Skip to main content

Theory and Modern Applications

Blending type approximation by τ-Baskakov-Durrmeyer type hybrid operators

Abstract

In this work, we construct a Durrmeyer type modification of the τ-Baskakov operators depends on two parameters \(\alpha >0\) and \(\tau \in [0,1]\). We derive the rate of approximation of these operators in a weighted space and also obtain a quantitative Voronovskaja type asymptotic formula as well as a Grüss Voronovskaya type approximation.

1 Introduction

Chen et al. [9] recently defined a new kind of Bernstein operators by assuming fixed τ in \(\mathbb{R}\) (the set of real numbers) and showed that newly defined τ-Bernstein operators are positive and linear with the choice of \(\tau \in [0,1]\). The Kantorovich variant of aforesaid operators was reported by Mohiuddine et al. [22] and investigated several approximation properties, and most recently their Stancu and Schurer types generalization have been constructed and studied by Mohiuddine and Özger [26] and Özger et al. [33].

Inspired from the τ-Bernstein operators, for τ in \([0,1]\) and \(m\in \mathbb{N}\) (the set of natural numbers), Aral and Erbay [7] constructed τ-Baskakov as follows:

$$\begin{aligned} \mathscr{B}_{m}^{(\tau )}(\zeta ;y)= \sum _{j=0}^{\infty }p_{m,j}^{( \tau )}(y) \zeta \biggl(\frac{j}{m} \biggr), \quad y\in [0,\infty ), \end{aligned}$$
(1.1)

where

$$\begin{aligned}& \begin{aligned} p_{m,j}^{(\tau )}(y)&=\frac{y^{j-1}}{(1+y)^{m+j-1}} \biggl[ \frac{\tau y}{(1+y)} \binom{m+j-1}{j}\\ &\quad {} -(1-\tau ) (1+y)\binom{m+j-3}{j-2}+(1- \tau )y\binom{m+j-1}{j} \biggr], \end{aligned} \\& \binom{m-3}{-2}=0\quad \text{and}\quad \binom{m-2}{-1}=0. \end{aligned}$$

Setting \(\tau =1\) in (1.1) leads to the Baskakov operators [8]. Later, İlarslan et al. [16] presented a generalization of the above operators (1.1) in Kantorovich sense. Such type of operators are also defined and studied by Nasiruzzaman et al. [31].

In [36], the authors considered an integral modification of a Szász–Mirakjan–Beta type operators and presented several approximation results for their operators. In 2015, Gupta [13] presented a general class of hybrid integral type operators and proved some significant approximation properties of the operators. Kajla and Agrawal [20] obtained an interesting generalization of Szász operators with the help of Charlier polynomials. By taking these operators into account, they studied a Voronovskaya type asymptotic formula and the degree of approximation. Goyal and Kajla [12] constructed an integral type modification of generalized Lupaş operators involving a parameter \(\alpha >0\) and derived the order of approximation for these operators. For further investigation concerning such types of operators as well as statistical approximation, we refer to [16, 11, 14, 15, 1721, 2325, 2730, 34, 35, 3739] and the references therein.

Motivated by the operators constructed in [7, 16, 31], in the next section, we give Durrmeyer type modification of (1.1) and obtain some basic properties for further study in the next sections. Section 3 is devoted to obtain Voronovskaja type results of our new operators. In Sect. 4, we obtain approximation theorems by considering weighted function. In the last section, we considered some terminology defined in [40] and establish a quantitative and Grüss Voronovskaja type approximation.

2 Construction of operators and basic results

It depends on two parameters \(\alpha >0\) and \(\tau \in [0,1]\). For \(\varLambda >0\) and \(C_{\varLambda }[0,\infty ):= \{ \zeta \in C[0,\infty ): \zeta (t)=O(t^{ \varLambda }),t\geq 0 \} \), we define the operators

$$\begin{aligned} \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)= \sum _{j=1}^{\infty }p_{m,j}^{(\tau )}(y) \int _{0}^{\infty }l_{m,j}^{\alpha }(t)\zeta (t)\,dt+p_{m,0}^{(\tau )}(y)\zeta (0), \end{aligned}$$
(2.1)

where

$$ l_{m,j}^{\alpha }(t)= \frac{1}{B(j\alpha , m\alpha +1)} \frac{t^{j\alpha -1}}{(1+t)^{j\alpha +m\alpha +1}} $$

and \(p_{m,j}^{(\tau )}(y)\) is defined as above.

Lemma 1

For the operators \(\mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)\), we have

