Some approximation results for generalized Kantorovich-type operators

Abstract

In this paper, we construct a new family of operators, prove some approximation results in A-statistical sense and establish some direct theorems for Kantorovich-type integral operators.

MSC: Primary 41A10, 41A25, 41A36.

1 Introduction and preliminaries

Statistical convergence [1] and its variants, extensions and generalizations have been proved to be an active area of recent research in summability theory, e.g., lacunary statistical convergence [2], λ-statistical convergence [3], A-statistical convergence [4], statistical A-summability [5], statistical summability $(C,1)$ [6], statistical summability $(H,1)$ [7], statistical summability $(\overline{N},p)$ [8] and statistical σ-summability [9]etc. Following the work of Gadjiv and Orhan [10], these statistical summability methods have been used in establishing many approximation theorems (e.g., [5, 11–20] and [21]). Recently, the statistical approximation properties have also been investigated for several operators. For instance, in [22] Butzer and Hahn operators; in [23] and [24]q-analogue of Stancu-Beta operators; in [25] Bleimann, Butzer and Hahn operators; in [26] Baskakov-Kantorovich operators; in [27] Szász-Mirakjan operators; in [28] analogues of Bernstein-Kantorovich operators; and in [29]q-Lagrange polynomials were defined and their statistical approximation properties were investigated. Most recently, the statistical summability of Walsh-Fourier series has been discussed in [30]. In this paper, we construct a new family of operators with the help of Erkuş-Srivastava polynomials, establish some A-statistical approximation properties and direct theorems.

Let us recall the following definitions.

Let ℕ denote the set of all natural numbers. Let $K\subseteq \mathbb{N}$ and $K_n=\{k\le n:k\in K\}$. Then the natural density of K is defined by $\delta (K)={lim}_n{n}^{-1}|K_n|$ if the limit exists, where $|K_n|$ denotes the cardinality of the set $K_n$. A sequence $x=(x_k)$ of real numbers is said to be statistically convergent to L (cf. Fast [1]) provided that for every $ϵ>0$ the set $\{k\in \mathbb{N}:|x_k-L|\ge ϵ\}$ has natural density zero, i.e., for each $ϵ>0$,

$$\underset{n}{lim}\frac{1}{n}\left|\{k\le n:|x_k-L|\ge ϵ\}\right|=0.$$

In this case, we write $\mathit{st}\text{-}{lim}_k{x}_k=L$. Note that every convergent sequence is statistically convergent but not conversely.

Let $A=(a_{nk})$, $n,k=1,2,3,\dots$ , be an infinite matrix. For a given sequence $x=(x_k)$, the A-transform of x is defined by $Ax=({(Ax)}_n)$, where ${(Ax)}_n={\sum }_{k=1}^{\mathrm{\infty }}{a}_{nk}{x}_k$, provided the series converges for each n. We say that A is regular if ${lim}_n{(Ax)}_n=L=limx$. Let A be a regular matrix.

We say that a sequence $x=(x_k)$ is A-statistically convergent to a number L (cf. Kolk [4]) if for every $ϵ>0$,

$$\underset{n}{lim}\sum _{k:|x_k-L|\ge ϵ}{a}_{nk}=0.$$

In this case, we denote this limit by ${\mathit{st}}_A\text{-}{lim}_n{x}_n=L$.

Note that for $A=C_1:=(c_{jn})$, the Cesàro matrix of order 1, A-statistical convergence reduces to the statistical convergence.

