Skip to main content

Generalization of Szász–Mirakjan–Kantorovich operators using multiple Appell polynomials

Abstract

The purpose of the present paper is to introduce and study a sequence of positive linear operators defined on suitable spaces of measurable functions on \([0,\infty )\) and continuous function spaces with polynomial weights. These operators are Kantorovich type generalization of Jakimovski–Leviatan operators based on multiple Appell polynomials. Using these operators, we approximate suitable measurable functions by knowing their mean values on a sequence of subintervals of \([0,\infty )\) that do not constitute a subdivision of it. We also discuss the rate of convergence of these operators using moduli of smoothness.

Introduction

A multiple polynomial system [10] \(\{s_{n_{1},n_{2}}(x)\}\) is called multiple Appell if it has a generating function of the form

$$\begin{aligned} H(t_{1},t_{2})e^{x(t_{1}+t_{2})}= \sum _{n_{1}=0}^{ \infty } \sum_{n_{2}=0}^{\infty } \frac{s_{n_{1},n_{2}}(x)}{n_{1}!n_{2}!}t_{1}^{n_{1}}t_{2}^{n_{2}}, \end{aligned}$$
(1.1)

where \(H(t_{1},t_{2})\) has a series expansion

$$\begin{aligned} H(t_{1},t_{2})= \sum _{n_{1}=0}^{\infty } \sum_{n_{2}=0}^{\infty } \frac{a_{n_{1},n_{2}}}{n_{1}!n_{2}!}t_{1}^{n_{1}}t_{2}^{n_{2}}, \end{aligned}$$
(1.2)

with \(H(0,0)=a_{0,0} \neq 0\) and \(\frac{a_{n_{1},n_{2}}}{H(1,1)}\geq 0\) for all \(n_{1},n_{2}\in \mathbb{N}\). Also, (1.1) and (1.2) converge for \(\vert t_{1} \vert \leq R_{1}\), \(\vert t_{2} \vert \leq R_{2}\) (\(R_{1},R_{2}>1\)). \(s_{n_{1},n_{2}}\) is a multiple polynomial system, and for every \(n_{1}+n_{2}\geq 1\), this satisfies the following relationship:

$$\begin{aligned} s'_{n_{1},n_{2}}(x)=n_{1}s_{n_{1}-1,n_{2}}(x)+n_{2}s_{n_{1},n_{2}-1}(x). \end{aligned}$$

Also, for the multiple Appell polynomial systems \(s_{n_{1},n_{2}}(x)\), there exists a sequence \(\{a_{n_{1},n_{2}}\}_{n_{1},n_{2}=0}^{\infty }\) with \(a_{0,0}\neq 0\) such that

$$\begin{aligned} s_{n_{1},n_{2}}(x)= \sum_{k_{1}=0}^{n_{1}} \sum_{k_{2}=0}^{n_{2}}\binom{n_{1}}{k_{1}} \binom{n_{2} }{k_{2}}a_{n_{1}-k_{1},n_{2}-k_{2}}x^{k_{1}+k_{2}}. \end{aligned}$$
(1.3)

Therefore, \(s_{n_{1},n_{2}}(x)\) is a polynomial in x of degree \(n_{1}+n_{2}\).

Using these Appell polynomials, Varma [18] defined a generalization of Szász operators [17] as follows:

$$\begin{aligned} S_{n}(f;x)=\frac{e^{-nx}}{H(1,1)} \sum _{n_{1}=0}^{ \infty } \sum_{n_{2}=0}^{\infty } \frac{s_{n_{1},n_{2}} (\frac{nx}{2})}{n_{1}!n_{2}!}f \biggl(\frac{n_{1}+n_{2}}{n} \biggr). \end{aligned}$$
(1.4)

He also defined the Kantorovich type modification of these operators as follows:

$$\begin{aligned} K_{n}^{*}(f;x)=\frac{ne^{-nx}}{H(1,1)} \sum_{n_{1}=0}^{ \infty } \sum _{n_{2}=0}^{\infty } \frac{s_{n_{1},n_{2}}(\frac{nx}{2})}{n_{1}!n_{2}!} \int _{ \frac{n_{1}+n_{2}}{n}}^{\frac{n_{1}+n_{2}+1}{n}}f(u)\,du, \end{aligned}$$
(1.5)

and obtained the rate of convergence of these operators in terms of the classical modulus of continuity. Alternatively, the operator given by (1.5) may be expressed as

$$\begin{aligned} K_{n}^{*}(f;x)= \int _{0}^{\infty }K_{n}(x,u)f(u)\,du, \end{aligned}$$

where \(K_{n}(x,u)=\frac{ne^{-nx}}{H(1,1)} \sum_{n_{1}=0}^{ \infty } \sum_{n_{2}=0}^{\infty } \frac{s_{n_{1},n_{2}}(\frac{nx}{2})}{n_{1}!n_{2}!} \chi _{[ \frac{n_{1}+n_{2}}{n},\frac{n_{1}+n_{2}+1}{n}]}(u)\), \(\chi _{[\frac{n_{1}+n_{2}}{n},\frac{n_{1}+n_{2}+1}{n}]}(u)\) being the characteristic function of \([\frac{n_{1}+n_{2}}{n},\frac{n_{1}+n_{2}+1}{n} ]\) on \([0,\infty )\).

The purpose of the present paper is to make a generalization of these operators that extends to the unbounded setting an idea given in [6] where the authors studied a modification of Kantorovich operators. We refer the reader to some of the related papers [1, 2, 79, 1114, 16].

