Global Smoothness and Approximation
by Generalized Discrete Singular Operators

George A. Anastassiou$^\ast $ Merve Kester$^\ast $

September 23, 2014.
Published online: January 23, 2015.

$^\ast $Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, U.S.A., e-mail: ganastss@memphis.edu, mkester@memphis.edu.

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.

MSC. 26A15, 26D15, 41A17, 41A25, 41A28, 41A35, 41A80

Keywords. Simultaneous global smoothness, simultaneous approximation with rates, generalized discrete singular operators, modulus of smoothness.

1 Introduction

This article is motivated mainly by [ 3 ] , [ 6 ] , Chapter 18, and [ 8 ] , where J. Favard in 1944 introduced the discrete version of Gauss-Weierstrass operator

\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}

$n\in \mathbb {N} ,$ 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.

We are also greatly motivated by [ 1 ] and [ 2 ] .

Furthermore, we are inspired by [ 4 ] and [ 5 ] 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.

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 $1\leq p\leq \infty .$

2 Background

In [ 6 ] , 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.

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

\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}

that is $\sum \limits _{j=0}^{r}\alpha _{j}=1.$ Let $f\colon \mathbb {R}\rightarrow \mathbb {R}$ be Borel measurable, they defined for $x\in \mathbb {R},$

\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}

They supposed $\Theta _{r,\xi }(f;x)\in \mathbb {R},$ $\forall x\in \mathbb {R}.$

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

\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}

where

\begin{equation} \Delta _{t}^{m}f(x):=\sum _{j=0}^{m}(-1)^{m-j}{\tbinom {m}{j}}f(x+jt), \label{b24} \end{equation}

see also [ 7 , p. 44 ] .

Denote

\begin{equation} \omega _{m}(f,h)_{\infty }=\omega _{m}(f,h). \label{b25} \end{equation}

Notice that

\begin{equation} \omega _{m}(cf,h)_{p}=\left\vert c\right\vert \omega _{m}(f,h)_{p} \label{b25*} \end{equation}

where $c$ is a real constant.

They gave the main global smoothness preservation result in [ 6 ] as follows:

Theorem 1

Let $h{\gt}0,$ $f\in C(\mathbb {R})$.

$\mathbf{i)}$ Suppose $\Theta _{r,\xi }(f;x)\in \mathbb {R},$ $\xi {\gt}0,$ $\forall x\in \mathbb {R}$ and $\omega _{m}(f,h){\lt}\infty .$ Then

\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}

$\mathbf{ii)}$ Suppose $f\in \left( C(\mathbb {R})\cap L_{p}(\mathbb {R})\right) $, $p\geq 1$. Then

\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}

Next, in [ 6 ] , 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.

In [ 6 ] , they obtained also the next differentiation result

Theorem 2

Let $f\in C^{n-1}(\mathbb {R})$, such that $f^{(n)}$ exists, $n,r\in \mathbb {N}$. Furthermore assume that for each $x\in \mathbb {R}$ the function $f^{(i)}(x+jt)\in L_{1}(\mathbb {R},\mu _{\xi })$ as a function of $\mathit{t,}$ for all $i=0,1,\ldots ,n-1;$ $j=1,\ldots ,r.$ 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

\begin{equation} |f^{(i)}(x+jt)|\leq g_{i,j}(t), \label{b4} \end{equation}

for $\mu _{\xi }$-almost all $t\in \mathbb {R}$, all $i=1,\ldots ,n;$ $j=1,2,\ldots ,r.$ Then $f^{(i)}(x+jt)$ defines a $\mu _{\xi }$-integrable function with respect to $t$ for each $x\in \mathbb {R}$, all $i=1,\ldots ,n;$ $j=1,\ldots ,r$, and

\begin{equation} \left( \Theta _{r,\xi }\left( f;x\right) \right) ^{(i)}=\Theta _{r,\xi }\big( f^{(i)};x\big) , \label{b5} \end{equation}

for all $x\in \mathbb {R}$, all $i=1,\ldots ,n$.

On the other hand, in [ 4 ] , the authors defined important special cases of $\Theta _{r,\xi }$ operators for discrete probability measures $\mu _{\xi }$ as follows:

Let $f\in C^{n}(\mathbb {R} )$, $n\in \mathbb {Z} ^{+},$ $0{\lt}\xi \leq 1$, $x\in \mathbb {R} $.