$$\begin{aligned}& (\mathrm{i}) \quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{0};y)=1; \\& (\mathrm{ii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{1};y)=y- \frac{2 y}{m}+ \frac{2 y \tau }{m}; \\& (\mathrm{iii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{2};y)= \frac{y^{2} (-3+m+4 \tau ) \alpha }{(m \alpha -1) }+ \frac{y (-2+m+2 \tau +(-4+m+4 \tau ) \alpha )}{m (m \alpha -1 )}; \\& (\mathrm{iv})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{3};y)= \frac{(1+m) y^{3} (-4+m+6 \tau ) \alpha ^{2}}{(m \alpha -2 ) (m \alpha -1 )}+ \frac{3 y^{2} \alpha (-3+m+4 \tau +(-5+m+6 \tau ) \alpha )}{(m \alpha -2 ) (m \alpha -1 )} \\& \hphantom{((\mathrm{iv})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{3};y)=}{}+ \frac{y (1+\alpha ) (m (2+\alpha )+4 (-1+\tau ) (1+2 \alpha ))}{m (m \alpha -2 ) (m \alpha -1 )}; \\& (\mathrm{v})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{4};y)= \frac{(1+m) (2+m) y^{4} (-5+m+8 \tau ) \alpha ^{3}}{(m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )}\\& \hphantom{(\mathrm{v})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{4};y)=}{}+ \frac{6 (1+m) y^{3} \alpha ^{2} (-4+m+6 \tau +(-6+m+8 \tau ) \alpha )}{(m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )} \\& \hphantom{(\mathrm{v})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{4};y)=}{}+ \frac{y^{2} \alpha (1+\alpha ) (11 (-3+m+4 \tau )+(-57+7 m+64 \tau ) \alpha )}{(m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )}\\& \hphantom{(\mathrm{v})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{4};y)=}{}+ \frac{y (1+\alpha ) (m (2+\alpha ) (3+\alpha )+4 (-1+\tau ) (3+4 \alpha (2+\alpha )))}{m (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )}; \\& (\mathrm{vi})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{5};y)= \frac{(1+m) (2+m) (3+m) y^{5} (-6+m+10 \tau ) \alpha ^{4}}{(m \alpha -4 ) (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )}\\& \hphantom{(\mathrm{vi})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{5};y)=}{}+ \frac{10 (1+m) (2+m) y^{4} \alpha ^{3} (-5+m+8 \tau +(-7+m+10 \tau ) \alpha )}{(m \alpha -4 ) (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )} \\& \hphantom{(\mathrm{vi})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{5};y)=}{}+ \frac{5 (1+m) y^{3} \alpha ^{2} (1+\alpha ) (7 (-4+m+6 \tau )+(-44+5 m+54 \tau ) \alpha )}{(m \alpha -4 ) (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )} \\& \hphantom{(\mathrm{vi})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{5};y)=}{}+ \frac{1}{(m \alpha -4 ) (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )}\\& \hphantom{(\mathrm{vi})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{5};y)=}{}\times\bigl[5 y^{2} \alpha (1+\alpha ) \bigl(m (2+\alpha ) (5+3 \alpha )+2 \tau (4+3 \alpha ) (5+7 \alpha )\\& \hphantom{(\mathrm{vi})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{5};y)=}{}-3 \bigl(10+\alpha (25+13 \alpha )\bigr)\bigr)\bigr] \\& \hphantom{(\mathrm{vi})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{5};y)=}{}+ \frac{y (1+\alpha ) (2+\alpha ) (m (3+\alpha ) (4+\alpha )+8 (-1+\tau ) (3+4 \alpha (2+\alpha )))}{m (m \alpha -4 ) (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )}; \\& (\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)= \frac{(1+m) (2+m) (3+m) (4+m) y^{6} (-7+m+12 \tau ) \alpha ^{5}}{(m \alpha -5 ) (m \alpha -4 ) (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )} \\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}+ \frac{15 (1+m) (2+m) (3+m) y^{5} \alpha ^{4} (-6+m+10 \tau +(-8+m+12 \tau ) \alpha )}{(m \alpha -5 ) (m \alpha -4 ) (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )} \\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}+ \frac{1}{(m \alpha -5 ) (m \alpha -4 ) (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )}\\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}\times\bigl[5 (1+m) (2+m) y^{4} \alpha ^{3} (1+\alpha ) \bigl(17 (-5+m+8 \tau )\\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}+(-125+13 m+164 \tau ) \alpha \bigr)\bigr] \\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}+ \frac{1}{(m \alpha -5 ) (m \alpha -4 ) (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )}\\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}\times\bigl[15 (1+m) y^{3} \alpha ^{2} (1+\alpha ) \bigl(15 (-4+m+6 \tau )\\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}+(-144+19 m+182 \tau ) \alpha +2 (-38+3 m+44 \tau ) \alpha ^{2} \bigr)\bigr] \\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}+ \frac{1}{(m \alpha -5 ) (m \alpha -4 ) (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )}\\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}\times \bigl[y^{2} \alpha \bigl(274 (-3+m+4 \tau )+675 (-5+m+6 \tau ) \alpha \\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}+85 (-57+7 m+64 \tau ) \alpha ^{2}+225 (-13+m+14 \tau ) \alpha ^{3}\\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}+(-633+31 m+664 \tau ) \alpha ^{4} \bigr) \bigr] \\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}+ \frac{1}{m (m \alpha -5 ) (m \alpha -4 ) (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )}\\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}\times\bigl[y (1+\alpha ) (2+\alpha ) \bigl(m (3+\alpha ) (4+\alpha ) (5+\alpha )\\& \hphantom{(\mathrm{vii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}(e_{6};y)=}{}+8 (-1+\tau ) (1+2 \alpha ) (3+2 \alpha ) (5+2 \alpha )\bigr)\bigr]. \end{aligned}$$

Lemma 2

From Lemma 1, we obtain

$$\begin{aligned}& (\mathrm{i})\quad \mathscr{A}_{m,\alpha }^{(\tau )} \bigl((t-y);y\bigr)=\frac{2 y (\tau -1)}{m}; \\& (\mathrm{ii}) \quad \mathscr{A}_{m,\alpha }^{(\tau )} \bigl((t-y)^{2};y\bigr)= \frac{y^{2} (4 (\tau -1)+m (1+\alpha ))}{m (m \alpha -1 )}+ \frac{y (2 (\tau -1)+4 (\tau -1) \alpha +m (1+\alpha ))}{m (m \alpha -1 )}; \\& (\mathrm{iii})\quad \mathscr{A}_{m,\alpha }^{(\tau )} \bigl((t-y)^{4};y\bigr)= \frac{1}{m (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )}\bigl[y^{4} \bigl(48 (\tau -1)+3 m^{2} \alpha (1+\alpha )^{2}\\& \hphantom{(\mathrm{iii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}((t-y)^{4};y)=}{}+2 m (1+\alpha ) \bigl(9+\alpha (-19+28 \tau -5 \alpha +8 \tau \alpha )\bigr) \bigr)\bigr] \\& \hphantom{(\mathrm{iii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}((t-y)^{4};y)=}{}+\frac{1}{m (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )} \bigl[y^{3} \bigl(72 (\tau -1)+144 (\tau -1) \alpha \\& \hphantom{(\mathrm{iii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}((t-y)^{4};y)=}{} +6 m^{2} \alpha (1+\alpha )^{2}+2 m (1+\alpha ) \bigl(9+\alpha (-19+28 \tau -5 \alpha +8 \tau \alpha )\bigr)\\& \hphantom{(\mathrm{iii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}((t-y)^{4};y)=}{}+2 m (1+\alpha ) \bigl(9+\alpha \bigl(-5-13 \alpha +2 \tau (7+8 \alpha )\bigr)\bigr) \bigr) \bigr] \\& \hphantom{(\mathrm{iii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}((t-y)^{4};y)=}{}+\frac{1}{m (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )} \bigl[y^{2} \bigl(48 (\tau -1)+144 (\tau -1) \alpha \\& \hphantom{(\mathrm{iii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}((t-y)^{4};y)=}{}+96 (\tau -1) \alpha ^{2}+3 m^{2} \alpha (1+\alpha )^{2}+m (1+\alpha ) (2+\alpha ) (3+\alpha ) \\& \hphantom{(\mathrm{iii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}((t-y)^{4};y)=}{}+2 m (1+\alpha ) \bigl(9+ \alpha \bigl(-5-13 \alpha +2 \tau (7+8 \alpha )\bigr)\bigr) \bigr) \bigr] \\& \hphantom{(\mathrm{iii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}((t-y)^{4};y)=}{}+ \frac{1}{m (m \alpha -3 ) (m \alpha -2 ) (m \alpha -1 )}\bigl[y \bigl(12 (-1+\tau )+44 (-1+\tau ) \alpha\\& \hphantom{(\mathrm{iii})\quad \mathscr{A}_{m,\alpha }^{(\tau )}((t-y)^{4};y)=}{}+48 (\tau -1) \alpha ^{2}+16 (\tau -1) \alpha ^{3}+m (1+\alpha ) (2+\alpha ) (3+\alpha ) \bigr)\bigr]. \end{aligned}$$