In this paper we will study the following sequence \((P_{n})\) of positive linear operators:

$$\begin{aligned} P_{n}(h;x)=\frac{ne^{-nx}}{(b_{n}-c_{n})H(1,1)} \sum _{n_{1}=0}^{ \infty } \sum_{n_{2}=0}^{\infty } \frac{s_{n_{1},n_{2}} (\frac{nx}{2})}{n_{1}!n_{2}!} \int _{\frac{n_{1}+n_{2}+c_{n}}{n}}^{\frac{n_{1}+n_{2}+b_{n}}{n}}h(u)\,du\quad (n \geq 1, x\geq 0). \end{aligned}$$
(1.6)

For every \(h\in \Im [0,\infty )\) (the space of all Borel measurable locally integrable functions \(g:[0,\infty )\rightarrow \mathbb{R}\) such that the antiderivative \(G(x)=\int _{0}^{x} g(t) \,dt\) (\(x\geq 0\)) belongs to \(\ell ([0,\infty ))\), the space of all functions \(b:[0,\infty )\rightarrow \mathbb{R}\) such that \(\vert b(x) \vert \leq Me^{rx}\) (\(x\geq 0\)) for some \(M\geq 0\) and \(r\in \mathbb{R}\)).

Here \((c_{n})_{n\geq 1}\) and \((b_{n})_{n\geq 1}\) are two sequences of real numbers satisfying \(0\leq c_{n}< b_{n}\leq 1\) for every \(n\geq 1\). If \(c_{n}=0\) and \(b_{n}=1\) for all \(n\geq 1\), then the \(P_{n}\)’s (operators in (1.6) turn into (1.5). By using \(P_{n}\)’s we can reconstruct some suitable continuous or integrable functions by knowing their mean values on subinterval of \([0,\infty )\) which do not necessarily constitute a subdivision of \([0,\infty )\).

We will study the approximation properties of \((P_{n})\) for every \(n\geq 1\) on several continuous and weighted continuous function spaces as well as on Lebesgue spaces. We also discuss the rate of convergence of these operators by using appropriate moduli of smoothness.

Generalizing Jakimovski–Leviatan operators

Throughout this paper the following notations are used:

\(\wp [0,\infty )\)::

The space of all continuous real-valued functions on \([0,\infty )\).

\(\wp _{b}[0,\infty )\)::

The subspace of all functions in \(\wp [0,\infty )\) which are bounded. This space endowed with the sup-norm and the natural pointwise ordering is a Banach lattice.

\(\wp _{\ast }[0,\infty )\)::

The space of all continuous functions converging at infinity. This space is a Banach sublattice of \(\wp _{b}([0,\infty ))\).

\(\wp _{0}[0,\infty )\)::

subspace of \(\wp _{\ast }[0,\infty )\), consisting of all those functions that vanish at infinity.

Moreover, for every \(m\geq 1\), we set \(r_{m}(y)=(1+y^{m})^{-1}\) (\(y\geq 0\)) and

$$\begin{aligned} G_{m}:= \Bigl\{ h\in \wp [0,\infty )\big| \sup_{y\geq 0}r_{m}(y) \bigl\vert h(y) \bigr\vert \in \mathbb{R} \Bigr\} ; \end{aligned}$$

\(G_{m}\) is a Banach lattice endowed with the pointwise ordering and the weighted norm

$$\begin{aligned} \Vert h \Vert _{m}:= \sup_{y\geq 0}r_{m}(y) \bigl\vert h(y) \bigr\vert \quad (h \in G_{m}). \end{aligned}$$

The following space is the Banach sublattices of \(G_{m}\):

$$\begin{aligned} G_{m}^{\ast }:= \Bigl\{ h\in G_{m}\big| \lim _{y\rightarrow \infty } r_{m}(y)h(y)\in \mathbb{R} \Bigr\} . \end{aligned}$$

For a given \(g\in \Im ([0,\infty ))\) and \(G(x)=\int _{0}^{x} g(t)\,dt\), the antiderivative of \(g(x)\), operators given in (1.6) may be expressed as

$$\begin{aligned} P_{n}(g;x) =&\frac{ne^{-nx}}{(b_{n}-c_{n})H(1,1)} \\ &{}\times\sum _{n_{1}=0}^{ \infty } \sum_{n_{2}=0}^{\infty } \frac{s_{n_{1},n_{2}} (\frac{nx}{2})}{n_{1}!n_{2}!} \biggl[G \biggl(\frac{n_{1}+n_{2}+b_{n}}{n} \biggr)-G \biggl( \frac{n_{1}+n_{2}+c_{n}}{n} \biggr) \biggr] \end{aligned}$$
(2.1)
$$\begin{aligned} =&\frac{n}{(b_{n}-c_{n})}S_{n}\bigl(\tau _{n}(G)\bigr) (x), \end{aligned}$$
(2.2)

where \(S_{n}\) is given by (1.4) and the mapping \(\tau _{n}\) is defined as

$$\begin{aligned} \tau _{n}(G) (x):=G \biggl(\frac{n_{1}+n_{2}+b_{n}}{n} \biggr)-G \biggl( \frac{n_{1}+n_{2}+c_{n}}{n} \biggr)\quad (x \geq 0), \end{aligned}$$

\(P_{n}(g)\) can also be written as

$$\begin{aligned} P_{n}(g) (x)= \int _{0}^{+\infty }g \,d\mu _{n,x}\quad (n \geq 1,x\geq 0), \end{aligned}$$
(2.3)

where

$$\begin{aligned} \mu _{n,x}=\frac{ne^{-nx}}{(b_{n}-c_{n})H(1,1)} \sum_{n_{1}=0}^{ \infty } \sum_{n_{2}=0}^{\infty } \frac{s_{n_{1},n_{2}} (\frac{nx}{2})}{n_{1}!n_{2}!}\mu _{n,k}, \end{aligned}$$

and each \(\mu _{n,k}\) denotes the Borel measure on \([0,\infty )\) having density, the characteristic function of \([\frac{n_{1}+n_{2}+c_{n}}{n}, \frac{n_{1}+n_{2}+b_{n}}{n} ]\) w.r.t. the Borel–Lebesgue measure on \([0,\infty )\). Throughout the paper the symbol \(c_{m}\) denotes the function \(x^{m}\) by setting \(c_{m}(x)=x^{m}\) for every \(m\geq 0\) and for every \(x\geq 0\). Particularly \(c_{0}=\mathbf{1}\), where 1 denotes the constant function on \([0,\infty )\) of constant value 1. Finally, we shall set \(\rho _{x}(y)=y-x\) (\(y\geq 0\)) and x is a fixed nonnegative real number.