$i)$ When

\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}

they defined the generalized discrete Picard operators as

\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}

$ii)$ When

\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}

they defined the generalized discrete Gauss-Weierstrass operators as

\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}

$iii)$ Let $\alpha \in \mathbb {N} $, and $\beta {\gt}\tfrac {1}{\alpha }.$ When

\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}

they defined the generalized discrete Poisson-Cauchy operators as

\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}

They observed that for $c$ constant they have

\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}

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 $\left\Vert f\right\Vert _{\infty ,\mathbb {R} }{\lt}\infty .$

$iv)$ When

\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}

they defined the generalized discrete non-unitary Picard operators as

\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}

Here $\mu _{\xi ,P}(\nu )$ has mass

\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}

They observed that

\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}

which is the probability measure $\left( \ref{b26}\right) $ defining the operators $P_{r,\xi }^{\ast }.$

$v)$ When

\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}

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

\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}

Here $\mu _{\xi ,W}(\nu )$ has mass

\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}

They observed that

\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}

which is the probability measure $\left( \ref{b28}\right) $ defining the operators $W_{r,\xi }^{\ast }.$

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} $.

We notice that

\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}

where

\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}

and

\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}

where

\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}

In [ 4 ] , for $k=1,...,n$, the authors defined the ratios of sums

\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}

\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}

and for $\alpha \in \mathbb {N} ,$ $\beta {\gt}\tfrac {n+r+1}{2\alpha },$ they introduced

\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}

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] $.

In [ 4 ] , the authors also proved

\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}

and

\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}

The authors introduced also

\begin{equation} \delta _{k}:=\sum _{j=1}^{r}\alpha _{j}j^{k},\quad k=1,\ldots ,n\in \mathbb {N}. \label{b54} \end{equation}

Additionally, in [ 4 ] , the authors defined the following error quantities:

\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}

\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}

Furthermore, they introduced the errors $(n\in \mathbb {N} )$:

\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}

and

\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}

Next, they obtained the inequalities

\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}

\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}

and

\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}

with

\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}

In [ 4 ] , they first gave the following simultaneous approximation results for unitary operators. They showed

Theorem 3

Let $f\in C^{n}(\mathbb {R})$ with $f^{(n)}\in C_{u}(\mathbb {R} )$ (uniformly continuous functions on $\mathbb {R} $)$.$

$\mathbf{i)}$ For $n\in \mathbb {N} ,$

\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}

and

\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}

$\mathbf{ii)}$ For $n=0,$

\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}

and

\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}

In the above inequalities (45)–(48), the ratios of sums in their right hand sides $\left( \text{R.H.S.}\right) $ are uniformly bounded with respect to $\xi \in \left( 0,1\right] .$

In [ 4 ] , they had also

Theorem 4

Let $f\in C^{n}(\mathbb {R})$ with $f^{(n)}\in C_{u}(\mathbb {R} ),$ $n\in \mathbb {\mathbb {N} }$, and $\beta {\gt}\tfrac {n+r+1}{2\alpha }.$

$\mathbf{i)}$ For $n\in \mathbb {N} ,$

\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}

$\mathbf{ii)}$ For $n=0,$

\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}

In the above inequalities (49)–(50), the ratios of sums in their R.H.S. are uniformly bounded with respect to $\xi \in \left( 0,1\right] .$

Next, they stated their results in [ 4 ] for the errors $E_{0,P},$ $E_{0,W},$ $E_{n,P},$ and $E_{n,W}.$ They had

Corollary 5

Let $f\in C_{u}(\mathbb {R} ).$ Then

$\mathbf{i)}$

\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}

$\mathbf{ii)}$

\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}

In [ 4 ] , for $E_{n,P}$ and $E_{n,W}$, the authors presented

Theorem 6

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

$\mathbf{i)}$

\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}

$\mathbf{ii)}$

\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}

In the above inequalities (53)–(54), the ratios of sums in their R.H.S. are uniformly bounded with respect to $\xi \in \left( 0,1\right] .$

In [ 5 ] , the authors represented simultaneous $L_{p}$ approximation results. They started with

Theorem 7

$\mathbf{i)}$ Let $f\in C^{n}(\mathbb {R}),$ with $f^{(n)}\in L_{p}\left( \mathbb {R}\right) ,$ $n\in \mathbb {N},$ $p,q{\gt}1$: $\tfrac {1}{p}+\tfrac {1}{q}=1$, and rest as above in this section$.$Then

\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}

where

\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}

which is uniformly bounded for all $\xi \in \left( 0,1\right] .$

Additionally, as $\xi \rightarrow 0^{+}$ we obtain that $R.H.S.$ of $(\ref{b9})$ goes to zero.