Remark 1

We have

$$\begin{aligned}& \lim_{m\rightarrow \infty } m \mathscr{F}_{m,\alpha }^{ \tau ,1}(y) = 2y(\tau -1), \\& \lim_{m\rightarrow \infty }m \mathscr{F}_{m,\alpha }^{ \tau ,2}(y) = \frac{y (1+y) (1+\alpha )}{\alpha }, \\& \lim_{m\rightarrow \infty } m^{2} \mathscr{F}_{m,\alpha }^{ \tau ,4}(y) = \frac{3 y^{2} (1+y)^{2} (1+\alpha )^{2}}{\alpha ^{2}}, \\& \lim_{m\rightarrow \infty } m^{3} \mathscr{F}_{m,\alpha }^{ \tau ,6}(y) = \frac{15 y^{3} (1+y)^{3} (1+\alpha )^{3}}{\alpha ^{3}}, \end{aligned}$$

where \(\mathscr{F}_{m,\alpha }^{\tau ,\nu }:=\mathscr{A}_{m,\alpha }^{(\tau )}((t-y)^{\nu };y)\), \(\nu =1,2,4,6\).

3 Direct results

Theorem 1

Suppose that \(\zeta \in C_{\varLambda }[0,\infty )\). Then \(\lim_{m\rightarrow \infty }\mathscr{A}_{m,\alpha }^{( \tau )}(\zeta ;y)=\zeta (y)\), uniformly in each compact subset of \([0,\infty )\).

3.1 Voronovskaja type theorem

Theorem 2

Suppose that \(\zeta \in C_{\varLambda }[0,\infty )\). If \(\zeta ''\)exists at a point \(y\in [0,\infty )\), then

$$ \lim_{m\to \infty }m \bigl[\mathscr{A}_{m,\alpha }^{( \tau )}( \zeta ;y)-\zeta (y) \bigr]=2y(\tau -1)\zeta ^{\prime }(y)+ \frac{1}{2} \frac{y (1+y) (1+\alpha )}{\alpha }\zeta ^{ \prime \prime }(y). $$

Proof

Applying Taylor’s expansion, one writes

$$\begin{aligned} \zeta (t)=\zeta (y)+\zeta '(y) (t-y)+ \frac{1}{2}\zeta ''(y) (t-y)^{2}+ \varpi (t,y) (t-y)^{2}, \end{aligned}$$
(3.1)

where \(\lim_{t\rightarrow y}\varpi (t,y)=0\). By using the linearity of the operator \(\mathscr{A}_{m,\alpha }^{(\tau )}\), we get

$$\begin{aligned} \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)-\zeta (y)&= \mathscr{A}_{m, \alpha }^{(\tau )}\bigl((t-y);y\bigr)\zeta '(y)+\frac{1}{2}\mathscr{A}_{m,\alpha }^{( \tau )} \bigl((t-y)^{2};y\bigr)\zeta ''(y) \\ &\quad {}+\mathscr{A}_{m,\alpha }^{(\tau )}\bigl(\varpi (t,y) (t-y)^{2};y\bigr). \end{aligned}$$

By using the Cauchy–Schwarz inequality in the last term of the last inequality, we obtain

$$\begin{aligned} m\mathscr{A}_{m,\alpha }^{(\tau )}\bigl(\varpi (t,y) (t-y)^{2};y\bigr)\leq \sqrt{ \mathscr{A}_{m,\alpha }^{(\tau )} \bigl(\varpi ^{2}(t,y);y\bigr)} \sqrt{m^{2} \mathscr{A}_{m,\alpha }^{(\tau )}\bigl((t-y)^{4};y\bigr)}. \end{aligned}$$
(3.2)

As \(\varpi ^{2}(y,y)=0\) and \(\varpi ^{2}(\cdot ,y)\in C_{\varLambda }[0,\infty )\), we have

$$\begin{aligned} \lim_{m\rightarrow \infty } \mathscr{A}_{m,\alpha }^{( \tau )} \bigl(\varpi ^{2}(t,y);y\bigr)= \varpi ^{2}(y,y)=0. \end{aligned}$$
(3.3)

Combining (3.2)–(3.3) and Remark 1, we have

$$\begin{aligned} \lim_{m\rightarrow \infty }m\mathscr{A}_{m,\alpha }^{( \tau )} \bigl(\varpi (t,y) (t-y)^{2};y\bigr)=0. \end{aligned}$$
(3.4)

Hence

$$ \lim_{m\to \infty }m \bigl[\mathscr{A}_{m,\alpha }^{( \tau )}( \zeta ;y)-\zeta (y) \bigr]=2y(\tau -1)\zeta ^{\prime }(y)+ \frac{1}{2} \frac{y (1+y) (1+\alpha )}{\alpha }\zeta ^{ \prime \prime }(y). $$

 □

Let \(\mu _{1}\geq 0\), \(\mu _{2}>0\) be fixed. We consider Lipschitz-type space (see [32]) as follows:

$$ \mathrm{Lip}_{M}^{(\mu _{1},\mu _{2})}(r):= \biggl\{ \zeta \in C[0,\infty ): \bigl\vert \zeta (t)-\zeta (y) \bigr\vert \leq M \frac{ \vert t-y \vert ^{r}}{(t+\mu _{1}y^{2}+\mu _{2}y)^{\frac{r}{2}}}; y, t \in (0,\infty ) \biggr\} , $$

where \(0< r\leq 1\).

Theorem 3

Let \(\zeta \in \mathrm{Lip}_{M}^{(\mu _{1},\mu _{2})}(r)\)and \(r\in (0,1]\). Then, for all \(y\in (0,\infty )\), we have

$$\begin{aligned} \bigl\vert \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)-\zeta (y) \bigr\vert \leq M \biggl( \frac{\mathscr{F}_{m,\alpha }^{\tau ,2}(y)}{\mu _{1}y^{2}+\mu _{2}y} \biggr)^{\frac{r}{2}}. \end{aligned}$$