Preliminaries

Lemma 1

For operators (1.6), the estimates of moments are as follows [18]:

$$\begin{aligned}& (\mathrm{i}) \quad P_{n}(1;y)=1; \\& (\mathrm{ii}) \quad P_{n}(t;y)=y+ \frac{1}{2} \frac{b_{n}+c_{n}}{n}+ \frac{H_{t_{1}}(1,1)+H_{t_{2}}(1,1)}{nH(1,1)}; \\& \begin{aligned} (\mathrm{iii}) \quad P_{n}\bigl(t^{2};y \bigr)={}&y^{2}+ \frac{y}{n} \biggl(1+ \frac{b_{n}+c_{n}}{n}+2 \frac{(H_{t_{1}}(1,1)+H_{t_{2}}(1,1))}{H(1,1)} \biggr) +\frac{b_{n}^{2}+c_{n}^{2}+b_{n}c_{n}}{3n^{2}} \\ &{}\times \frac{1}{n^{2}H(1,1)}\bigl\{ H_{t_{1}}(1,1) (1+b_{n}+c_{n})+H_{t_{2}}(1,1) (1+b_{n}+c_{n}) \\ &{}+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr\} ; \end{aligned} \\& (\mathrm{iv}) \quad P_{n}\bigl(\rho _{y}(t);y\bigr)= \frac{1}{2}\frac{b_{n}+c_{n}}{n}+ \frac{H_{t_{1}}(1,1)+H_{t_{2}}(1,1)}{nH(1,1)}; \\& \begin{aligned} (\mathrm{v}) \quad P_{n}\bigl(\rho _{y}^{2}(t);y\bigr)= {}&\frac{y}{n}+ \frac{1}{n^{2}H(1,1)}\bigl\{ H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1) \\ &{}+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr\} \\ &{}+ \frac{b_{n}^{2}+c_{n}^{2}+b_{n}c_{n}}{n^{2}}+\frac{1}{n}(b_{n}+c_{n}) \biggl(\frac{H_{t_{1}}(1,1)+H_{t_{2}}(1,1)}{nH(1,1)} \biggr). \end{aligned} \end{aligned}$$

Proposition 1

For every\(\rho >0\), let\(f_{\rho }(x)=e^{-\rho x}\)where\(x\geq 0\). Then

$$\begin{aligned} S_{n}(f_{\rho }) (x)= \frac{H(e^{\frac{-\rho }{n}},e^{\frac{-\rho }{n}})}{H(1,1)}\exp \bigl(nx \bigl(e^{\frac{-\rho }{n}}-1 \bigr) \bigr). \end{aligned}$$
(3.1)

Proof

$$\begin{aligned} S_{n}(f_{\rho }) (x)= \frac{e^{-nx}}{H(1,1)} \sum_{n_{1}=0}^{ \infty } \sum _{n_{2}=0}^{\infty } \frac{s_{n_{1},n_{2}} (\frac{nx}{2})}{n_{1}!n_{2}!}e^{-\rho (\frac{n_{1}+n_{2}}{n} )}. \end{aligned}$$
(3.2)

Putting \(x= \frac{nx}{2}\) and \(t_{1}=t_{2}=e^{(\frac{-\rho }{n})}\) in (1.1), we get

$$\begin{aligned} \sum_{n_{1}=0}^{\infty } \sum_{n_{2}=0}^{ \infty }\frac{s_{n_{1},n_{2}} (\frac{nx}{2})}{n_{1}!n_{2}!}e^{-\rho (\frac{n_{1}+n_{2}}{n} )}=H \bigl(e^{\frac{-\rho }{n}},e^{\frac{-\rho }{n}}\bigr)e^{\frac{nx}{2} ( e^{\frac{-\rho }{n}}+e^{\frac{-\rho }{n}} )}. \end{aligned}$$
(3.3)

Using (3.3) in (3.2), we have the result. □

Proposition 2

$$\begin{aligned} P_{n}(f_{\rho })=\frac{n}{\rho (b_{n}-c_{n})} \bigl(e^{ \frac{-\rho c_{n}}{n}}-e^{\frac{-\rho b_{n}}{n}} \bigr)S_{n}(f_{\rho }) \end{aligned}$$
(3.4)

for every\(n\geq 1\).

Moreover, for every\(n\geq 1\)and\(\rho >0\),

$$\begin{aligned} P_{n}(f_{\rho })\leq S_{n}(f_{\rho }). \end{aligned}$$
(3.5)

Proof

(3.4) holds after a straightforward computation, and the proof of (3.5) is as follows:

$$\begin{aligned} \frac{n}{\rho (b_{n}-c_{n})} \bigl(e^{\frac{-\rho c_{n}}{n}}-e^{ \frac{-\rho b_{n}}{n}} \bigr)\leq \frac{n}{\rho (b_{n}-c_{n})} \bigl(1-e^{-( \frac{\rho b_{n}}{n}-\frac{\rho c_{n}}{n})} \bigr)\leq 1 \end{aligned}$$

since \(1-e^{-x}\leq x \) (\(x\geq 0\)). □

Theorem 1

The operator\(P_{n}\)for every\(n\geq 1\)defined by (1.6) has the following properties:

  1. (i)

    \(P_{n}\)is a positive and continuous linear operator from\(\wp _{b}([0,\infty ))\)to\(\wp _{b}([0,\infty ))\)and\(\Vert P_{n} \Vert _{\wp _{b}([0,\infty ))}=1\).

  2. (ii)

    \(P_{n}(\wp _{0}([0,\infty )))\subset \wp _{0}([0,\infty ))\).