2 Construction of a new operator and its properties

The well-known (two-variable) polynomials $g_n^{(\alpha ,\beta )}(x,y)$, which are generated by

$${(1-xz)}^{-\alpha }{(1-yz)}^{-\beta }=\sum _{n=0}^{\mathrm{\infty }}{g}_n^{(\alpha ,\beta )}(x,y)z^n(|z|<min\{{|x|}^{-1},{|y|}^{-1}\}),$$

(2.1)

are the Lagrange polynomials which occur in certain problems in statistics [31]. Recently, Chan [32] introduced and systematically investigated the multivariable extension of the classical Lagrange polynomials $g_n^{(\alpha ,\beta )}(x,y)$. These multivariable Lagrange polynomials, which are popularly known in the literature as the Chan-Chyan-Srivastava polynomials, are generated by (see [32] and [33])

$$\begin{array}{r}\prod _{j=1}^r\left\{{(1-x_{j}z)}^{-{\alpha }_j}\right\}=\sum _{n=0}^{\mathrm{\infty }}{g}_n^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r)z^n,\\ {\alpha }_j\in \mathbb{C}(j=1,2,\dots ,r);|z|<min\{{|x_1|}^{-1},\dots ,{|x_r|}^{-1}\}.\end{array}$$

(2.2)

Clearly, the defined generating function (2.2) yields the explicit representation given by [[34], p.140, Eq. (6)]

$$g_n^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r)=\sum _{k_1+\cdots +k_r=n}{({\alpha }_1)}_{k_1}\cdots {({\alpha }_r)}_{k_r}\frac{x_1^{k_1}}{!k_1}\cdots \frac{x_r^{k_r}}{!k_r}$$

(2.3)

or, equivalently, by [[14], p.522, Eq. (17)]

$$\begin{array}{r}{g}_n^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r)\\ =\sum _{n_{r-1}=0}^n\sum _{n_{r-2}=0}^{n_{r-1}}\cdots \sum _{n_1=0}^{n_2}\frac{{({\alpha }_1)}_{n_1}{({\alpha }_2)}_{n_2-n_1}\cdots {({\alpha }_r)}_{n-n_r-1}}{n_1!(n_2-n_1)!\cdots (n-n_{r-1})!}{x}_1^{n_1}{x}_2^{n_2-n_1}\cdots x_r^{n-n_{r-1}}.\end{array}$$

(2.4)

On the other hand, Altin and Erkuş [34] presented a multivariable extension of the so-called Lagrange-Hermite polynomials generated by

$$\begin{array}{r}\prod _{j=1}^r\left\{{(1-x_j{z}^j)}^{-{\alpha }_j}\right\}=\sum _{n=0}^{\mathrm{\infty }}{h}_n^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r)z^n,\\ {\alpha }_j\in \mathbb{C}(j=1,2,\dots ,r);|z|<min\{{|x_1|}^{-1},\dots ,{|x_r|}^{-1/r}\}.\end{array}$$

(2.5)

The case $r=2$ of the polynomials given by (2.5) corresponds to the familiar (two-variable) Lagrange-Hermite polynomials considered by Dattoli et al. [23].

The multivariable polynomials

$$U_{n,{\ell }_1,\dots ,{\ell }_r}^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r),$$

which are defined by the following generating function [[32], p.268, Eq. (3)]:

$$\begin{array}{r}\prod _{j=1}^r\left\{{(1-x_j{z}^{{\ell }_j})}^{-{\alpha }_j}\right\}=\sum _{n=0}^{\mathrm{\infty }}{U}_{n,{\ell }_1,\dots ,{\ell }_r}^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r)z^n,\\ {\alpha }_j\in \mathbb{C}(j=1,2,\dots ,r);{\ell }_j\in \mathbb{N}(j=1,2,\dots ,r);|z|<min\{{|x_1|}^{-1/{\ell }_1},\dots ,{|x_r|}^{-1/{\ell }_r}\},\end{array}$$

(2.6)

are a unification (and generalization) of several known families of multivariable polynomials including (for example) Chan-Chyan-Srivastava polynomials

$$g_n^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r)$$

defined by (2.2) (see [35] for details). Obviously, the Chan-Chyan-Srivastava polynomials

$$g_n^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r)$$

follow as a special case of the polynomials due to Erkuş and Srivastava [35]

$$U_{n,{\ell }_1,\dots ,{\ell }_r}^{{\alpha }_1,\dots ,{\alpha }_r}(x_1,\dots ,x_r),$$

when

$${\ell }_j=1(j=1,\dots ,r),$$

where (as well as in what follows)

$$\mathbb{N}=\{1,2,3,\dots \}\text{and}{\mathbb{N}}_0=\{0,1,2,\dots \}=\mathbb{N}\cup \{0\}.$$