Proof

Using Hölder’s inequality with \(p=\frac{2}{r}\), \(q=\frac{2}{2-r}\), we obtain

$$\begin{aligned}& \bigl\vert \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)-\zeta (y) \bigr\vert \\& \quad = \sum_{j=1}^{\infty }p_{m,j}^{(\tau )}(y) \int _{0}^{\infty }l_{m,j}^{\alpha }(t) \bigl\vert \zeta (t)-\zeta (y) \bigr\vert \,dt+p_{m,0}^{(\tau )}(y) \bigl\vert \zeta (0)-\zeta (y) \bigr\vert \\& \quad \leq \sum_{j=1}^{\infty }p_{m,j}^{(\tau )}(y) \biggl( \int _{0}^{\infty }l_{m,j}^{\alpha }(t) \bigl\vert \zeta (t)-\zeta (y) \bigr\vert ^{ \frac{2}{r}}\,dt \biggr)^{\frac{r}{2}} +p_{m,0}^{(\tau )}(y) \bigl\vert \zeta (0)- \zeta (y) \bigr\vert \\& \quad \leq \Biggl\{ \sum_{j=1}^{\infty }p_{m,j}^{(\tau )}(y) \int _{0}^{\infty }l_{m,j}^{\alpha }(t) \bigl\vert \zeta (t) - \zeta (y) \bigr\vert ^{ \frac{2}{r}}\,dt +p_{m,0}^{(\tau )}(y) \bigl\vert \zeta (0)-\zeta (y) \bigr\vert ^{ \frac{2}{r}} \Biggr\} ^{\frac{r}{2}}\\& \qquad {}\times \Biggl( \sum_{j=0}^{\infty }p_{m,j}^{(\tau )}(y) \Biggr)^{\frac{2-r}{2}} \\& \quad = \Biggl\{ \sum_{j=1}^{\infty }p_{m,j}^{(\tau )} \int _{0}^{\infty }l_{m,j}^{\alpha }(t) \bigl\vert \zeta (t)-\zeta (y) \bigr\vert ^{\frac{2}{r}}\,dt+ p_{m,0}^{( \tau )}(y) \bigl\vert \zeta (0)-\zeta (y) \bigr\vert ^{\frac{2}{r}} \Biggr\} ^{\frac{r}{2}} \\& \quad \leq M \Biggl( \sum_{j=1}^{\infty }p_{m,j}^{(\tau )}(y) \int _{0}^{\infty }l_{m,j}^{\alpha }(t) \frac{(t-y)^{2}}{(t+\mu _{1}y^{2}+\mu _{2}y)}\,dt +p_{m,0}^{(\tau )}(y) \frac{y^{2}}{(\mu _{1}y^{2}+\mu _{2}y)} \Biggr)^{\frac{r}{2}} \\& \quad \leq \frac{M}{(\mu _{1}y^{2}+\mu _{2}y)^{\frac{r}{2}}} \Biggl( \sum_{j=1}^{\infty }p_{m,j}^{(\tau )}(y) \int _{0}^{\infty }l_{m,j}^{\alpha }(t) (t-y)^{2}\,dt+y^{2}p_{m,0}^{(\tau )}(y) \Biggr)^{\frac{r}{2}} \\& \quad = \frac{M}{(\mu _{1}y^{2}+\mu _{2}y)^{\frac{r}{2}}} \bigl( \mathscr{A}_{m,\alpha }^{(\tau )} \bigl((t-y)^{2};y\bigr) \bigr)^{\frac{r}{2}} = \frac{M}{(\mu _{1}y^{2}+\mu _{2}y)^{\frac{r}{2}}} \bigl(\mathscr{F}_{m, \alpha }^{\tau ,2}(y)\bigr)^{\frac{r}{2}}. \end{aligned}$$

Thus, the proof is completed. □

4 Weighted approximation

Suppose \(H_{\xi }[0,\infty )\) is the space of all real valued functions on \([0,\infty )\) satisfies the relation \(|\zeta (y)|\leq N_{\zeta }\xi (y)\), where \(\xi (y)=1+y^{2}\) is a weight function and \(N_{\zeta }\) is a positive constant depending only on ζ. Let \(C_{\xi }[0,\infty )\) be the space of all continuous functions in \(H_{\xi }[0,\infty )\) endowed with the norm considered by

$$ \Vert \zeta \Vert _{\xi }:= \sup_{y\in [0,\infty )} \frac{ \vert \zeta (y) \vert }{\xi (y)} $$

and

$$ C_{\xi }^{0}[0,\infty ):= \biggl\{ \zeta \in C_{\xi }[0, \infty ):\lim_{y\rightarrow \infty }\frac{ \vert \zeta (y) \vert }{\xi (y)} \text{ exists and is finite} \biggr\} . $$

Theorem 4

For each \(\zeta \in C_{\xi }^{0}[0,\infty )\)and \(r>0\), we have

$$ \lim_{m\rightarrow \infty } \sup_{y\in [0,\infty )} \frac{ \vert \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)-\zeta (y) \vert }{(1+y^{2})^{1+r}}=0. $$

Proof

Let \(y_{0}>0\) be arbitrary but fixed. Then we get

$$\begin{aligned} \sup_{y\in [0,\infty )} \frac{ \vert \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)-\zeta (y) \vert }{(1+y^{2})^{1+r}} \leq & \sup _{y\leq y_{0}} \frac{ \vert \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)-\zeta (y) \vert }{(1+y^{2})^{1+r}} + \sup_{y>y_{0}} \frac{ \vert \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)-\zeta (y) \vert }{(1+y^{2})^{1+r}} \\ \leq & \sup_{y\leq y_{0}} \bigl\{ \bigl\vert \mathscr{A}_{m,\alpha }^{( \tau )}( \zeta ;y)-\zeta (y) \bigr\vert \bigr\} + \sup_{y>y_{0}} \frac{ \vert \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y) \vert }{(1+y^{2})^{1+r}} \\ &{}+ \sup_{y>y_{0}} \frac{ \vert \zeta (y) \vert }{(1+y^{2})^{1+r}}. \end{aligned}$$

Since \(|\zeta (t)|\leq \|\zeta \|_{\xi }(1+t^{2})\), \(\forall t\geq 0\)

$$\begin{aligned} \sup_{y\in [0,\infty )} \frac{ \vert \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)-\zeta (y) \vert }{(1+y^{2})^{1+r}} \leq & \bigl\Vert \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)-\zeta (y) \bigr\Vert _{C[0,y_{0}]} + \Vert \zeta \Vert _{\xi }\sup _{y>y_{0}} \frac{ \vert \mathscr{A}_{m,\alpha }^{(\tau )}(1+t^{2};y) \vert }{(1+y^{2})^{1+r}} \\ &{}+ \sup_{y>y_{0}} \frac{ \Vert \zeta \Vert _{\xi }}{(1+y^{2})^{r}} \\ =& I_{1}+I_{2}+I_{3},\quad \text{say}. \end{aligned}$$
(4.1)

