corollary_thmwrapper " id="mc1">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">26</span>
  </div>
  <div class="corollary_thmcontent">
  <p>Let \(f^{(i)}\in C_{u}(\mathbb {R} ),\) \(i=0,1,\ldots ,\rho ,\) \(\rho \in \mathbb {Z} ^{+},\) and \(0{\lt}\xi \leq 1.\) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there.</i> Then <i class="itshape">for all </i>\(i=0,1,\ldots ,\rho ,\) we have </p>
<p>\(\mathbf{i)}\)</p>
<div class="equation" id="m56">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert \left( E_{0,P}(f,x)\right) ^{\left( i\right) }\right\vert \leq \Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\omega _{r}(f^{\left( i\right) },\left\vert \nu \right\vert )e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{1+2\xi e^{-\frac{1}{\xi }}}\Bigg) +\left\vert f^{\left( i\right) }(x)\right\vert \left\vert m_{\xi ,P}-1\right\vert , \label{m56} \end{equation}
  </div>
  <span class="equation_label">156</span>
</p>
</div>
<p>\(\mathbf{ii)}\)</p>
<div class="equation" id="m57">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert \left( E_{0,W}(f,x)\right) ^{\left( i\right) }\right\vert \leq \Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\omega _{r}(f^{\left( i\right) },\left\vert \nu \right\vert )e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\Bigg) +\left\vert f^{\left( i\right) }(x)\right\vert \left\vert m_{\xi ,W}-1\right\vert . \label{m57} \end{equation}
  </div>
  <span class="equation_label">157</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000039">
  <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 \(\left( \ref{b55}\right)\), \(\left( \ref{b56}\right)\), also by \(\left( E_{0,P}(f,x)\right) ^{\left( i\right) }\! \! =\! \! E_{0,P}(f^{\left( i\right) },x)\) and \(\left( E_{0,W}(f,x)\right) ^{\left( i\right) } \) \(=E_{0,W}(f^{\left( i\right) },x).\) <div class="proof_wrapper" id="a0000000040">
  <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="theorem_thmwrapper " id="mt8">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">27</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\in C^{n+\rho }(\mathbb {R}),\) \(n\in \mathbb {N} ,\) \(\rho \in \mathbb {Z}^{+}\) and \(f^{(n+i)}\in C_{u}(\mathbb {R} ),\) \(i=0,1,\ldots ,\rho ,\) \(0{\lt}\xi \leq 1\), and \(\left\Vert f^{\left( i\right) }\right\Vert _{\infty ,\mathbb {R} }{\lt}\infty .\) <i class="itshape">We consider the assumptions of Theorem </i>\(\ref{mt3}\)<i class="itshape"> valid for </i>\(n=\rho .\) Then <i class="itshape">for all </i>\(i=0,1,\ldots ,\rho ,\) we have </p>
<p>\(\mathbf{i)}\)</p>
<div class="displaymath" id="m58">
  \begin{align} & \left\Vert \left( E_{n,P}(f,x)\right) ^{\left( i\right) }\right\Vert _{\infty ,x}\leq \label{m58} \\ & \leq \tfrac {\omega _{r}(f^{(n+i)},\xi )}{n!}\Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }|\nu |^{n}\left( 1+\frac{|\nu |}{\xi }\right) ^{r}e^{\tfrac {-\left\vert \nu \right\vert }{_{\xi }}}}{1+2\xi e^{-\frac{1}{\xi }}}\Bigg) +\left\Vert f^{\left( i\right) }\right\Vert _{\infty ,\mathbb {R} }\left\vert m_{\xi ,P}-1\right\vert , \nonumber \end{align}
</div>
<p>and </p>
<p>\(\mathbf{ii)}\)</p>
<div class="displaymath" id="m59">
  \begin{align} & \left\Vert \left( E_{n,W}(f,x)\right) ^{\left( i\right) }\right\Vert _{\infty ,x}\leq \label{m59} \\ & \leq \tfrac {\omega _{r}(f^{(n+i)},\xi )}{n!}\Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }|\nu |^{n}\left( 1+\frac{|\nu |}{\xi }\right) ^{r}e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\Bigg) +\left\Vert f^{\left( i\right) }\right\Vert _{\infty ,\mathbb {R} }\left\vert m_{\xi ,W}-1\right\vert . \nonumber \end{align}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000041">
  <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 \(\left( \ref{b57}\right)\), \(\left( \ref{b58}\right)\), also by \(\left( E_{n,P}(f,x)\right) ^{\left( i\right) }\! \! =\! \! E_{n,P}(f^{\left( i\right) },x)\) and \(\left( E_{n,W}(f,x)\right) ^{\left(\! \!  i\right) }\) \(=E_{n,W}(f^{\left( i\right) },x).\) <div class="proof_wrapper" id="a0000000042">