Moreover, the Lagrange-Hermite polynomials

$$h_n^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r)$$

follow as a special case of the polynomials [35]

$$U_{n,{\ell }_1,\dots ,{\ell }_r}^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r),$$

when

$${\ell }_j=1(j=1,\dots ,r).$$

The generating function (2.6) yields the following explicit representation ([[35], p.268, Eq. (4)]):

$$U_{n,{\ell }_1,\dots ,{\ell }_r}^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r)=\sum _{{\ell }_1{k}_1+\cdots +{\ell }_r{k}_r=n}{({\alpha }_1)}_{k_1}\cdots {({\alpha }_r)}_{k_r}\frac{x_1^{k_1}}{k_1!}\cdots \frac{x_r^{k_r}}{k_r!},$$

(2.7)

which, in the special case when

$${\ell }_j=1(j=1,\dots ,r),$$

corresponds to (2.3).

The following relationship is established between the polynomials due to Erkuş and Srivastava [35] and the Chan-Chyan-Srivastava polynomials by applying the generating functions (2.2) and (2.6) in [36].

$$\begin{array}{r}\sum _{n=0}^{\mathrm{\infty }}{U}_{n,{\ell }_1,\dots ,{\ell }_r}^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r)z^n\\ =\prod _{i=1}^r\left\{{(1-x_i{z}^{{\ell }_i})}^{-{\alpha }_i}\right\}=\prod _{i=1}^r\prod _{j=1}^{{\ell }_i}\left\{{(1-{\omega }^{(i,j)}z)}^{-{\alpha }_i}\right\}\\ =\sum _{n=0}^{\mathrm{\infty }}{g}_n^{({\alpha }_1,\dots ,{\alpha }_1,\dots ,{\alpha }_r,\dots ,{\alpha }_r)}({\omega }_{11},\dots ,{\omega }_{1{\ell }_1},\dots ,{\omega }_{r1},\dots ,{\omega }_{r{\ell }_r})z^n,\end{array}$$

(2.8)

where it is tacitly assumed that the following set:

$${\omega }^{(i,j)}:1\le i\le r\text{and}1\le j\le {\ell }_i({\ell }_i\in \mathbb{N};i=1,\dots ,r),$$

which depends upon the ${\ell }_i$ distinct values of the factor $x_i^{\frac{1}{{\ell }_i}}$ occurring in the expression

$$1-{\left(x_i^{\frac{1}{{\ell }_i}}z\right)}^{{\ell }_i}(i=1,\dots ,r),$$

exists such that

$${(1-x_i{z}^{{\ell }_i})}^{-{\alpha }_i}=\prod _{j=1}^{{\ell }_i}\left\{{(1-{\omega }^{(i,j)}z)}^{-{\ell }_j}\right\}(i=1,\dots ,r).$$

Thus, by assertion (2.8), we obtain the desired relationship as follows:

$$U_{n,{\ell }_1,\dots ,{\ell }_r}^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r)=g_n^{({\alpha }_1,\dots ,{\alpha }_1,\dots ,{\alpha }_r,\dots ,{\alpha }_r)}({\omega }^{(1,1)},\dots ,{\omega }^{(1,{\ell }_1)},\dots ,{\omega }^{(r,1)},\dots ,{\omega }^{(r,{\ell }_r)}).$$