Applying Theorem 1, therefore for a given \(\epsilon >0\), \(m_{1}\in \mathbb{N}\), such that

$$\begin{aligned} I_{1}= \bigl\Vert \mathscr{A}_{m,\alpha }^{(\tau )}( \zeta ;y)-\zeta (y) \bigr\Vert _{C[0,y_{0}]} < \frac{\epsilon }{3},\quad \text{for all } m\geq m_{1}. \end{aligned}$$
(4.2)

Since \(\lim_{m\to \infty }\sup_{y>y_{0}} \frac{\mathscr{A}_{m,\alpha }^{(\tau )}(1+t^{2};y)}{1+y^{2}}=1\), it follows that \(m_{2}\in \mathbb{N}\) such that

$$ \sup_{y>y_{0}} \frac{\mathscr{A}_{m,\alpha }^{(\tau )}(1+t^{2};y)}{1+y^{2}}\leq \frac{(1+y_{0}^{2})^{r}}{ \Vert \zeta \Vert _{\xi }}\cdot \frac{\epsilon }{3}+1, \quad \text{for all } m\geq m_{2}. $$

Hence,

$$\begin{aligned} I_{2} &= \Vert \zeta \Vert _{\xi }\sup _{y>y_{0}} \frac{ \vert \mathscr{A}_{m,\alpha }^{(\tau )}(1+t^{2};y) \vert }{(1+y^{2})^{1+r}} \leq \frac{ \Vert \zeta \Vert _{\xi }}{(1+y_{0}^{2})^{r}} \sup _{y>y_{0}} \frac{ \vert \mathscr{A}_{m,\alpha }^{(\tau )}(1+t^{2};y) \vert }{1+y^{2}} \\ &\leq \frac{ \Vert \zeta \Vert _{\xi }}{(1+y_{0}^{2})^{r}} + \frac{\epsilon }{3}, \quad \text{for all } m\geq m_{2}. \end{aligned}$$
(4.3)

Let us choose \(y_{0}\) to be so large that

$$\begin{aligned} \frac{ \Vert \zeta \Vert _{\xi }}{(1+y_{0}^{2})^{r}} < \frac{\epsilon }{6}, \end{aligned}$$

then

$$\begin{aligned} I_{3}=\sup_{y>y_{0}} \frac{ \Vert \zeta \Vert _{\xi }}{(1+y^{2})^{r}} \leq \frac{ \Vert \zeta \Vert _{\xi }}{(1+y_{0}^{2})^{r}} < \frac{\epsilon }{6}. \end{aligned}$$
(4.4)

Let \(m_{0}=\max \{m_{1},m_{2}\}\), then by combining (4.2)–(4.4)

$$\begin{aligned} \sup_{y\in [0,\infty )} \frac{ \vert \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)-\zeta (y) \vert }{(1+y^{2})^{1+r}} < \epsilon , \quad \text{for all } m \geq m_{0}. \end{aligned}$$

Hence the proof is done. □

Theorem 5

Let \(\zeta \in C_{\xi }^{0}[0,\infty )\). Then we have

$$\begin{aligned} \lim_{m\rightarrow \infty } \bigl\Vert \mathscr{A}_{m,\alpha }^{( \tau )}(\zeta )-\zeta \bigr\Vert _{\xi }=0. \end{aligned}$$
(4.5)

Proof

To prove (4.5), by [10], it is sufficient to show the following:

$$\begin{aligned} \lim_{m\rightarrow \infty } \bigl\Vert \mathscr{A}_{m,\alpha }^{( \tau )}\bigl(t^{\nu };y \bigr)-e_{\nu } \bigr\Vert _{\xi }=0,\quad \nu =0,1,2. \end{aligned}$$
(4.6)

Since \(\mathscr{A}_{m,\alpha }^{(\tau )}(1;y)=1\), so (4.6) holds true for \(\nu =0\).

From Lemma 1, we obtain

$$ \bigl\Vert \mathscr{A}_{m,\alpha }^{(\tau )}(t;y)-y \bigr\Vert _{\xi }= \sup_{y\geq 0}\frac{1}{1+y^{2}} \biggl\vert y+\frac{2y(\tau -1)}{m}-y \biggr\vert \leq \sup_{y\geq 0} \biggl(\frac{y}{1+y^{2}} \biggr)\frac{2 \vert \tau -1 \vert }{m}. $$
(4.7)

Thus, \(\lim_{m\rightarrow \infty }\|\mathscr{A}_{m,\alpha }^{( \tau )}(t;y)-y\|_{\xi }=0\).

Finally, we obtain

$$\begin{aligned}& \bigl\Vert \mathscr{A}_{m,\alpha }^{(\tau )} \bigl(t^{2};y\bigr)-y^{2} \bigr\Vert _{\xi } \\& \quad = \sup_{y\geq 0}\frac{1}{1 + y^{2}} \biggl\vert \frac{y^{2} (-3+m+4 \tau ) \alpha }{(m \alpha -1) }+ \frac{y (-2+m+2 \tau +(-4+m+4 \tau ) \alpha )}{m (m \alpha -1 )}-y^{2} \biggr\vert \\& \quad \leq \sup_{y\geq 0}\frac{y^{2}}{1+y^{2}} \biggl\vert \frac{(m+m (-3+4 \alpha ) \rho )}{m (m \alpha -1 )} \biggr\vert \\& \qquad {}+ \sup_{y\geq 0}\frac{y}{1+y^{2}} \biggl\vert \frac{(m+m \alpha +2 (\tau -1 ) (1+2 \alpha ))}{m (m\alpha -1 )} \biggr\vert , \end{aligned}$$
(4.8)

which implies that \(\lim_{m\rightarrow \infty }\|\mathscr{A}_{m,\alpha }^{( \tau )}(t^{2};y)-y^{2}\|_{\xi }=0\). □

5 Some Voronoskaja type approximation theorem

To examine the degree of approximation of functions in \(C_{\xi }[0,\infty )\), Yüksel and Ispir [40] presented the weighted modulus of smoothness \(\varOmega (\zeta ;\sigma )\) as follows:

$$\begin{aligned} \varOmega (\zeta ;\sigma ) = \sup_{0\leq h < \sigma , y\in [0,\infty )} \frac{ \vert \zeta (y+h)-\zeta (y) \vert }{(1+h^{2})(1+y^{2})} \end{aligned}$$
(5.1)

for \(\zeta \in C_{\xi }[0,\infty )\). It was proved in [40] that, if \(\zeta \in C_{\xi }^{0}[0,\infty )\), then \(\varOmega (\cdot;\sigma )\) has the properties

$$ \lim_{\sigma \to 0}\varOmega (\zeta ;\sigma ) = 0 $$

and

$$\begin{aligned} \varOmega (\zeta ;\lambda \sigma )\leq 2(1+\lambda ) \bigl(1+\sigma ^{2}\bigr) \varOmega (\zeta ;\sigma ), \quad \lambda >0. \end{aligned}$$
(5.2)