Proof

(i) For any \(f\in \wp _{b}([0,\infty ))\), there exists an \(M_{f}\) depending on f such that \(\vert f \vert \leq M_{f}\). Therefore, for every \(n\geq 1\),

$$\begin{aligned} \bigl\vert P_{n}(f) \bigr\vert \leq & \frac{ne^{-nx}}{(b_{n}-c_{n})H(1,1)} \sum _{n_{1}=0}^{\infty } \sum_{n_{2}=0}^{\infty } \frac{s_{n_{1},n_{2}} (\frac{nx}{2})}{n_{1}!n_{2}!} \int _{\frac{n_{1}+n_{2}+c_{n}}{n}}^{\frac{n_{1}+n_{2}+b_{n}}{n}} \bigl\vert f(u) \bigr\vert \,du \\ \leq & M_{f}, \end{aligned}$$

and \(P_{n}(1)=1\). Hence \(\Vert P_{n} \Vert _{\wp _{b}([0,\infty ))}=1\).

(ii) For fixed \(h\in \wp _{0}([0,\infty ))\) and \(\epsilon \geq 0\), there exists \(u\geq 0\) such that \(\vert h(x) \vert \leq \frac{\epsilon }{4}\) for any \(x\geq u-1\). Now, since \(T_{m}(x)e^{-nx}\rightarrow 0\) as \(x\rightarrow \infty \), where \(T_{m}(x)\) is any polynomial of degree \(m\in \mathbb{N}\) in x for every m, therefore there exists \(v>u\) such that, for every \(x\geq v\),

$$\begin{aligned} T_{m}(x)e^{-nx}\leq \frac{\epsilon }{4 \Vert h \Vert _{\infty }(n[u]+1)^{2}} \end{aligned}$$

for any \(m=0,1,\ldots,2n[u]\), where \([u]\) denotes the integer part of u.

Using (1.3), we have \(\frac{s_{n_{1},n_{2}}(\frac{nx}{2})}{H(1,1)}\) is a polynomial of degree \(n_{1}+n_{2}\). Therefore, for every \(x\geq v\), we have

$$\begin{aligned} \bigl\vert P_{n}(h;x) \bigr\vert \leq & \frac{n}{b_{n}-c_{n}} \sum _{n_{1}=0}^{n[u]} \sum _{n_{2}=0}^{n[u]} e^{-nx} \frac{s_{n_{1},n_{2}}(\frac{nx}{2})}{n_{1}!n_{2}!H(1,1)} \int _{ \frac{n_{1}+n_{2}+c_{n}}{n}}^{\frac{n_{1}+n_{2}+b_{n}}{n}} \bigl\vert h(t) \bigr\vert \,dt \\ &{}+ \frac{n}{b_{n}-c_{n}} \sum_{n_{1}=n[u]+1}^{\infty } \sum_{n_{2}=0}^{n[u]} e^{-nx} \frac{s_{n_{1},n_{2}}(\frac{nx}{2})}{n_{1}!n_{2}!H(1,1)} \int _{ \frac{n_{1}+n_{2}+c_{n}}{n}}^{\frac{n_{1}+n_{2}+b_{n}}{n}} \bigl\vert h(t) \bigr\vert \,dt \\ &{}+ \frac{n}{b_{n}-c_{n}} \sum_{n_{1}=0}^{n[u]} \sum_{n_{2}=n[u]+1}^{\infty } e^{-nx} \frac{s_{n_{1},n_{2}}(\frac{nx}{2})}{n_{1}!n_{2}!H(1,1)} \int _{ \frac{n_{1}+n_{2}+c_{n}}{n}}^{\frac{n_{1}+n_{2}+b_{n}}{n}} \bigl\vert h(t) \bigr\vert \,dt \\ &{}+ \frac{n}{b_{n}-c_{n}} \sum_{n_{1}=n[u]+1}^{\infty } \sum_{n_{2}=n[u]+1}^{\infty } e^{-nx} \frac{s_{n_{1},n_{2}}(\frac{nx}{2})}{n_{1}!n_{2}!H(1,1)} \int _{ \frac{n_{1}+n_{2}+c_{n}}{n}}^{\frac{n_{1}+n_{2}+b_{n}}{n}} \bigl\vert h(t) \bigr\vert \,dt \\ \leq & \frac{\epsilon }{4}+ \frac{\epsilon }{4} \sum _{n_{1}=0}^{ \infty } \sum_{n_{2}=0}^{\infty } e^{-nx} \frac{s_{n_{1},n_{2}}(\frac{nx}{2})}{n_{1}!n_{2}!H(1,1)} \\ &{}+\frac{\epsilon }{4} \sum_{n_{1}=0}^{\infty } \sum _{n_{2}=0}^{\infty } e^{-nx} \frac{s_{n_{1},n_{2}}(\frac{nx}{2})}{n_{1}!n_{2}!H(1,1)} \\ &{}+\frac{\epsilon }{4} \sum_{n_{1}=0}^{\infty } \sum _{n_{2}=0}^{\infty } e^{-nx} \frac{s_{n_{1},n_{2}}(\frac{nx}{2})}{n_{1}!n_{2}!H(1,1)} \\ =&\epsilon . \end{aligned}$$

 □

Theorem 2

If\(f\in \wp _{*}([0,\infty ))\), then\(\lim_{n\rightarrow \infty } P_{n}(f)=f\)uniformly on\([0,\infty )\).

Moreover, if\(f\in \wp _{b}([0,\infty ))\), then\(\lim_{n\rightarrow \infty } P_{n}(f)=f\)uniformly on every compact subset of\([0,\infty )\).