Now by using the Erkuş-Srivastava multivariable polynomials given by (2.2), we introduce the following family of positive linear operators on $C[0,1]$:

$$\begin{array}{r}{T}_n^{{\omega }^{(1,1)},\dots ,{\omega }^{(1,{\ell }_1)},\dots ,{\omega }^{(r,1)},\dots ,{\omega }^{(r,{\ell }_r)}}(f;x)\\ =\prod _{i=1}^r{\left\{(1-x_i{z}^{{\ell }_i})\right\}}^n\sum _{m=0}^{\mathrm{\infty }}{U}_{m,{\ell }_1,\dots ,{\ell }_r}^{({\alpha }_1,\dots ,{\alpha }_r)}(x_1,\dots ,x_r)z^m{\int }_{\frac{k_r}{n+k_r-1}}^{\frac{k_r+1}{n+k_r-1}}f(t)dt\\ =\prod _{i=1}^r\prod _{j=1}^{{\ell }_i}{\left\{(1-{\omega }^{(i,j)}z)\right\}}^n\sum _{m=0}^{\mathrm{\infty }}{g}_m({\omega }^{(1,1)},\dots ,{\omega }^{(1,{\ell }_1)},\dots ,{\omega }^{(r,1)},\dots ,{\omega }^{(r,{\ell }_r)})z^n\\ \times (n+k_r-1){\int }_{\frac{k_r}{n+k_r-1}}^{\frac{k_r+1}{n+k_r-1}}f(t)dt,\end{array}$$

(2.9)

where

$${\alpha }_j\in \mathbb{C}(j=1,2,\dots ,r);{\ell }_j\in \mathbb{N}(j=1,2,\dots ,r);|z|<min\{{|x_1|}^{-1/{\ell }_1},\dots ,{|x_r|}^{-1/{\ell }_r}\}.$$

Throughout this paper, we assume that

$${\omega }^{(i,j)}={\left\{{\omega }_{(n)}^{(i,j)}\right\}}_{n\in \mathbb{N}},1\le i\le r\text{ and }1\le j\le {\ell }_i({\ell }_i\in \mathbb{N};i=1,\dots ,r),$$

are sequences of real numbers such that

$$0<{\omega }^{(i,j)}<1.$$

For convenience, taking $r=1$, ${\ell }_i=2$, ${\alpha }_1={\alpha }_2=n$ in (2.9), we have

$$\begin{array}{r}{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f;x)\\ ={(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=0}^{\mathrm{\infty }}{g}_m^{(n,n)}({\omega }^{(1,1)},{\omega }^{(1,2)})x^m{\int }_{\frac{k}{n+k-1}}^{\frac{k+1}{n+k-1}}f(t)dt\\ ={(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=0}^{\mathrm{\infty }}\{\sum _{k_1=m}\frac{{({\omega }_n^{(1,1)})}^{k_1}}{k_1!}{(n)}_{k_1}{\int }_{\frac{k}{n+k-1}}^{\frac{k+1}{n+k-1}}f(t)dt\}{x}^m.\end{array}$$

(2.10)

Lemma 2.1 For each $x\in [0,1]$ and $n\in \mathbb{N}$,

$$T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_0;x)=1(f_0(x)=1).$$

Lemma 2.2 For each $x\in [0,1]$ and $n\in \mathbb{N}$,

$$T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_1;x)\le x{\omega }_n^{(1,1)}+\frac{1}{2n}(f_1(x)=x).$$