For \(\zeta \in C_{\xi }^{0}[0,\infty )\), it follows from (5.1) and (5.2) that

$$\begin{aligned} \bigl\vert \zeta (t)-\zeta (y) \bigr\vert \leq & \bigl(1+(t-y)^{2} \bigr) \bigl(1+y^{2}\bigr) \varOmega \bigl( \zeta ; \vert t-y \vert \bigr) \\ \leq & 2 \biggl(1+\frac{ \vert t-y \vert }{\sigma } \biggr) \bigl(1+\sigma ^{2} \bigr) \varOmega (\zeta ;\sigma ) \bigl(1+(t-y)^{2} \bigr) \bigl(1+y^{2}\bigr). \end{aligned}$$
(5.3)

In the next theorem, we compute the degree of approximation of ζ by the operator \(\mathscr{A}_{m,\alpha }^{(\tau )}\) in the weighted space of continuous functions \(C_{\xi }^{0}[0,\infty )\) in terms of the weighted modulus of smoothness \(\varOmega (\cdot;\sigma )\), \(\sigma >0\).

5.1 Quantitative Voronovskaya type theorem

Theorem 6

Suppose that \(\zeta \in C_{\xi }^{0}[0,\infty )\)such that \(\zeta '(y), \zeta ''(y)\in C_{\xi }^{0}[0,\infty )\). Then, for sufficiently large m and each \(y\in [0,\infty )\),

$$\begin{aligned}& \biggl\vert m \biggl\{ \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)- \zeta (y)- \zeta '(y)\mathscr{A}_{m,\alpha }^{(\tau )} \bigl((t-y);y \bigr) - \frac{\zeta ''(y)}{2!} \mathscr{A}_{m,\alpha }^{(\tau )} \bigl((t-y)^{2};y \bigr) \biggr\} \biggr\vert \\& \quad = O(1) \varOmega \bigl( \zeta '';\sqrt{1/m} \bigr). \end{aligned}$$

Proof

By Taylor’s formula

$$\begin{aligned} \zeta (t) &= \zeta (y) + \zeta '(y) (t-y) + \frac{\zeta ''(\eta )}{2!} (t-y)^{2} \\ &= \zeta (y) + \zeta '(y) (t-y) + \frac{\zeta ''(y)}{2!} (t-y)^{2} + h_{2}(t,y), \end{aligned}$$
(5.4)

where \(\eta \in (y,t)\) and hence

$$\begin{aligned} h_{2}(t,y) = \frac{\zeta ''(\eta )-\zeta ''(y)}{2!} (t-y)^{2}. \end{aligned}$$
(5.5)

In view of the inequality (5.3) of the weighted modulus of continuity, we obtain

$$\begin{aligned} \bigl\vert {\zeta ''(\eta )-\zeta ''(y)} \bigr\vert &\leq \bigl(1+(\eta -y)^{2}\bigr) \bigl(1+y^{2}\bigr) \varOmega \bigl(\zeta ''; \vert \eta -y \vert \bigr) \\ &\leq \bigl(1+(t-y)^{2}\bigr) \bigl(1+y^{2}\bigr) \varOmega \bigl(\zeta ''; \vert t-y \vert \bigr) \\ &\leq 2\bigl(1+(t-y)^{2}\bigr) \bigl(1+y^{2}\bigr) \biggl(1+\frac{ \vert t-y \vert }{\sigma } \biggr) \bigl(1+ \sigma ^{2}\bigr) \varOmega \bigl(\zeta '';\sigma \bigr), \end{aligned}$$
(5.6)

but

$$\begin{aligned} \biggl(1+\frac{ \vert t-y \vert }{\sigma } \biggr) \bigl(1+(t-y)^{2}\bigr) \leq \textstyle\begin{cases} 2(1+\sigma ^{2}), & \vert t-y \vert < \sigma , \\ 2\frac{(t-y)^{4}}{\sigma ^{4}}(1+\sigma ^{2}), & \vert t-y \vert \geq \sigma , \end{cases}\displaystyle \end{aligned}$$

that is,

$$\begin{aligned} \biggl(1+\frac{ \vert t-y \vert }{\sigma } \biggr) \bigl(1+(t-y)^{2}\bigr) \leq 2 \biggl(1+ \frac{(t-y)^{4}}{\sigma ^{4}} \biggr) \bigl(1+\sigma ^{2}\bigr). \end{aligned}$$
(5.7)

Combining (5.5)–(5.7) and choosing \(0<\sigma <1\), we obtain

$$\begin{aligned} \bigl\vert h_{2}(t,y) \bigr\vert \leq 2 \bigl(1+\sigma ^{2}\bigr)^{2}\bigl(1+y^{2}\bigr) \varOmega \bigl(\zeta ''; \sigma \bigr) \biggl(1+ \frac{(t-y)^{4}}{\sigma ^{4}} \biggr) (t-y)^{2}. \end{aligned}$$
(5.8)

Operating \(\mathscr{A}_{m,\alpha }^{(\tau )}\) and Lemma 2 on both sides of (5.4), we get

$$\begin{aligned} &\biggl\vert \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)- \zeta (y) - \zeta '(y) \mathscr{A}_{m,\alpha }^{(\tau )}(t-y;y) - \frac{\zeta ''(y)}{2!} \mathscr{A}_{m,\alpha }^{(\tau )} \bigl((t-y)^{2};y \bigr) \biggr\vert \\ &\quad \leq \mathscr{A}_{m,\alpha }^{(\tau )} \bigl( \bigl\vert h_{2}(t,y) \bigr\vert ;y\bigr). \end{aligned}$$
(5.9)

Applying Remark 1 and using Eq. (5.8), we get

$$\begin{aligned} &\mathscr{A}_{m,\alpha }^{(\tau )} \bigl( \bigl\vert h_{2}(t,y) \bigr\vert ;y \bigr) \\ &\quad \leq 2\bigl(1+ \sigma ^{2}\bigr)^{2}\bigl(1+y^{2}\bigr)\varOmega \bigl( \zeta '';\sigma \bigr)\mathscr{A}_{m, \alpha }^{(\tau )} \biggl( \biggl((t-y)^{2}+ \frac{(t-y)^{6}}{\sigma ^{4}} \biggr);y \biggr) \\ &\quad = 2\bigl(1+\sigma ^{2}\bigr)^{2}\bigl(1+y^{2} \bigr)\varOmega \bigl(\zeta '';\sigma \bigr) \biggl( \mathscr{A}_{m,\alpha }^{(\tau )} \bigl((t-y)^{2};y \bigr) + \frac{1}{\sigma ^{4}}\mathscr{A}_{m,\alpha }^{(\tau )} \bigl((t-y)^{6};y \bigr) \biggr) \\ &\quad = 2\bigl(1+\sigma ^{2}\bigr)^{2}\bigl(1+y^{2} \bigr)\varOmega \bigl(\zeta '';\sigma \bigr) \biggl( O(1/m) + \frac{1}{\sigma ^{4}}O\bigl(1/m^{3}\bigr) \biggr). \end{aligned}$$