  <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="theorem_thmwrapper " id="mt9">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">28</span>
  </div>
  <div class="theorem_thmcontent">
  <p>\(\mathbf{i)}\) <i class="itshape">Let </i>\(f\in C^{n+\rho }(\mathbb {R}), \) <i class="itshape">with</i> \(f^{(n+i)}\in L_{p}\left( \mathbb {R}\right) ,\) \(n\in \mathbb {N},\) \(i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.\) Let \(p,q{\gt}1:\tfrac {1}{p}+\tfrac {1}{q}=1,\) \(np\neq 1.\) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\rm \ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there</i>. Then <i class="itshape">for all </i>\(i=0,1,\ldots ,\rho ,\)</p>
<div class="displaymath" id="m60">
  \begin{align} & \big\Vert \left( E_{n,P}(f,x)\right) ^{\left( i\right) }\big\Vert _{p} \label{m60} \\ & \leq \frac{\xi ^{\frac{1}{p}}\omega _{r}(f^{(n+i)},\xi )_{p}\left( \sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{\xi }}\right) ^{\frac{1}{q}}}{((n-1)!)(q(n-1)+1)^{\frac{1}{q}}\left( rp+1\right) ^{\frac{1}{p}}}\cdot \! \!  \left[ \tfrac {\left( \sum \limits _{\nu =-\infty }^{\infty }\left( \left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp+1}-1\right) \left\vert \nu \right\vert ^{np-1}e^{\frac{-\left\vert \nu \right\vert }{\xi }}\right) }{1+2\xi e^{-\frac{1}{\xi }}}^{\frac{1}{p}}\right]\nonumber \\ & \quad +\left\Vert f^{(i)}(x)\right\Vert _{p}\left\vert m_{\xi ,P}-1\right\vert \nonumber \end{align}
</div>
<p>holds. </p>
<p>\(\mathbf{ii)}\) <i class="itshape">Let </i>\(f\in C^{n+\rho }(\mathbb {R}),\) <i class="itshape">with</i> \(f^{(n+i)}\in L_{1}\left( \mathbb {R}\right) ,\) \(n\in \mathbb {N-}\left\{  1\right\}  ,\) \(i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.\) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\rm \ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there. Then for all </i> \(i=0,1,\ldots ,\rho \),</p>
<div class="displaymath" id="m61">
  \begin{align}  \big\Vert \left( E_{n,P}(f,x)\right) ^{\left( i\right) }\big\Vert _{1} \label{m61} & \leq \tfrac {\xi \omega _{r}(f^{(n+i)},\xi )_{1}}{(n-1)!\left( r+1\right) } \cdot \left[ \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( \left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{r+1}-1\right) \left\vert \nu \right\vert ^{n-1}e^{\frac{-\left\vert \nu \right\vert }{\xi }}}{1+2\xi e^{-\frac{1}{\xi }}}\right] \\ & \quad +\left\Vert f^{(i)}(x)\right\Vert _{1}\left\vert m_{\xi ,P}-1\right\vert \nonumber \end{align}
</div>
<p>holds. </p>
<p>\(\mathbf{iii)}\) <i class="itshape">Let </i> \(f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R}\right) \right) ,\) \(i=0,1,\ldots ,\rho \in \mathbb {Z}^{+};\) \(p,q{\gt}1\) such that \(\tfrac {1}{p}+\tfrac {1}{q}=1.\) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\rm \ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there. Then for all </i>\(i=0,1,\ldots ,\rho ,\)</p>
<div class="displaymath" id="m62">
  \begin{align}  \big\Vert \left( E_{0,P}(f,x)\right) ^{\left( i\right) }\big\Vert _{p} & \leq \omega _{r}(f^{\left( i\right) },\xi )_{p}\left( \textstyle \sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{\xi }}\right) ^{\frac{1}{q}} \label{m62} \cdot \left[ \tfrac {\left( \sum \limits _{\nu =-\infty }^{\infty }\left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp}e^{\frac{-\left\vert \nu \right\vert }{\xi }}\right) }{1+2\xi e^{-\frac{1}{\xi }}}^{\frac{1}{p}}\right] \\ & \quad +\big\Vert f^{(i)}(x)\big\Vert _{p}\left\vert m_{\xi ,P}-1\right\vert \nonumber \end{align}
</div>
<p>holds. </p>
<p>\(\mathbf{iv)}\) <i class="itshape">Let </i> \(f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{1}\left( \mathbb {R}\right) \right) ,\) \(i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.\) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\rm \ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there. Then for all </i>\(i=0,1,\ldots ,\rho ,\)</p>
<div class="displaymath" id="m63">