$\mathbf{ii)}$ When $p=1,$ let $f\in C^{n}(\mathbb {R})$, $f^{(n)}\in L_{1}(\mathbb {R}),$ and $n\in \mathbb {N} -\{ 1\} .$ Then

\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}

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.

$\mathbf{iii)}$ When $n=0$, let $f\in \left( C\left( \mathbb {R} \right) \cap L_{p}(\mathbb {R})\right) ,$ $p,q{\gt}1$ such that $\tfrac {1}{p}+\tfrac {1}{q}=1$ and the rest as above in this section. Then

\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}

where

\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}

which is uniformly bounded for all $\xi \in \left( 0,1\right] .$

Hence, as $\xi \rightarrow 0^{+},$ we obtain that $P_{r,\xi }^{\ast }\rightarrow $ unit operator $I$ in the $L_{p}$ norm for $p{\gt}1$.

$\mathbf{iv)}$ When $n=0$ and $p=1,$ let $f\in \left( C\left( \mathbb {R} \right) \cap L_{1}(\mathbb {R})\right) $ and the rest as above in this section. Then the inequality

\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}

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 $\xi \rightarrow 0^{+}.$

Next, the authors presented their quantitative results for the Gauss-Weierstrass operators, see [ 5 ] . They started with

Theorem 8

\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}

where

\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}

which is uniformly bounded for all $\xi \in \left( 0,1\right] .$

Additionally, as $\xi \rightarrow 0^{+}$ we obtain that $R.H.S.$ of $(\ref{b13})$ goes to zero.

$\mathbf{ii)}$ For $p=1$, let $f\in C^{n}(\mathbb {R})$, $f^{(n)}\in L_{1}(\mathbb {R}),$ and $n\in \mathbb {N} -\{ 1\} .$ Then

\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}

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.

$\mathbf{iii)}$ For $n=0$, let $f\in \left( C\left( \mathbb {R} \right) \cap L_{p}(\mathbb {R})\right) ,$ $p,q{\gt}1$ such that $\tfrac {1}{p}+\tfrac {1}{q}=1$ and the rest as above in this section. Then

\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}

where

\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}

which is uniformly bounded for all $\xi \in \left( 0,1\right] .$

Hence, as $\xi \rightarrow 0^{+},$ we obtain that $W_{r,\xi }^{\ast }\rightarrow $ unit operator $I$ in the $L_{p}$ norm for $p{\gt}1$.

$\mathbf{iv)}$ For $n=0$ and $p=1$, let $f\in \left( C\left( \mathbb {R} \right) \cap L_{1}(\mathbb {R})\right) $ and the rest as above in this section. Then the inequality

\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}

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 $\xi \rightarrow 0^{+}.$

For the Poisson-Cauchy operators, in [ 5 ] , the authors showed

Theorem 9

\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}

where

\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}

is uniformly bounded for all $\xi \in \left( 0,1\right] .$

Additionally, as $\xi \rightarrow 0^{+},$ we obtain that $R.H.S.$ of $(\ref{b17})$ goes to zero.

$\mathbf{ii)}$ When $p=1$, let $f\in C^{n}(\mathbb {R})$, $f^{(n)}\in L_{1}(\mathbb {R}),$ $\beta {\gt}\tfrac {r+n+1}{2\alpha },$ and $n\in \mathbb {N} -\{ 1\} .$ Then

\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}

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.

$\mathbf{iii)}$When $n=0$, let $f\in \left( C\left( \mathbb {R} \right) \cap L_{p}(\mathbb {R})\right) ,$ $p,q{\gt}1$ such that $\tfrac {1}{p}+\tfrac {1}{q}=1,$ $\beta {\gt}\tfrac {p\left( r+2\right) +1}{2\alpha },$ and the rest as above in this section. Then

\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}

where

\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}

which is uniformly bounded for all $\xi \in \left( 0,1\right] .$

Hence, as $\xi \rightarrow 0^{+},$ we obtain that $Q_{r,\xi }^{\ast }\rightarrow $ unit operator $I$ in the $L_{p}$ norm for $p{\gt}1$.