By choosing \(\sigma = \sqrt{1/m}\), we get

$$\begin{aligned} m \mathscr{A}_{m,\alpha }^{(\tau )} \bigl( \bigl\vert h_{2}(t,y) \bigr\vert ;y \bigr) = O(1) \varOmega \bigl(\zeta '';\sqrt{1/m} \bigr). \end{aligned}$$
(5.10)

Hence, from (5.9) and (5.10), we get

$$\begin{aligned}& \biggl\vert m \biggl\{ \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y)- \zeta (y) \zeta '(y) \mathscr{A}_{m,\alpha }^{(\tau )}(t-y;y)- \frac{\zeta ''(y)}{2!} \mathscr{A}_{m,\alpha }^{(\tau )} \bigl((t-y)^{2};y \bigr) \biggr\} \biggr\vert \\& \quad = O(1) \varOmega \bigl(\zeta ''; \sqrt{1/m} \bigr),\quad \text{as } m\to \infty . \end{aligned}$$

This completes the proof. □

5.2 Grüss Voronovskaya type theorem

Theorem 7

Suppose that ζ, g and \(\zeta g\in C_{\xi }^{0}[0,\infty )\)such that \(\zeta '\), \(g'\), \((\zeta g)'\), \(\zeta ''\), \(g''\)and \((\zeta g)'' \in C_{\xi }^{0}[0,\infty )\). Then, for each \(y\in [0,\infty )\),

$$\begin{aligned} \lim_{m\to \infty } m \bigl\{ \mathscr{A}_{m,\alpha }^{(\tau )} \bigl(( \zeta g);y\bigr)-\mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y) \mathscr{A}_{m, \alpha }^{(\tau )}(g;y) \bigr\} = \zeta '(y)g'(y) \frac{y (1+y) (1+\alpha )}{\alpha }. \end{aligned}$$

Proof

Since \((\zeta g)(y)=\zeta (y)g(y)\), \((\zeta g)'(y) = \zeta '(y)g(y) + \zeta (y)g'(y) \) and \((\zeta g)''(y) = \zeta ''(y)g(y) + 2\zeta '(y)g'(y) + \zeta (y)g''(y)\), we may write

$$\begin{aligned}& \mathscr{A}_{m,\alpha }^{(\tau )}\bigl((\zeta g);y\bigr)- \mathscr{A}_{m,\alpha }^{( \tau )}(\zeta ;y)\mathscr{A}_{m,\alpha }^{(\tau )}(g;y) \\& \quad = \biggl\{ \mathscr{A}_{m,\alpha }^{(\tau )}\bigl((\zeta g);y\bigr) - \zeta (y).g(y) - (\zeta g)'(y)\mathscr{A}_{m,\alpha }^{(\tau )}(t-y;y) - \frac{(\zeta g)''(y)}{2!} \mathscr{A}_{m,\alpha }^{(\tau )} \bigl((t-y)^{2};y\bigr) \biggr\} \\& \qquad {}- g(y) \biggl\{ \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y) - \zeta (y) - \zeta '(y)\mathscr{A}_{m,\alpha }^{(\tau )}(t-y;y) - \frac{\zeta ''(y)}{2!}\mathscr{A}_{m,\alpha }^{(\tau )}\bigl((t-y)^{2};y \bigr) \biggr\} \\& \qquad {}- \mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y) \biggl\{ \mathscr{A}_{m, \alpha }^{(\tau )}(g;y) - g(y) - g'(y) \mathscr{A}_{m,\alpha }^{(\tau )}(t-y;y) - \frac{g''(y)}{2!} \mathscr{A}_{m,\alpha }^{(\tau )}\bigl((t-y)^{2};y\bigr) \biggr\} \\& \qquad {}+ \frac{1}{2!}\mathscr{A}_{m,\alpha }^{(\tau )} \bigl((t-y)^{2};y\bigr) \bigl\{ \zeta (y)g''(y)+2 \zeta '(y)g'(y) - g''(y) \mathscr{A}_{m,\alpha }^{( \tau )}(\zeta ;y) \bigr\} \\& \qquad {}+ \mathscr{A}_{m,\alpha }^{(\tau )}(t-y;y) \bigl\{ \zeta (y)g'(y) - g'(y) \mathscr{A}_{m,\alpha }^{(\tau )}( \zeta ;y) \bigr\} . \end{aligned}$$

Now, by using Lemma 2 and Theorems 1 and 6, we get

$$\begin{aligned} \lim_{m\to \infty } m \bigl\{ \mathscr{A}_{m,\alpha }^{(\tau )} \bigl(( \zeta g);y\bigr)-\mathscr{A}_{m,\alpha }^{(\tau )}(\zeta ;y) \mathscr{A}_{m, \alpha }^{(\tau )}(g;y) \bigr\} = \zeta '(y)g'(y) \frac{y (1+y) (1+\alpha )}{\alpha }, \end{aligned}$$

which proves our theorem. □

References

  1. Acar, T.: Asymptotic formulas for generalized Szász–Mirakyan operators. Appl. Math. Comput. 263, 233–239 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Acar, T., Mohiuddine, S.A., Mursaleen, M.: Approximation by \((p,q)\)-Baskakov–Durrmeyer–Stancu operators. Complex Anal. Oper. Theory 12, 1453–1468 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Acu, A.M., Gupta, V.: Direct results for certain summation-integral type Baskakov–Szász operators. Results Math. 72, 1161–1180 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Acu, A.M., Hodiş, S., Raşa, I.: A survey on estimates for the differences of positive linear operators. Constr. Math. Anal. 1(2), 113–127 (2018)

    Google Scholar 

  5. Ansari, K.J., Mursaleen, M., Rahman, S.: Approximation by Jakimovski–Leviatan operators of Durrmeyer type involving multiple Appell polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(2), 1007–1024 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ansari, K.J., Rahman, S., Mursaleen, M.: Approximation and error estimation by modified Păltănea operators associating Gould–Hopper polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2827–2851 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Aral, A., Erbay, H.: Parametric generalization of Baskakov operators. Math. Commun. 24, 119–131 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Baskakov, V.A.: A sequence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk SSSR 113, 249–251 (1957)

    MathSciNet  MATH  Google Scholar 

  9. Chen, X., Tan, J., Liu, Z., Xie, J.: Approximation of functions by a new family of generalized Bernstein operators. J. Math. Anal. Appl. 450, 244–261 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gadjiev, A.D., On, P.P.: Korovkin type theorems. Math. Notes 20(5), 781–786 (1976)