  \begin{align}  \big\Vert \left( E_{0,P}(f,x)\right) ^{\left( i\right) }\big\Vert _{1} & \leq \Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{r}e^{\frac{-\left\vert \nu \right\vert }{\xi }}}{1+2\xi e^{-\frac{1}{\xi }}}\Bigg) \omega _{r}(f^{\left( i\right) },\xi )_{1} \label{m63}+\big\Vert f^{(i)}(x)\big\Vert _{1}\left\vert m_{\xi ,P}-1\right\vert \end{align}
</div>
<p>holds. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000043">
  <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 Theorem \(\ref{bt13}\) and by \(\left( E_{n,P}(f,x)\right) ^{\left( i\right) }=E_{n,P}(f^{\left( i\right) },x)\) for \(n\in \mathbb {Z} ^{+}.\) <div class="proof_wrapper" id="a0000000044">
  <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="theorem_thmwrapper " id="mt10">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">29</span>
  </div>
  <div class="theorem_thmcontent">
  <p>\(\mathbf{i)}\) <i class="itshape">Let </i>\(f\in C^{n+\rho }(\mathbb {R}),\) <i class="itshape">with</i> \(f^{(n+i)}\in L_{p}\left( \mathbb {R}\right) ,\) \(n\in \mathbb {N},\) \(i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.\) Let \(p,q{\gt}1:\tfrac {1}{p}+\tfrac {1}{q}=1,\) \(np\neq 1.\) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\rm \ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there.</i> Then <i class="itshape">for all </i>\(i=0,1,\ldots ,\rho ,\)</p>
<div class="displaymath" id="m64">
  \begin{align} & \big\Vert \left( E_{n,W}(f,x)\right) ^{\left( i\right) }\big\Vert _{p}\leq \label{m64}\\ & \leq \frac{\xi ^{\frac{1}{p}}\omega _{r}(f^{(n+i)},\xi )_{p}\left( \sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{\xi }}\right) ^{\frac{1}{q}}}{((n-1)!)(q(n-1)+1)^{\frac{1}{q}}\left( rp+1\right) ^{\frac{1}{p}}}\cdot \! \!  \left[ \tfrac {\left( \sum \limits _{\nu =-\infty }^{\infty }\left( \left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{rp+1}-1\right) \left\vert \nu \right\vert ^{np-1}e^{\frac{-\nu ^{2}}{\xi }}\right) }{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}^{\frac{1}{p}}\right] \nonumber \\ & \quad +\big\Vert f^{(i)}(x)\big\Vert _{p}\left\vert m_{\xi ,W}-1\right\vert \nonumber \end{align}
</div>
<p>holds. </p>
<p>\(\mathbf{ii)}\) <i class="itshape">Let </i>\(f\in C^{n+\rho }(\mathbb {R}),\) <i class="itshape">with</i> \(f^{(n+i)}\in L_{1}\left( \mathbb {R}\right) ,\) \(n\in \mathbb {N-}\left\{  1\right\}  ,\) \(i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.\) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\rm \ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there. Then for all </i> \(i=0,1,\ldots ,\rho \), we have</p>
<div class="displaymath" id="m65">
  \begin{align}  \big\Vert \left( E_{n,W}(f,x)\right) ^{\left( i\right) }\big\Vert _{1} \label{m65} & \leq \tfrac {\xi \omega _{r}(f^{(n+i)},\xi )_{1}}{(n-1)!\left( r+1\right) } \cdot \left[ \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( \left( 1+\tfrac {\left\vert \nu \right\vert }{\xi }\right) ^{r+1}-1\right) \left\vert \nu \right\vert ^{n-1}e^{\frac{-\nu ^{2}}{\xi }}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\right] \\ & \quad +\big\Vert f^{(i)}(x)\big\Vert _{1}\left\vert m_{\xi ,W}-1\right\vert \nonumber \end{align}
</div>
<p>holds. </p>
<p>\(\mathbf{iii)}\) <i class="itshape">Let </i> \(f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R}\right) \right) ,\) \(i=0,1,\ldots ,\rho \in \mathbb {Z}^{+};\) \(p,q{\gt}1\) such that \(\tfrac {1}{p}+\tfrac {1}{q}=1.\) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\rm \ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there. Then for all </i>\(i=0,1,\ldots ,\rho ,\)</p>
<div class="displaymath" id="m66">
  \begin{align} & \big\Vert \left( E_{0,W}(f,x)\right) ^{\left( i\right) }\big\Vert _{p}\leq \label{m66} \\ & \leq \omega _{r}(f^{\left( i\right) },\xi )_{p}\left( \sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{\xi }}\right) ^{\frac{1}{q}}\! \! \cdot \! \!  \left[ \tfrac {\left( \sum \limits _{\nu =-\infty }^{\infty }\left( 1+\tfrac {\left\vert \nu \right\vert }{\xi }\right) ^{rp}e^{\frac{-\nu ^{2}}{\xi }}\right) ^{\frac{1}{p}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\right] \!  \! +\! \! \big\Vert f^{(i)}(x)\big\Vert _{p}\left\vert m_{\xi ,W}\! -\! 1\right\vert \nonumber \end{align}