Proof

To prove the first part, it suffices to show that \(P_{n}(f)\rightarrow f\) for every \(f\in \wp _{0}([0,\infty ))\) or, in fact, for each function \(f_{\rho }\) defined in Proposition 1 since the subspace generated by them is dense in \(\wp _{0}([0,\infty ))\) and the sequence \((P_{n})_{n\geq 1}\) is equibounded on \(\wp _{0}([0,\infty ))\). Now, by using (3.1), for every \(x\geq 0\) and \(n\geq 1\), we get

$$\begin{aligned}& \bigl\vert P_{n}(f_{\rho }) (x)-f_{\rho }(x) \bigr\vert \\& \quad \leq \biggl\vert \frac{n}{\rho (b_{n}-c_{n})} \bigl(e^{-\rho \frac{c_{n}}{n}}-e^{-\rho \frac{b_{n}}{n}} \bigr)-1 \biggr\vert S_{n}(f_{ \rho }) (x)+ \bigl\vert S_{n}(f_{\rho }) (x)-f_{\rho }(x) \bigr\vert . \end{aligned}$$

Now since

$$\begin{aligned} S_{n}(f_{\rho }) (x) =&\frac{e^{-nx}}{H(1,1)} \sum _{n_{1}=0}^{ \infty } \sum_{n_{2}=0}^{\infty } \frac{s_{n_{1},n_{2}} (\frac{nx}{2})}{n_{1}!n_{2}!}e^{-\rho (\frac{n_{1}+n_{2}}{n} )} \\ \leq &\frac{e^{-nx}}{H(1,1)} \sum_{n_{1}=0}^{\infty } \sum_{n_{2}=0}^{\infty } \frac{s_{n_{1},n_{2}} (\frac{nx}{2})}{n_{1}!n_{2}!} \\ =&1. \end{aligned}$$

We have

$$\begin{aligned}& \bigl\vert P_{n}(f_{\rho }) (x)-f_{\rho }(x) \bigr\vert \\& \quad \leq \biggl\vert \frac{n}{\rho (b_{n}-c_{n})} \bigl(e^{-\rho \frac{c_{n}}{n}}-e^{-\rho \frac{b_{n}}{n}} \bigr)-1 \biggr\vert + \bigl\vert S_{n}(f_{ \rho }) (x)-f_{\rho }(x) \bigr\vert , \end{aligned}$$

and by using the inequality \(\frac{n}{\rho (b_{n}-c_{n})} (e^{-\rho \frac{c_{n}}{n}}-e^{- \rho \frac{b_{n}}{n}} )\leq 1\) (proved in Proposition 2), we get

$$\begin{aligned}& \bigl\vert P_{n}(f_{\rho }) (x)-f_{\rho }(x) \bigr\vert \\& \quad \leq \biggl(1-\frac{n}{\rho (b_{n}-c_{n})} \bigl(e^{-\rho \frac{c_{n}}{n}}-e^{-\rho \frac{b_{n}}{n}} \bigr) \biggr) + \bigl\vert S_{n}(f_{ \rho }) (x)-f_{\rho }(x) \bigr\vert \\& \quad \leq \biggl(1-\frac{n}{\rho (b_{n}-c_{n})} \bigl(e^{-\rho \frac{c_{n}}{n}}-e^{-\rho \frac{b_{n}}{n}} \bigr) \biggr)+ \bigl\Vert S_{n}(f_{ \rho })-f_{\rho } \bigr\Vert _{\infty }. \end{aligned}$$

Now from [[5], p. 845], we have

$$ \biggl(1-\frac{n}{\rho (b_{n}-c_{n})} \bigl(e^{-\rho \frac{c_{n}}{n}}-e^{- \rho \frac{b_{n}}{n}} \bigr) \biggr)\leq \frac{\rho }{n}. $$

Therefore

$$\begin{aligned} \bigl\vert P_{n}(f_{\rho }) (x)-f_{\rho }(x) \bigr\vert \leq \frac{\rho }{n}+ \bigl\Vert S_{n}(f_{ \rho })-f_{\rho } \bigr\Vert _{\infty }. \end{aligned}$$

Now since the sequence \((S_{n}(f_{\rho }))_{n\geq 1}\) converges uniformly to \(f_{\rho }\) by Proposition 1, the result is achieved.

For the second part of the theorem, we see that, from Lemma 1, \(\lim_{n\rightarrow \infty }P_{n}(g)=g\) uniformly on compact subsets of \([0,\infty )\) for every \(g\in \{\mathbf{1},c_{1},c_{2}\}\subset G_{2}^{\ast }\), the result holds from [[3], Theorem 3.5]. □

Estimating the rate of convergence

We now present some estimates of the rate of convergence of \((P_{n}(h))_{n\geq 1}\) to h by using the moduli of smoothness of first and second order \(\omega (h,\gamma )\) and \(\omega _{2}(h,\gamma )\). For the definitions of \(\omega (h,\gamma )\) and \(\omega _{2}(h,\gamma )\), we refer the reader to [[4], Sect. 5.1].

Theorem 3

Let\(h\in \wp _{b}([0,\infty ))\), \(n\geq 1\), and\(y\geq 0\). Then

$$\begin{aligned}& \bigl\vert P_{n}(h) (y)-h(y) \bigr\vert \\& \quad \leq \biggl(\frac{b_{n}+c_{n}}{2\sqrt{n}}+ \frac{H_{t_{1}}(1,1)+H_{t_{2}}(1,1)}{\sqrt{n}H(1,1)} \biggr)\omega \biggl(h, \frac{1}{\sqrt{n}} \biggr) \\& \quad \quad {} + \biggl[1+\frac{1}{2} \biggl(y+\frac{1}{nH(1,1)}\bigl\{ H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1) \\& \quad\quad{} +H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr\} \\& \quad \quad {} +\frac{b_{n}^{2}+c_{n}^{2}+b_{n}c_{n}}{n}+(b_{n}+c_{n}) \biggl( \frac{H_{t_{1}}(1,1)+H_{t_{2}}(1,1)}{H(1,1)} \biggr) \biggr) \biggr]\omega _{2} \biggl(h, \frac{1}{\sqrt{n}} \biggr). \end{aligned}$$