$\mathbf{iv)}$ 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

\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}

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 $\xi \rightarrow 0^{+}.$

Next in [ 5 ] , they stated their results for the errors $E_{0,P},$ $E_{0,W},$ $E_{n,P},$ and $E_{n,W}$ as follows

Theorem 10

$\mathbf{i)}$ 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

\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}

holds. Additionally, as $\xi \rightarrow 0^{+},$ we obtain that $R.H.S.$ of $(\ref{b59})$ goes to zero.

$\mathbf{ii)}$ When $p=1,$ let $f\in C^{n}(\mathbb {R}),f\in L_{1}(\mathbb {R} )$, $f^{(n)}\in L_{1}(\mathbb {R}),$ and $n\in \mathbb {N} -\{ 1\} .$ Then

\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}

holds. Additionally, as $\xi \rightarrow 0^{+},$ we obtain that $R.H.S.$ of $(\ref{b60})$ goes to zero.

$\mathbf{iii)}$ When $n=0,$ let $p,q{\gt}1$ such that $\tfrac {1}{p}+\tfrac {1}{q}=1,$ $f\in L_{p}(\mathbb {R} )$, and the rest as above in this section. Then

\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}

holds. Hence, as $\xi \rightarrow 0^{+},$ we obtain that $R.H.S.$ of $(\ref{b61})$ goes to zero.

$\mathbf{iv)}$ When $n=0$ and $p=1,$ the inequality

\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}

holds. Hence, as $\xi \rightarrow 0^{+},$ we obtain that $R.H.S.$ of $(\ref{b62})$ goes to zero.

Next in [ 5 ] , the authors gave quantitative results for $E_{n,W}(f,x)$

Theorem 11

$\mathbf{i)}$ 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

\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}

holds. Additionally, as $\xi \rightarrow 0^{+},$ we obtain that $R.H.S.$ of $(\ref{b63})$ goes to zero.

$\mathbf{ii)}$ For $p=1,$ let $f\in C^{n}(\mathbb {R}),f\in L_{1}(\mathbb {R} )$, $f^{(n)}\in L_{1}(\mathbb {R}),$ and $n\in \mathbb {N} -\{ 1\} .$ Then

\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}

holds. Additionally, as $\xi \rightarrow 0^{+},$ we obtain that $R.H.S.$ of $(\ref{b64})$ goes to zero.

$\mathbf{iii)}$ For $n=0,$ let $p,q{\gt}1$ such that $\tfrac {1}{p}+\tfrac {1}{q}=1,$ $f\in L_{p}(\mathbb {R} )$, and the rest as above in this section. Then

\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}

holds. Hence, as $\xi \rightarrow 0^{+},$ we obtain that $R.H.S.$ of $(\ref{b65})$ goes to zero.

$\mathbf{iv)}$ For $n=0$ and $p=1,$ the inequality

\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}

holds. Hence, as $\xi \rightarrow 0^{+},$ we obtain that $R.H.S.$ of $(\ref{b66})$ goes to zero.

3 Main Results

We start with global smoothness preservation properties of the operators $P_{r,\xi }^{\ast },$ $W_{r,\xi }^{\ast },$ and $\Theta _{r,\xi }^{\ast }.$

Theorem 12

Let $h{\gt}0$ and $0{\lt}\xi \leq 1.$

$\mathbf{i)}$ 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

\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}

\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}

and

\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}

$\mathbf{ii)}$ Suppose $f\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R} \right) \right) ,$ $p\geq 1.$ Then

\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}

\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}

and

\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}

Proof â–¼

By Theorem $\ref{bt1}.$

Proof â–¼

For $r=1,$ we get $\alpha _{0}=0$ and $\alpha _{1}=1.$ Hence, we obtain

\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}

\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}

and for $\beta {\gt}\tfrac {1}{\alpha },$ $\alpha \in \mathbb {N} $

\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}

Therefore, by Theorem $\ref{mt1},$ we have

Theorem 13

Let $h{\gt}0$ and $0{\lt}\xi \leq 1.$

$\mathbf{i)}$ 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

\begin{equation} \omega _{m}(P_{\xi }^{\ast }f,h)\leq \omega _{m}(f,h), \label{m10} \end{equation}

\begin{equation} \omega _{m}(W_{\xi }^{\ast }f,h)\leq \omega _{m}(f,h), \label{m11} \end{equation}

and

\begin{equation} \omega _{m}(Q_{\xi }^{\ast }f,h)\leq \omega _{m}(f,h). \label{m12} \end{equation}

Inequalities (90), (91), and (92) are sharp, that is attained by $f\left( x\right) =g\left( x\right) =x^{m}.$