Proof Let each $x\in [0,1]$ be fixed. Then from (2.10) we get

$$\begin{array}{r}{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_1;x)\\ ={(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=0}^{\mathrm{\infty }}(n+k_1-1)\left\{\sum _{k_1=m}\frac{{({\omega }_n^{(1,1)})}^{k_1}}{k_1!}{(n)}_{k_1}{\int }_{\frac{k}{n+k-1}}^{\frac{k+1}{n+k-1}}tdt\right\}{x}^m\\ ={(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=0}^{\mathrm{\infty }}(n+k_1-1)\left\{\sum _{k_1=m}\frac{{({\omega }_n^{(1,1)})}^{k_1}}{k_1!}{(n)}_{k_1}{\left[\frac{t^2}{2}\right]}_{\frac{k}{n+k-1}}^{\frac{k+1}{n+k-1}}\right\}{x}^m\\ ={(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=0}^{\mathrm{\infty }}\sum _{k_1=m}\frac{(2k_1+1)}{2(n+k-1)}\frac{{({\omega }_n^{(1,1)})}^{k_1}}{k_1!}{(n)}_{k_1}{x}^m\\ =x{\omega }_n^{(1,1)}x{(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=1}^{\mathrm{\infty }}\sum _{k_1=1}^m\frac{{({\omega }_n^{(1,1)})}^{k_1-1}}{k_1-1!}{(n)}_{k_1-1}{x}^{m-1}\\ ={(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=0}^{\mathrm{\infty }}\sum _{k_1=m}\frac{1}{2(n+k-1)}\frac{{({\omega }_n^{(1,1)})}^{k_1}}{k_1!}{(n)}_{k_1}{x}^m\\ \le {\omega }_n^{(1,1)}x+\frac{1}{2n},0<{\omega }_n^{(1,1)}<1,{\omega }_n^{(1,1)}\to 1.\end{array}$$

 □

Lemma 2.3 For each $x\in [0,1]$ and $n\in \mathbb{N}$,

$$T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_2;x)\le x^2{\left({\omega }_n^{(1,1)}\right)}^2+\frac{2x({\omega }_n^{(1,1)})}{n}+\frac{1}{3n^2}(f_2(x)=x^2).$$

Proof Let each $x\in [0,1]$ be fixed. Then from (2.10) we get

$$\begin{array}{r}{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_1;x)\\ ={(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=0}^{\mathrm{\infty }}(n+k_1-1)\left\{\sum _{k_1=m}\frac{{({\omega }_n^{(1,1)})}^{k_1}}{k_1!}{(n)}_{k_1}{\int }_{\frac{k}{n+k-1}}^{\frac{k+1}{n+k-1}}{t}^{2}dt\right\}{x}^m\\ ={(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=0}^{\mathrm{\infty }}(n+k_1-1)\left\{\sum _{k_1=m}\frac{{({\omega }_n^{(1,1)})}^{k_1}}{k_1!}{(n)}_{k_1}{\left[\frac{t^3}{3}\right]}_{\frac{k}{n+k-1}}^{\frac{k+1}{n+k-1}}\right\}{x}^m\\ ={(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=0}^{\mathrm{\infty }}\sum _{k=m}(n+k-1)\frac{{({\omega }_n^{(1,1)})}^{k_1}}{(k_1)!}{(n)}_{k_1}\\ \times \{\frac{k_1^2}{{(n+k_1-1)}^3}+\frac{k_1}{{(n+k_1-1)}^3}+\frac{1}{3{(n+k_1-1)}^3}\}{x}^m\\ =x{\omega }_n^{(1,1)}{(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=1}^{\mathrm{\infty }}\sum _{k=1}^m\left(\frac{k}{n+k-1}\right)\left\{\frac{{({\omega }_n^{(1,1)})}^{k_1-1}}{(k_1-1)!}{(n)}_{k_1-1}\right\}{x}^{m-1}\\ +x{\omega }_n^{(1,1)}{(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=1}^{\mathrm{\infty }}\sum _{k=1}^m\left(\frac{1}{n+k-1}\right)\\ \times \left\{\frac{{({\omega }_n^{(1,1)})}^{k_1-1}}{(k_1-1)!}{(n)}_{k_1-1}\right\}{x}^{m-1}\\ +x{\omega }_n^{(1,1)}{(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=1}^{\mathrm{\infty }}\sum _{k=1}^m\left(\frac{1}{3{(n+k-1)}^2}\right)\left\{\frac{{({\omega }_n^{(1,1)})}^{k_1}}{k_1!}{(n)}_{k_1}\right\}{x}^m\\ \le x^2{\left({\omega }_n^{(1,1)}\right)}^2{(1-{\omega }^{(1,1)}x)}^n{(1-{\omega }^{(1,2)}x)}^n\sum _{m=2}^{\mathrm{\infty }}\sum _{k=2}^m\left(\frac{(n+k-2)}{n+k-1}\right)\\ \times \left\{\frac{{({\omega }_n^{(1,1)})}^{k_1-2}}{(k_1-2)!}{(n)}_{k_1-2}\right\}{x}^{m-2}+\frac{2x{\omega }_n^{(1,1)}}{n}+\frac{1}{3n^2}\\ \le x^2{\left({\omega }_n^{(1,1)}\right)}^2+\frac{2x{\omega }_n^{(1,1)}}{n}+\frac{1}{3n^2},\\ T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_2;x)-f_2(x)\le x^2({\left({\omega }_n^{(1,1)}\right)}^2-1)+\frac{2x{\omega }_n^{(1,1)}}{n}+\frac{1}{3n^2}.\end{array}$$