</div>
<p>holds. </p>
<p>\(\mathbf{iv)}\) <i class="itshape">Let </i> \(f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{1}\left( \mathbb {R}\right) \right) ,\) \(i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.\) <i class="itshape">We consider the assumptions of Theorem </i>\(\mathit{\rm \ref{mt3}}\)<i class="itshape"> as valid for </i>\(n=\rho \)<i class="itshape"> there. Then for all </i>\(i=0,1,\ldots ,\rho ,\)</p>
<div class="displaymath" id="m67">
  \begin{align} & \big\Vert \left( E_{0,W}(f,x)\right) ^{\left( i\right) }\big\Vert _{1}\leq \label{m67} \\ & \leq \Bigg( \tfrac {\sum \limits _{\nu =-\infty }^{\infty }\left( 1+\frac{\left\vert \nu \right\vert }{\xi }\right) ^{r}e^{\frac{-\nu ^{2}}{\xi }}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1}\Bigg) \omega _{r}(f^{\left( i\right) },\xi )_{1} +\big\Vert f^{(i)}(x)\big\Vert _{1}\left\vert m_{\xi ,W}-1\right\vert \nonumber \end{align}
</div>
<p>holds. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000045">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> By Theorem \(\ref{bt15}\) and by \(\left( E_{n,W}(f,x)\right) ^{\left( i\right) }=E_{n,W}(f^{\left( i\right) },x)\) for \(n\in \mathbb {Z} ^{+}.\) <div class="proof_wrapper" id="a0000000046">
  <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><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="Abel">1</a></dt>
  <dd><p><a href ="http://www.cs.ubbcluj.ro/~studia-m/2011-2/abel-final.pdf"> <i class="sc">U. Abel</i>, <i class="itshape">Asymptotic expansions for the Favard operators and their left quasi-interpolants</i>, Studia Univ. Babeş-Bolyai Math., <b class="bf">56</b> (2011) no. 2, pp.&#160;199–206. <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="Abel-Butzer">2</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/s00365-011-9134-y"> <i class="sc">U. Abel</i> and <i class="sc">P.L. Butzer</i>, <i class="itshape">Complete asymptotic expansion for generalized Favard operators</i>, Constructive Approximation, <b class="bf">35</b> (2012), pp.&#160;73–88. <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="Anastass">3</a></dt>
  <dd><p><i class="sc">G.A. Anastassiou</i>, <i class="itshape">On discrete Gauss-Weierstrass and Picard operators</i>, Panamer. Math. J., <b class="bf">23</b> (2013) no. 2, pp.&#160;79–86. </p>
</dd>
  <dt><a name="Anastass-Kester1">4</a></dt>
  <dd><p><a href ="http://www.cs.ubbcluj.ro/~studia-m/2015-1/2015_1/05-anastassiou-final.pdf"> <i class="sc">G.A. Anastassiou</i> and <i class="sc">M. Kester</i>, <i class="itshape">Quantitative uniform approximation by generalized discrete singular operators</i>, Studia Univ. Babeş-Bolyai Math., <b class="bf">60</b> (2015) no. 1, pp.&#160;39–60. <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="Anastass-Kester2">5</a></dt>
  <dd><p><i class="sc">G.A. Anastassiou</i> and <i class="sc">M. Kester</i>, \(L_{p}\)<i class="itshape"> approximation with rates by generalized discrete singular operators,</i> Communications in Applied Analysis, accepted for publication, 2014. </p>
</dd>
  <dt><a name="Anastass-Mezei">6</a></dt>
  <dd><p><i class="sc">G.A. Anastassiou</i> and <i class="sc">R.A. Mezei</i>, <i class="itshape">Approximation by Singular Integrals</i>, Cambridge Scientific Publishers, Cambrige, UK, 2012. </p>
</dd>
  <dt><a name="Devore-Lorentz">7</a></dt>
  <dd><p><i class="sc">R.A. DeVore</i> and <i class="sc">G.G. Lorentz</i>, <i class="itshape">Constructive Approximation</i>, Springer-Verlag, Vol. 303, Berlin, New York, 1993. </p>
</dd>
  <dt><a name="Favard">8</a></dt>
  <dd><p><i class="sc">J. Favard</i>, <i class="itshape">Sur les multiplicateurs d’interpolation</i>, J. Math. Pures Appl., IX, <b class="bf">23</b> (1944), pp.&#160;219–247. </p>
</dd>
</dl>


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