$\mathbf{ii)}$ Suppose $f\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R} \right) \right) ,$ $p\geq 1.$ Then

\begin{equation} \omega _{m}(P_{\xi }^{\ast }f,h)_{p}\leq \omega _{m}(f,h)_{p}, \label{m13} \end{equation}

\begin{equation} \omega _{m}(W_{\xi }^{\ast }f,h)_{p}\leq \omega _{m}(f,h)_{p}, \label{m14} \end{equation}

and

\begin{equation} \omega _{m}(Q_{\xi }^{\ast }f,h)_{p}\leq \omega _{m}(f,h)_{p}. \label{m15} \end{equation}

Proof â–¼

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

\begin{equation} \omega _{m}(g,h)=\omega _{m}(x^{m},h)=m!h^{m}. \label{m16} \end{equation}

On the other hand, we have

\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}

Thus, we get

\begin{equation} \omega _{m}(g,h)=\omega _{m}(P_{\xi }^{\ast }g,h). \label{m18} \end{equation}

Similarly, we obtain

\begin{equation} \omega _{m}(g,h)=\omega _{m}(W_{\xi }^{\ast }g,h), \label{m19} \end{equation}

and

\begin{equation} \omega _{m}(g,h)=\omega _{m}(Q_{\xi }^{\ast }g,h). \label{m20} \end{equation}

100

Proof â–¼

Next, we present the following theorem for the non-unitary operators $P_{r,\xi }$ and $W_{r,\xi }$

Theorem 14

Let $h{\gt}0$ and $0{\lt}\xi \leq 1.$

$\mathbf{i)}$ 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

\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}

101

\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}

102

$\mathbf{ii)}$ Suppose $f\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R} \right) \right) ,$ $p\geq 1.$ Then

\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}

103

\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}

104

Proof â–¼

By $\left( \ref{b25*}\right) $, $\left( \ref{b40.1}\right) ,$ $\left( \ref{b40.3}\right) $ and Theorem $\ref{mt1}$, we have

\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}

105

\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}

106

and

\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}

107

\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}

108

Additionally, in [ 3 ] , it was shown that

\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}

109

and

\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}

110

Thus, by (105)–(110), we obtain the inequalities (101)–(104).

Proof â–¼

Now, we give our results for the derivatives of the unitary 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) $ mentioned above. First, we get

Theorem 15

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

\begin{equation} |f^{(i)}(x+j\nu )|\leq g_{i,j}(\nu ), \label{m21} \end{equation}

111

for $\mu _{\xi }$-almost all $\nu \in \mathbb {R} $, all $i=1,\ldots ,n;$ $j=1,2,\ldots ,r.$ Then, $f^{(i)}(x+j\nu )$ defines a $\mu _{\xi }$-integrable function with respect to $\nu $ for each $x\in \mathbb {R} $, all $i=1,\ldots ,n;$ $j=1,\ldots ,r$.

$\mathbf{i)}$ When

\[ \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}} \]

we get

\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}

112

for all $x\in \mathbb {R} $, and for all $i=1,\ldots ,n$.

$\mathbf{ii)}$ When

\[ \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{_{\xi }}}} \]

we have

\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}

113

for all $x\in \mathbb {R} $, and for all $i=1,\ldots ,n$.

$\mathbf{iii)}$ Let $\alpha \in \mathbb {N} $, and $\beta {\gt}\tfrac {1}{\alpha }.$ When

\[ \mu _{\xi }(\nu )=\tfrac {\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}\text{,} \]

we obtain

\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}

114

for all $x\in \mathbb {R} $, and for all $i=1,\ldots ,n$.

Proof â–¼

By Theorem $\ref{bt2}.$

Proof â–¼

Next, we present our results for the derivatives of non-unitary operators $P_{r,\xi }\left( f;x\right) $ and $W_{r,\xi }\left( f;x\right) .$

Proposition 16

Let the assumptions of the Theorem $\ref{mt3}$ be valid.

$\mathbf{i)}$ When

\[ \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{1+2\xi e^{\frac{-1}{\xi }}} \]

we get

\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}

115

for all $x\in \mathbb {R} $, and for all $i=1,\ldots ,n$.

$\mathbf{ii)}$ When

\[ \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sqrt{\pi \xi }\left( 1-{\rm erf}\left( \frac{1}{\sqrt{\xi }}\right) \right) +1} \]

we have

\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}

116

for all $x\in \mathbb {R} $, and for all $i=1,\ldots ,n$.