(2.11)

On the other hand, since

$$0\le T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}({(y-x)}^2;x)=T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_2;x)-2x{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_1;x)+x^2,$$

it follows from Lemma 2.1 and Lemma 2.2 that

$$T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_2;x)-f_2(x)\ge 2x^2({\left({\omega }_n^{(1,1)}\right)}^2-1).$$

(2.12)

Combining (2.11) and (2.12), we have

$$|T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_2;x)-f_2(x)|\le x^2(1-{\left({\omega }_n^{(1,1)}\right)}^2)+\frac{2x{\omega }_n^{(1,1)}}{n}+\frac{1}{3n^2}.$$

Then, taking supremum over $x\in [0,1]$, we have

$$\parallel T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_2;x)-f_2(x)\parallel \le x^2(1-{\left({\omega }_n^{(1,1)}\right)}^2)+\frac{2x{\omega }_n^{(1,1)}}{n}+\frac{1}{3n^2}.$$

(2.13)

 □

Remark 2.1

$$\begin{array}{rl}{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(t-x;x)& =x{\omega }_n^{(1,1)}-x+\frac{1}{2n}\\ =x({\omega }_n^{(1,1)}-1)\frac{1}{2n}.\end{array}$$

Remark 2.2 Let $x\in [0,1]$, since $T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}$ is linear, we get

$$\begin{array}{r}{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}({(y-x)}^2;x)\\ =T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_2;x)-2x{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_1;x)+x^2{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_0;x)\\ \le x^2{\left({\omega }^{(1,1)}\right)}^2+\frac{2x{\omega }^{(1,1)}}{n}-2x^2{\omega }^{(1,1)}+x^2-2x[x{\omega }_n^{(1,1)}+\frac{1}{2n}]+x^2\\ \le x^2{\left({\omega }^{(1,1)}\right)}^2+\frac{2x{\omega }^{(1,1)}}{n}-2x^2{\omega }^{(1,1)}+x^2+\frac{1}{3n^2}-\frac{x}{n}.\end{array}$$

3 A-statistical approximation

Let $C[a,b]$ be a linear space of all real-valued continuous functions f on $[a,b]$, and let T be a linear operator which maps $C[a,b]$ into itself. We say that T is positive if for every non-negative $f\in C[a,b]$, we have $T(f,x)\ge 0$ for all $x\in [a,b]$ . We know that $C[a,b]$ is a Banach space with the norm

$${\parallel f\parallel }_{C[a,b]}:=\underset{x\in [a,b]}{sup}|f(x)|,f\in C[a,b].$$

For typographical convenience, we will write $\parallel \cdot \parallel$ in place of ${\parallel \cdot \parallel }_{C[a,b]}$ if no confusion arises.