Proof

Since (2.3) holds, by [[15], Theorem 2.2.1] and Lemma 1, for every \(\gamma >0\),

$$\begin{aligned}& \bigl\vert P_{n}(h) (y)-h(y) \bigr\vert \\& \quad \leq \bigl\vert P_{n}(1) (y)-1 \bigr\vert \bigl\vert h(y) \bigr\vert +\frac{1}{\gamma } \bigl\vert P_{n}(\rho _{y}) (y) \bigr\vert \omega (h,\gamma )+ \biggl[P_{n}(1) (y)+\frac{1}{2\gamma ^{2}}P_{n}\bigl( \rho _{y}^{2} \bigr)y) \biggr]\omega _{2}(h,\gamma ) \\& \quad =\frac{1}{\gamma } \biggl(\frac{b_{n}+c_{n}}{2n}+ \frac{H_{t_{1}}(1,1)+H_{t_{2}}(1,1)}{nH(1,1)} \biggr) \omega (h,\gamma ) \\& \quad \quad {}+ \biggl[1+\frac{1}{2\gamma ^{2}} \biggl(\frac{y}{n}+ \frac{1}{n^{2}H(1,1)}\bigl\{ H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1) \\& \quad\quad{} +H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr\} \\& \quad \quad {} +\frac{b_{n}^{2}+c_{n}^{2}+b_{n}c_{n}}{n^{2}}+\frac{1}{n}(b_{n}+c_{n}) \biggl(\frac{H_{t_{1}}(1,1)+H_{t_{2}}(1,1)}{nH(1,1)} \biggr) \biggr) \biggr] \omega _{2}(h,\gamma ). \end{aligned}$$

Putting \(\gamma =\frac{1}{\sqrt{n}}\), we get the result. □

Quantitative estimates

Lemma 2

Let\(0\leq c_{n}\leq b_{n}\leq 1\) (\(n\geq 1\)), \(h\in \wp _{b}([0,\infty ))\), and\(G(x)=\int _{0}^{x} g(t)\,dt\) (≥0). Then, for every\(x\geq 0\)and\(n\geq 1\),

$$\begin{aligned} \biggl\vert \frac{n}{c_{n}-a_{n}}\tau _{n}(G) (x)-g(x) \biggr\vert \leq \omega \biggl(g,\frac{b_{n}-c_{n}}{n} \biggr). \end{aligned}$$
(5.1)

Moreover, for every\(\gamma >0\),

$$\begin{aligned} \omega \bigl(\tau _{n}(G),\gamma \bigr)\leq \frac{b_{n}-c_{n}}{n}\omega \biggl(g, \gamma +\frac{b_{n}-c_{n}}{n} \biggr). \end{aligned}$$
(5.2)

Proof

Let \(x\geq 0\) be fixed and \(n\geq 1\); then by applying Lagrange’s theorem to the function G in the interval \([x+\frac{c_{n}}{n},x+\frac{b_{n}}{n} ]\), we have

$$\begin{aligned} \frac{n}{b_{n}-c_{n}}\tau _{n}(G) (x)=g(\xi _{n,x}), \end{aligned}$$

where \(\xi _{n,x}\) is a point in \([x+\frac{c_{n}}{n},x+\frac{b_{n}}{n} ]\).

Now,

$$\begin{aligned} \biggl\vert \frac{n}{b_{n}-c_{n}}\tau _{n}(G) (x)-g(x) \biggr\vert = \bigl\vert g(\xi _{n,x})-g(x) \bigr\vert \leq \omega \bigl(g, \vert \xi _{n,x}-x \vert \bigr)\leq \omega \biggl(g, \frac{b_{n}-c_{n}}{n} \biggr). \end{aligned}$$

Now, for \(x,y\geq 0\) such that \(\vert x-y \vert \leq \gamma \) where \(\gamma \geq 0\) is a fixed number, again by Lagrange’s theorem,

$$\begin{aligned} \bigl\vert \tau _{n}(G) (x)-\tau _{n}(G) (y) \bigr\vert =\frac{b_{n}-c_{n}}{n} \bigl\vert g(\xi _{n,x})-g( \zeta _{n,y}) \bigr\vert \leq \frac{b_{n}-c_{n}}{n} \omega \bigl(g, \vert \xi _{n,x}-\zeta _{n,y} \vert \bigr), \end{aligned}$$

where \(\zeta _{n,y}\) is some element in the interval \([y+\frac{c_{n}}{n},y+\frac{b_{n}}{n} ]\), and hence the result since

$$\begin{aligned} \vert \xi _{n,x}-\zeta _{n,y} \vert \leq \vert x-y \vert +\frac{b_{n}-c_{n}}{n}\leq \gamma + \frac{b_{n}-c_{n}}{n}. \end{aligned}$$

 □

Theorem 4

Consider\(g\in \wp _{b}([0,\infty ))\), \(n\geq 1\), and\(x\geq 0\). Then

$$\begin{aligned}& \bigl\vert P_{n}(g) (x)-g(x) \bigr\vert \\& \quad \leq \biggl(2+\sqrt{x+\frac{1}{nH(1,1)} \bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1)} \bigr) \biggr) \\& \quad\quad{}\times \omega \biggl(g,\frac{\sqrt{n}+b_{n}-c_{n}}{n} \biggr). \end{aligned}$$
(5.3)

Furthermore, ifgis differentiable on\([0,\infty )\)and\(g'\in \wp _{b}([0,\infty ))\), then

$$\begin{aligned}& \bigl\vert P_{n}(g) (x)-g(x) \bigr\vert \\& \quad \leq \biggl(\sqrt{\frac{x}{n}+ \frac{1}{n^{2}H(1,1)}\bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1)} \bigr) \biggr) \\& \quad \quad {}\times \biggl(1+\sqrt{x+\frac{1}{nH(1,1)}\bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr)} \biggr) \\& \quad \quad {}\times \omega \biggl(g',\frac{\sqrt{n}+b_{n}-c_{n}}{n} \biggr)+ \bigl\Vert g' \bigr\Vert _{\infty } \frac{b_{n}-c_{n}}{n}. \end{aligned}$$
(5.4)