Proof â–¼

By $\left( \ref{b40.1}\right) $ and $\left( \ref{b40.3}\right) $ we have

\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}

117

and

\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}

118

Thus by Theorem $\ref{mt3},$ we get

\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}

and

\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}

Proof â–¼

We have the following application of the Theorem $\ref{mt3}$ for the case of $r=1.$

Proposition 17

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

\begin{equation} |f^{(i)}(x+\nu )|\leq g_{i}(\nu ), \label{m25} \end{equation}

121

for $\mu _{\xi }$-almost all $\nu \in \mathbb {R} $, all $i=1,\ldots ,n.$ Then, $f^{(i)}(x+\nu )$ defines a $\mu _{\xi }$-integrable function with respect to $\nu $ for each $x\in \mathbb {R} $, and for all $i=1,\ldots ,n$.

$\mathbf{i)}$ When

\[ \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\left\vert \nu \right\vert }{_{\xi }}}} \]

we get

\begin{equation} \left( P_{\xi }^{\ast }\left( f;x\right) \right) ^{(i)}=P_{\xi }^{\ast }( f^{(i)};x) , \label{m26} \end{equation}

122

for all $x\in \mathbb {R} $, and for all $i=1,\ldots ,n$.

$\mathbf{ii)}$ When

\[ \mu _{\xi }(\nu )=\tfrac {e^{\frac{-\nu ^{2}}{_{\xi }}}}{\sum \limits _{\nu =-\infty }^{\infty }e^{\frac{-\nu ^{2}}{_{\xi }}}} \]

we have

\begin{equation} \left( W_{\xi }^{\ast }\left( f;x\right) \right) ^{(i)}=W_{\xi }^{\ast }( f^{(i)};x) , \label{m27} \end{equation}

123

for all $x\in \mathbb {R} $, and for all $i=1,\ldots ,n$.

$\mathbf{iii)}$ Let $\alpha \in \mathbb {N} $, and $\beta {\gt}\tfrac {1}{\alpha }.$ When

\[ \mu _{\xi }(\nu )=\tfrac {\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}{\sum \limits _{\nu =-\infty }^{\infty }\left( \nu ^{2\alpha }+\xi ^{2\alpha }\right) ^{-\beta }}\text{,} \]

we obtain

\begin{equation} \left( Q_{\xi }^{\ast }\left( f;x\right) \right) ^{(i)}=Q_{\xi }^{\ast }( f^{(i)};x) , \label{m28} \end{equation}

124

for all $x\in \mathbb {R} $, and for all $i=1,\ldots ,n$.

We obtain

Theorem 18

Let $h{\gt}0$ and the assumptions of the Theorem $\ref{mt3}$ be valid.

$\mathbf{i)}$ Suppose that $\omega _{m}(f^{(i)},h){\lt}\infty ,$ for all $i=0,1,...,n.$ Then

\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}

125

\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}

126

and

\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}

127

$\mathbf{ii)}$ 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

\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}

128

\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}

129

and

\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}

130

Proof â–¼

By Theorems $\ref{mt1}$, $\ref{mt3}.$

Proof â–¼

Next, we state our results for the non-unitary operators

Theorem 19

Let $h{\gt}0$ and the assumptions of the Theorem $\ref{mt3}$ be valid.

$\mathbf{i)}$ Suppose that $\omega _{m}(f^{(i)},h){\lt}\infty ,$ for all $i=0,1,...,n.$ Then

\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}

131

and

\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}

132

$\mathbf{ii)}$ 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

\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}

133

and

\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}

134

Proof â–¼

By $\left( \ref{b40.1}\right) ,$ $\left( \ref{b40.3}\right) ,$ $\left( \ref{m20.7}\right) ,$ $\left( \ref{m20.8}\right) ,$ and Theorem $\ref{mt4}.$

Proof â–¼

For the case of $r=1$ we have

Proposition 20

Let $h{\gt}0$ and the assumptions of the Proposition $\ref{mp1}$ be valid.