    Google Scholar 

  11. Goyal, M., Gupta, V., Agrawal, P.N.: Quantitative convergence results for a family of hybrid operators. Appl. Math. Comput. 271, 893–904 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Goyal, M., Kajla, A.: Blending-type approximation by generalized Lupaş–Durrmeyer-type operators. Bol. Soc. Mat. Mex. 25, 551–566 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gupta, V.: Direct estimates for a new general family of Durrmeyer type operators. Boll. Unione Mat. Ital. 7(4), 279–288 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gupta, V., Rassias, M.T.: Moments of Linear Positive Operators and Approximation. Springer, Berlin (2019)

    Book  MATH  Google Scholar 

  15. Gupta, V., Rassias, T.M.: Direct estimates for certain Szász type operators. Appl. Math. Comput. 251, 469–474 (2015)

    MathSciNet  MATH  Google Scholar 

  16. İlarslan, H.G.İ., Erbay, H., Aral, A.: Kantorovich-type generalization of parametric Baskakov operators. Math. Methods Appl. Sci. 42, 6580–6587 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kadak, U., Mohiuddine, S.A.: Generalized statistically almost convergence based on the difference operator which includes the \((p,q)\)-gamma function and related approximation theorems. Results Math. 73, 9 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Kajla, A., Acar, T.: A new modification of Durrmeyer type mixed hybrid operators. Carpath. J. Math. 34, 47–56 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Kajla, A., Acu, A.M., Agrawal, P.N.: Baskakov–Szász type operators based on inverse Pólya–Eggenberger distribution. Ann. Funct. Anal. 8, 106–123 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kajla, A., Agrawal, P.N.: Szász–Durrmeyer type operators based on Charlier polynomials. Appl. Math. Comput. 268, 1001–1014 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Kilicman, A., Mursaleen, M.A., Al-Abied, A.A.H.A.: Stancu type Baskakov–Durrmeyer operators and approximation properties. Mathematics 8, 1164 (2020). https://doi.org/10.3390/math8071164

    Article  Google Scholar 

  22. Mohiuddine, S.A., Acar, T., Alotaibi, A.: Construction of a new family of Bernstein–Kantorovich operators. Math. Methods Appl. Sci. 40, 7749–7759 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mohiuddine, S.A., Alamri, B.A.S.: Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 1955–1973 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mohiuddine, S.A., Asiri, A., Hazarika, B.: Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems. Int. J. Gen. Syst. 48(5), 492–506 (2019)

    Article  MathSciNet  Google Scholar 

  25. Mohiuddine, S.A., Hazarika, B., Alghamdi, M.A.: Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems. Filomat 33(14), 4549–4560 (2019)

    Article  MathSciNet  Google Scholar 

  26. Mohiuddine, S.A., Özger, F.: Approximation of functions by Stancu variant of Bernstein–Kantorovich operators based on shape parameter α. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, 70 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mursaleen, M., Al-Abeid, A.A.H., Ansari, K.J.: Approximation by Jakimovski–Leviatan–Păltănea operators involving Sheffer polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(2), 1251–1265 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mursaleen, M., Rahman, S., Ansari, K.J.: Approximation by generalized Stancu type integral operators involving Sheffer polynomials. Carpath. J. Math. 34(2), 215–228 (2018)

    MathSciNet  MATH  Google Scholar 

  29. Mursaleen, M., Rahman, S., Ansari, K.J.: On the approximation by Bézier–Pǎltǎnea operators based on Gould–Hopper polynomials. Math. Commun. 24, 147–164 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Mursaleen, M., Rahman, S., Ansari, K.J.: Approximation by Jakimovski–Leviatan–Stancu–Durrmeyer type operators. Filomat 33, 1517–1530 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nasiruzzaman, M., Rao, N., Wazir, S., Kumar, R.: Approximation on parametric extension of Baskakov–Durrmeyer operators on weighted spaces. J. Inequal. Appl. 2019, 103 (2019)

    Article  MathSciNet  Google Scholar 

  32. Özarslan, M.A., Aktuǧlu, H.: Local approximation properties for certain King type operators. Filomat 27(1), 173–181 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Özger, F., Srivastava, H.M., Mohiuddine, S.A.: Approximation of functions by a new class of generalized Bernstein–Schurer operators. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, 173 (2020)

    Article  MathSciNet  Google Scholar 

  34. Srivastava, H.M., Gupta, V.: A certain family of summation-integral type operators. Math. Comput. Model. 37(12–13), 1307–1315 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Srivastava, H.M., Gupta, V.: Rate of convergence for the Bézier variant of the Bleimann–Butzer–Hahn operators. Appl. Math. Lett. 18, 849–857 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Srivastava, H.M., Ícoz, G., Çekim, B.: Approximation properties of an extended family of the Szász–Mirakjan–Beta-type operators. Axioms 8, Article ID 111 (2019)

    Article  Google Scholar 

  37. Srivastava, H.M., Mursaleen, M., Alotaibi, A.M., Nasiruzzaman, Md., Al-Abied, A.A.H.: Some approximation results involving the q-Szász–Mirakjan–Kantorovich type operators via Dunkl’s generalization. Math. Methods Appl. Sci. 40, 5437–5452 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Srivastava, H.M., Özger, F., Mohiuddine, S.A.: Construction of Stancu-type Bernstein operators based on Bézier bases with shape parameter λ. Symmetry 11, Article ID 316 (2019)

    Article  MATH  Google Scholar 

  39. Srivastava, H.M., Zeng, X.M.: Approximation by means of the Szász–Bézier integral operators. Int. J. Pure Appl. Math. 14(3), 283–294 (2004)

    MathSciNet  MATH  Google Scholar 

  40. Yüksel, I., Ispir, N.: Weighted approximation by a certain family of summation integral-type operators. Comput. Math. Appl. 52(10–11), 1463–1470 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (RG-36-130-38). The authors, therefore, acknowledge with thanks DSR for technical and financial support.

Availability of data and materials

Not applicable.

Funding

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (RG-36-130-38).

Author information

Authors and Affiliations

Authors

Contributions

The authors contributed equally and significantly in writing this paper. The authors read and approved the final manuscript.

Corresponding author

Correspondence to M. Mursaleen.

Ethics declarations

Competing interests

The authors declare they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mohiuddine, S.A., Kajla, A., Mursaleen, M. et al. Blending type approximation by τ-Baskakov-Durrmeyer type hybrid operators. Adv Differ Equ 2020, 467 (2020). https://doi.org/10.1186/s13662-020-02925-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-020-02925-1

MSC

Keywords