Theorem 3.1 Let $A=(a_{jn})$ be a non-negative regular summability matrix. Then

$${\mathit{st}}_A\text{-}\underset{n}{lim}{\omega }_n^{(1,1)}=1$$

(3.1)

if and only if for all $f\in C[0,1]$,

$${\mathit{st}}_A\text{-}\underset{n}{lim}\parallel T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f)-f\parallel =0.$$

(3.2)

Proof Suppose that (3.2) holds for all $f\in C[0,1]$. Then we have

$${\mathit{st}}_A\text{-}\underset{n}{lim}\parallel T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_1)-f_1\parallel =0$$

(3.3)

since $f_1\in C[0,1]$. By Lemma 2.2, we have

$$\parallel T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_1)-f_1\parallel =1-{\omega }_n^{(1,1)}.$$

(3.4)

By (3.3) and (3.4), we immediately get

$${\mathit{st}}_A\text{-}\underset{n}{lim}{\omega }_n^{(1,1)}=1.$$

Conversely, suppose that (3.1) holds. Then from Lemma 2.1 we have ${lim}_n\parallel T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_0)-f_0\parallel =0$. Hence

$${\mathit{st}}_A\text{-}\underset{n}{lim}\parallel T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_0)-f_0\parallel =0(f_0(x)=1).$$

(3.5)

Also from Lemma 2.2 it follows that

$$\parallel T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_1)-f_1\parallel =1-{\omega }_n^{(1,1)}.$$

Therefore, by using (3.1), we get

$${\mathit{st}}_A\text{-}\underset{n}{lim}\parallel T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_1)-f_1\parallel =0(f_1(x):=x).$$

(3.6)

Now we claim that

$${\mathit{st}}_A\text{-}\underset{n}{lim}\parallel T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_2)-f_2\parallel =0(f_2(x):=x^2).$$

(3.7)

By Lemma 2.3, we have

$$\parallel T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_2)-f_2\parallel \le 2(1-{\omega }_n^{(1,1)})+\frac{2{\omega }_n^{(1,1)}}{n}+\frac{1}{3n^2}.$$

(3.8)

Now, for a given $ϵ>0$, we define the following sets:

$$\begin{array}{c}D:=\{n:\parallel T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f_2)-f_2\parallel \ge ϵ\},\hfill \\ D_1:=\{n:1-{\omega }_n^{(1,1)}\ge \frac{ϵ}{4}\},\hfill \\ D_2:=\{n:\frac{{{\omega }^{(1,1)}}_n}{n}\ge \frac{ϵ}{2}\},\hfill \\ D_3:=\{n:\frac{1}{n^2}\ge ϵ\}.\hfill \end{array}$$

From (3.8), it is easy to see that $D\subseteq D_1\cup D_2\cup D_3$. Then, for each $j\in \mathbb{N}$, we get

$$\sum _{n\in D}{a}_{jn}\le \sum _{n\in D_1}{a}_{jn}+\sum _{n\in D_2}{a}_{jn}+\sum _{n\in D_3}{a}_{jn}.$$

(3.9)

Using (3.3), we get

$${\mathit{st}}_A\text{-}\underset{n}{lim}(1-{\omega }_n^{(1,1)})=0$$

and

$${\mathit{st}}_A\text{-}\underset{n}{lim}\frac{{\omega }_n^{(1,1)}}{n}=0.$$

Now, using the above facts and taking the limit as $j\to \mathrm{\infty }$ in (3.9), we conclude that

$$\underset{j}{lim}\sum _{n\in D}{a}_{jn}=0,$$

which gives (3.7). Now, combining (3.5)-(3.7), and using the statistical version of the Korovkin approximation theorem (see Gadjiv and Orhan [10], Theorem 1), we get the desired result.

This completes the proof of the theorem. □

In a similar manner, we can extend Theorem 3.1 to the $(i,j)$-dimensional case for the operators $T_n^{{\omega }^{(1,1)},\dots ,{\omega }^{(1,{\ell }_1)},\dots ,{\omega }^{(r,1)},\dots ,{\omega }^{(r,{\ell }_r)}}(f;x)$ given by (2.9) as follows.