$\mathbf{i)}$ Assume that $\omega _{m}(f^{(i)},h){\lt}\infty ,$ for all $i=0,1,...,n.$ Then

\begin{equation} \omega _{m}(( P_{\xi }^{\ast }f) ^{(i)},h)\leq \omega _{m}(f^{(i)},h), \label{m35} \end{equation}

135

\begin{equation} \omega _{m}(( W_{\xi }^{\ast }f) ^{(i)},h)\leq \omega _{m}(f^{(i)},h), \label{m36} \end{equation}

136

and

\begin{equation} \omega _{m}(( Q_{\xi }^{\ast }f) ^{(i)},h)\leq \omega _{m}(f^{(i)},h). \label{m37} \end{equation}

137

$\mathbf{ii)}$ 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

\begin{equation} \omega _{m}(( P_{\xi }^{\ast }f) ^{(i)},h)_{p}\leq \omega _{m}(f^{(i)},h)_{p}, \label{m38} \end{equation}

138

\begin{equation} \omega _{m}(( W_{\xi }^{\ast }f) ^{(i)},h)_{p}\leq \omega _{m}(f^{(i)},h)_{p}, \label{m39} \end{equation}

139

and

\begin{equation} \omega _{m}(( Q_{\xi }^{\ast }f) ^{(i)},h)_{p}\leq \omega _{m}(f^{(i)},h)_{p}. \label{m40} \end{equation}

140

Proof â–¼

By Theorem $\ref{mt2}$ and Proposition $\ref{mp1}.$

Proof â–¼

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

Theorem 21

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 $0{\lt}\xi \leq 1$. We consider the assumptions of Theorem $\ref{mt3}$ valid for $n=\rho $ there.

\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}

\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}

and

\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}

where $\beta {\gt}\tfrac {n+r+1}{2\alpha },$ $\alpha \in \mathbb {N} .$

Proof â–¼

By Theorems $\ref{bt3},$ $\ref{bt5}.$

Proof â–¼

Next we have

Theorem 22

Let $f\in C^{n+\rho }(\mathbb {R}),$ with $f^{(n+i)}\in L_{p}\left( \mathbb {R}\right) ,$ $n\in \mathbb {N},$ $i=0,1,\ldots ,$ $\rho \in \mathbb {Z}^{+}.$ Let $p,q{\gt}1:\tfrac {1}{p}+\tfrac {1}{q}=1. $ We consider the assumptions of Theorem $\mathit{\rm \ref{mt3}}$ as valid for $n=\rho $ there. Then

$\mathbf{i)}$

\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}

$\mathbf{ii)}$

\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}

$\mathbf{iii)}$ for $\beta {\gt}\tfrac {p\left( n+r\right) +1}{2\alpha },$ $\alpha \in \mathbb {N} $

\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}

Proof â–¼

By Theorems 7-9.

Proof â–¼

Now, we give our results for the special case of $n=0.$

Proposition 23

Let $f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R}\right) \right) ,$ $i=0,1,\ldots ,\rho \in \mathbb {Z}^{+};$ $p,q{\gt}1$ such that $\tfrac {1}{p}+\tfrac {1}{q}=1.$ We consider the assumptions of Theorem $\mathit{\ref{mt3}}$ as valid for $n=\rho $ there. Then for all $i=0,1,\ldots ,\rho ,$ we have

$\mathbf{i)}$

\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}

147

$\mathbf{ii)}$

\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}

148

$\mathbf{iii)}$ for $\beta {\gt}\tfrac {p\left( r+2\right) +1}{2\alpha },$ $\alpha \in \mathbb {N} $

\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}

149

Proof â–¼

By Theorems 7-9.

Proof â–¼

For the special case of $p=1,$ we obtain

Theorem 24

Let $f\in C^{n+\rho }(\mathbb {R}),$ with $f^{(n+i)}\in L_{1}\left( \mathbb {R}\right) ,$ $n\in \mathbb {N-}\left\{ 1\right\} ,$ $i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.$ We consider the assumptions of Theorem $\mathit{\ref{mt3}}$ as valid for $n=\rho $ there. Then for all $i=0,1,\ldots ,\rho $, we have

$\mathbf{i)}$

\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}

$\mathbf{ii)}$

\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}

$\mathbf{iii)}$ for $\beta {\gt}\tfrac {n+r+1}{2\alpha },$ $\alpha \in \mathbb {N} $

\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}

Proof â–¼

By Theorems 7–9.

Proof â–¼

For $p=1$ and $n=0,$ we give

Proposition 25

Let $f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{1}\left( \mathbb {R}\right) \right) ,$ $i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.$ We consider the assumptions of Theorem $\mathit{\rm \ref{mt3}}$ as valid for $n=\rho $ there. Then for all $i=0,1,\ldots ,\rho ,$ we have

$\mathbf{i)}$

\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}

153

$\mathbf{ii)}$

\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}

154

$\mathbf{iii)}$ for $\beta {\gt}\tfrac {r+3}{2\alpha },$ $\alpha \in \mathbb {N} $

\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}

155

Proof â–¼

By Theorems 7–9.