Proof

From [[4], Theorem 5.2.4], it follows that, for every \(\gamma >0\),

$$\begin{aligned}& \bigl\vert S_{n}(g) (x)-g(x) \bigr\vert \\& \quad \leq \biggl(1+\frac{1}{\gamma }\sqrt{\frac{x}{n}+ \frac{1}{n^{2}H(1,1)}\bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr)} \biggr) \\& \quad\quad{}\times \omega (g,\gamma ). \end{aligned}$$
(5.5)

From this and from (5.1) and (5.2), we have

$$\begin{aligned}& \bigl\vert P_{n}(g) (x)-g(x) \bigr\vert \\& \quad \leq \frac{n}{b_{n}-c_{n}} \bigl\vert S_{n}\bigl(\tau _{n}(G)\bigr) (x)-\tau _{n}(G) (x) \bigr\vert + \biggl\vert \frac{n}{b_{n}-c_{n}}\tau _{n}(G) (x)-g(x) \biggr\vert \\& \quad \leq \biggl(1+\frac{1}{\gamma }\sqrt{\frac{x}{n}+ \frac{1}{n^{2}H(1,1)}\bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr)} \biggr) \\& \quad \quad {}\times\frac{n}{b_{n}-c_{n}}\omega \bigl(\tau _{n}(G),\gamma \bigr)+ \omega \biggl(g,\frac{b_{n}-c_{n}}{n} \biggr) \\& \quad \leq \biggl(1+ \frac{1}{\gamma }\sqrt{\frac{x}{n}+ \frac{1}{n^{2}H(1,1)}\bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr)} \biggr) \\& \quad \quad {}\times\omega \biggl(g,\gamma +\frac{b_{n}-c_{n}}{n} \biggr)+\omega \biggl(g, \frac{b_{n}-c_{n}}{n} \biggr) \\& \quad \leq \biggl(2+\frac{1}{\gamma }\sqrt{\frac{x}{n}+ \frac{1}{n^{2}H(1,1)}\bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr)} \biggr) \\& \quad \quad{}\times \omega \biggl(g,\gamma +\frac{b_{n}-c_{n}}{n} \biggr). \end{aligned}$$

Taking \(\gamma = \frac{1}{\sqrt{n}}\), we have (5.3).

And for (5.4), assume that g is differentiable on \([0,\infty )\) and \(g'\in \wp _{b}([0,\infty ))\); then

$$\begin{aligned} \omega \biggl(g,\frac{b_{n}-c_{n}}{n} \biggr)\leq \bigl\Vert g' \bigr\Vert _{\infty } \frac{b_{n}-c_{n}}{n}. \end{aligned}$$
(5.6)

Moreover, from [[4], Theorem 5.2.4], we have, for \(\gamma >0\),

$$\begin{aligned}& \bigl\vert S_{n}(g) (x)-g(x) \bigr\vert \\& \quad \leq \sqrt{\frac{x}{n}+\frac{1}{n^{2}H(1,1)}\bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr)} \\& \quad \quad {}\times\biggl(1+\frac{1}{\gamma }\sqrt{\frac{x}{n}+ \frac{1}{n^{2}H(1,1)}\bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr)} \biggr) \\& \quad \quad {}\times\omega \bigl(g',\gamma \bigr). \end{aligned}$$
(5.7)

Since g is differentiable, \(\tau _{n}(G)\) is also differentiable with bounded and continuous derivative, and for every \(x\geq 0\) and \(n\geq 1\),

$$\begin{aligned} \tau _{n}(G)'(x)=g \biggl(x+\frac{b_{n}}{n} \biggr)-g \biggl(x+ \frac{c_{n}}{n} \biggr). \end{aligned}$$

Now, if \(x,y\geq 0\) are two elements such that \(\vert x-y \vert \leq \gamma \), then by Lagrange’s theorem, there exist \(\eta _{n,x}\in [x+\frac{c_{n}}{n},x+\frac{b_{n}}{n} ]\) and \(\xi _{n,y}\in [y+\frac{c_{n}}{n},y+\frac{b_{n}}{n} ]\) such that

$$\begin{aligned} \bigl\vert \tau _{n}(G)'(x)-\tau _{n}(G)'(y) \bigr\vert =& \biggl\vert g \biggl(x+\frac{b_{n}}{n} \biggr)-g \biggl(x+ \frac{c_{n}}{n} \biggr)-g \biggl(y+\frac{b_{n}}{n} \biggr)+g \biggl(y+ \frac{c_{n}}{n} \biggr) \biggr\vert \\ =&\frac{b_{n}-c_{n}}{n} \bigl\vert g'(\eta _{n,x})-g'( \xi _{n,y}) \bigr\vert \leq \frac{b_{n}-c_{n}}{n}\omega \biggl(g',\gamma +\frac{b_{n}-c_{n}}{n} \biggr). \end{aligned}$$