Proof â–¼

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

Corollary 26

Let $f^{(i)}\in C_{u}(\mathbb {R} ),$ $i=0,1,\ldots ,\rho ,$ $\rho \in \mathbb {Z} ^{+},$ and $0{\lt}\xi \leq 1.$ We consider the assumptions of Theorem $\mathit{\ref{mt3}}$ as valid for $n=\rho $ there. Then for all $i=0,1,\ldots ,\rho ,$ we have

$\mathbf{i)}$

\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}

156

$\mathbf{ii)}$

\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}

157

Proof â–¼

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) } $ $=E_{0,W}(f^{\left( i\right) },x).$

Proof â–¼

Theorem 27

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 $\left\Vert f^{\left( i\right) }\right\Vert _{\infty ,\mathbb {R} }{\lt}\infty .$ We consider the assumptions of Theorem $\ref{mt3}$ valid for $n=\rho .$ Then for all $i=0,1,\ldots ,\rho ,$ we have

$\mathbf{i)}$

\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}

and

$\mathbf{ii)}$

\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}

Proof â–¼

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 $\left( E_{n,W}(f,x)\right) ^{\left(\! \! i\right) }$ $=E_{n,W}(f^{\left( i\right) },x).$

Proof â–¼

Theorem 28

$\mathbf{i)}$ Let $f\in C^{n+\rho }(\mathbb {R}), $ with $f^{(n+i)}\in L_{p}\left( \mathbb {R}\right) ,$ $n\in \mathbb {N},$ $i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.$ Let $p,q{\gt}1:\tfrac {1}{p}+\tfrac {1}{q}=1,$ $np\neq 1.$ We consider the assumptions of Theorem $\mathit{\rm \ref{mt3}}$ as valid for $n=\rho $ there. Then for all $i=0,1,\ldots ,\rho ,$

\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}

holds.

$\mathbf{ii)}$ Let $f\in C^{n+\rho }(\mathbb {R}),$ with $f^{(n+i)}\in L_{1}\left( \mathbb {R}\right) ,$ $n\in \mathbb {N-}\left\{ 1\right\} ,$ $i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.$ We consider the assumptions of Theorem $\mathit{\rm \ref{mt3}}$ as valid for $n=\rho $ there. Then for all $i=0,1,\ldots ,\rho $,

\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}

holds.

$\mathbf{iii)}$ Let $f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{p}\left( \mathbb {R}\right) \right) ,$ $i=0,1,\ldots ,\rho \in \mathbb {Z}^{+};$ $p,q{\gt}1$ such that $\tfrac {1}{p}+\tfrac {1}{q}=1.$ We consider the assumptions of Theorem $\mathit{\rm \ref{mt3}}$ as valid for $n=\rho $ there. Then for all $i=0,1,\ldots ,\rho ,$

\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}

holds.

$\mathbf{iv)}$ Let $f^{(i)}\in \left( C\left( \mathbb {R} \right) \cap L_{1}\left( \mathbb {R}\right) \right) ,$ $i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.$ We consider the assumptions of Theorem $\mathit{\rm \ref{mt3}}$ as valid for $n=\rho $ there. Then for all $i=0,1,\ldots ,\rho ,$

\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}

holds.

Proof â–¼

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 $n\in \mathbb {Z} ^{+}.$

Proof â–¼

Theorem 29

$\mathbf{i)}$ Let $f\in C^{n+\rho }(\mathbb {R}),$ with $f^{(n+i)}\in L_{p}\left( \mathbb {R}\right) ,$ $n\in \mathbb {N},$ $i=0,1,\ldots ,\rho \in \mathbb {Z}^{+}.$ Let $p,q{\gt}1:\tfrac {1}{p}+\tfrac {1}{q}=1,$ $np\neq 1.$ We consider the assumptions of Theorem $\mathit{\rm \ref{mt3}}$ as valid for $n=\rho $ there. Then for all $i=0,1,\ldots ,\rho ,$

\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}

holds.

\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}

holds.

\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}

holds.

\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}

holds.

Proof â–¼

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 $n\in \mathbb {Z} ^{+}.$

Proof â–¼

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Global Smoothness and Approximation by Generalized Discrete Singular Operators

1 Introduction

2 Background

3 Main Results

Bibliography

Global Smoothness and Approximation
by Generalized Discrete Singular Operators