Hence,

$$\begin{aligned} \omega \bigl(\tau _{n}(G)', \gamma \bigr)\leq \frac{b_{n}-c_{n}}{n}\omega \biggl(g', \gamma + \frac{b_{n}-c_{n}}{n} \biggr), \end{aligned}$$
(5.8)

and by (5.6), (5.7), and (5.8),

$$\begin{aligned}& \bigl\vert P_{n}(g) (x)-g(x) \bigr\vert \\& \quad \leq \frac{n}{b_{n}-c_{n}} \bigl\vert S_{n}\bigl(\tau _{n}(G)\bigr) (x)-\tau _{n}(G) (x) \bigr\vert + \biggl\vert \frac{n}{b_{n}-c_{n}}\tau _{n}(G) (x)-g(x) \biggr\vert \\& \quad \leq \frac{n}{b_{n}-c_{n}} \\& \quad\quad{}\times \sqrt{\frac{x}{n}+\frac{1}{n^{2}H(1,1)} \bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr)} \\& \quad \quad {}\times\biggl(1+\frac{1}{\gamma }\sqrt{\frac{x}{n}+ \frac{1}{n^{2}H(1,1)}\bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr)} \biggr) \\& \quad \quad {}\times\omega \bigl(\tau _{n}(G)',\gamma \bigr)+ \omega \biggl(g,\frac{n}{b_{n}-c_{n}} \biggr) \\& \quad \leq \sqrt{\frac{x}{n}+\frac{1}{n^{2}H(1,1)}\bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr)} \\& \quad \quad {}\times\biggl(1+\frac{1}{\gamma }\sqrt{\frac{x}{n}+ \frac{1}{n^{2}H(1,1)}\bigl(H_{t_{1}}(1,1)+H_{t_{2}}(1,1)+H_{t_{1}t_{1}}(1,1)+H_{t_{2}t_{2}}(1,1)+2H_{t_{1}t_{2}}(1,1) \bigr)} \biggr) \\& \quad \quad {}\times\omega \biggl(g',\gamma +\frac{b_{n}-c_{n}}{n} \biggr)+ \bigl\Vert g' \bigr\Vert _{\infty } \frac{b_{n}-c_{n}}{n}. \end{aligned}$$

In particular, for \(\gamma = \frac{1}{n^{\frac{1}{2}}}\), we have (5.4). □

References

  1. 1.

    Ahasan, M., Mursaleen, M.: Generalized Szász–Mirakjan type operators via q-calculus and approximation properties. Appl. Math. Comput. 371, 124916 (2020)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Alotaibi, A., Mursaleen, M.: Dunkl generalization of q-Szász–Mirakjan–Kantrovich type operators and approximation. J. Comput. Anal. Appl. 27(1), 66–76 (2019)

    Google Scholar 

  3. 3.

    Altomare, F.: Korovkin-type theorems and approximation by positive linear operators. Surv. Approx. Theory 5, 92–164 (2010)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Altomare, F., Campiti, M.: Korovkin Type Approximation Theory and Its Applications. De Gruyter Studies in Mathematics, vol. 17. de Gruyter, Berlin (1994)

    Google Scholar 

  5. 5.

    Altomare, F., Cappelletti, M., Leonessa, V.: On a generalization of Szász–Mirakjan–Kantorovich operators. Results Math. 63, 837–863 (2012)

    Article  Google Scholar 

  6. 6.

    Altomare, F., Leonessa, V.: On a sequence of positive linear operators associated with a continuous selection of Borel measures. Mediterr. J. Math. 3, 363–382 (2006)

    MathSciNet  Article  Google Scholar 

  7. 7.

    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)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Gupta, P., Agrawal, P.N.: Jakimovski–Leviatan operators of Durrmeyer type involving Appell polynomials. Turk. J. Math. 42, 1457–1470 (2018)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Gupta, P., Agrawal, P.N.: Quantitative Voronovskaja and Grüss Voronovskaja-type theorems for operators of Kantorovich type involving multiple Appell polynomials. Iran. J. Sci. Technol. Trans. A, Sci. 43, 1679–1687 (2019)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Lee, D.W.: On multiple Appell polynomials. Proc. Am. Math. Soc. 139, 2133–2141 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Mursaleen, M., Ahasan, M.: The Dunkl generalization of Stancu type q-Szász–Mirakjan–Kantorovich operators and some approximation results. Carpath. J. Math. 34(3), 363–370 (2018)

    MATH  Google Scholar 

  12. 12.

    Mursaleen, M., Alotaibi, A., Ansari, K.J.: On a Kantorovich variant of Szász–Mirakjan operators. J. Funct. Spaces 2016, 1035253 (2016)

    MATH  Google Scholar 

  13. 13.

    Mursaleen, M., Ansari, K.J., Khan, A.: Approximation by Kantorovich type q-Bernstein Stancu operators. Complex Anal. Oper. Theory 11, 85–107 (2017)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Mursaleen, M., Rahman, S., Alotaibi, A.: Dunkl generalization of q-Szász–Mirakjan–Kantorovich operators which preserve some test functions. J. Inequal. Appl. 2016, 317 (2016)

    Article  Google Scholar 

  15. 15.

    Păltănea, R.: Approximation Theory Using Positive Linear Operators. Birkhäuser, Boston (2004)

    Google Scholar 

  16. 16.

    Srivastava, H.M., Mursaleen, M., Alotaibi, A., Md, N., 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(15), 5437–5452 (2017)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Szász, O.: Generalization of Bernstein’s polynomials to the infinite interval. J. Res. Natl. Bur. Stand. 45, 239–245 (1950)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Verma, S.: On a generalization of Szász operators by multiple Appell polynomials. Stud. Univ. Babeş–Bolyai, Math. 58(3), 361–369 (2013)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

Pooja Gupta and Ramu Dubey gratefully acknowledge the Department of Mathematics, J.C. Bose University of Science and Technology, YMCA, Faridabad-121 006, Haryana, India. Also, all the authors would like to express sincere thanks to the referees for their valuable suggestions and remarks towards the improvement of the paper.

Availability of data and materials

Nil.

Funding

Nil.

Author information

Affiliations

Authors

Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Vishnu Narayan Mishra.

Ethics declarations

Competing interests

The authors declare that 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

Verify currency and authenticity via CrossMark

Cite this article

Swarup, C., Gupta, P., Dubey, R. et al. Generalization of Szász–Mirakjan–Kantorovich operators using multiple Appell polynomials. J Inequal Appl 2020, 156 (2020). https://doi.org/10.1186/s13660-020-02423-8

Download citation

MSC

  • 41A36

Keywords

  • Szász operator
  • Multiple Appell polynomials
  • Moduli of smoothness