Theorem 3.2 Let $A=(a_{jn})$ be a non-negative regular summability matrix. Then

$${\mathit{st}}_A\text{-}\underset{n}{lim}{\omega }_n^{(i,j)}=1$$

if and only if for all $f\in C[0,1]$,

$${\mathit{st}}_A\text{-}\underset{n}{lim}\parallel T_n^{{\omega }^{(1,1)},\dots ,{\omega }^{(1,{\ell }_1)},\dots ,{\omega }^{(r,1)},\dots ,{\omega }^{(r,{\ell }_r)}}(f)-f\parallel =0.$$

Remark 3.1 If in Theorem 3.2 we replace $A=(a_{jn})$ by the identity matrix, we immediately get the following theorem which is a classical case of Theorem 3.2.

Theorem 3.3 ${lim}_n{\omega }_n^{(i,j)}=1$ if and only if for all $f\in C[0,1]$, the sequence

$$T_n^{{\omega }^{(1,1)},\dots ,{\omega }^{(1,{\ell }_1)},\dots ,{\omega }^{(r,1)},\dots ,{\omega }^{(r,{\ell }_r)}}(f)$$

is uniformly convergent to f on $[0,1]$.

Finally, we display an example which satisfies all the hypotheses of Theorem 3.2, but not of Theorem 3.3. Therefore, this indicates that our A-statistical approximation in Theorem 3.2 is stronger than its classical case.

Take $A=C_1:=(c_{jn})$, the Cesàro matrix of order 1 and

$${\omega }^{(i,j)}:={\left({\omega }_n^{(i,j)}\right)}_{n\in \mathbb{N}}(j=1,\dots ,r-1)$$

are sequences of real numbers defined by

$${\omega }_n^{(i,j)}:=\{\begin{array}{cc}\frac{1}{2}\hfill & \text{if }n=m^2(m\in \mathbb{N});\hfill \\ 1-\frac{1}{n+ij}\hfill & \text{otherwise}.\hfill \end{array}$$

(3.10)

We then observe that

$$0<{\omega }_n^{(i,j)}<1(n\in \mathbb{N})$$

and also that

$${\mathit{st}}_A\text{-}\underset{n}{lim}{\omega }_n^{(i,j)}=1.$$

Therefore, by Theorem 3.2, we have that for all $f\in C[0,1]$,

$${\mathit{st}}_A\text{-}\underset{n}{lim}\parallel T_n^{{\omega }^{(1,1)},\dots ,{\omega }^{(1,{\ell }_1)},\dots ,{\omega }^{(r,1)},\dots ,{\omega }^{(r,{\ell }_r)}}(f)-f\parallel =0.$$

However, since the sequence ${\omega }_n^{(i,j)}$ defined by (3.10) is non-convergent, Theorem 3.3 does not hold in this case.

4 Direct theorems

By $C_B[0,1]$, we denote the space of all real-valued continuous bounded functions f on the interval $[0,1]$, the norm $\parallel \cdot \parallel$ on the space $C_B[0,1]$ is given by

$$\parallel f\parallel =\underset{0\le x\le 1}{sup}|f(x)|.$$

Peetre’s K-functional is defined by

$$K_2(f,\delta )=inf\left[\{\parallel f-g\parallel +\delta \parallel g^{″}\parallel :g\in W^2\}\right],$$

where

$$W^2=\{g\in C_B[0,1]:g^{\prime },g^{″}\in C_B[0,1]\}.$$

By [14] there exists a positive constant $c>0$ s.t.

$$K_2(f,\delta )\le c{w}_2(f,{\delta }^{1/2}),\delta >0,$$

where the second-order modulus of smoothness is

$$w_2(f,\sqrt{\delta })=\underset{0\le h\le \sqrt{\delta }}{sup}\underset{0\le x\le 1}{sup}|f(x+2h)-2f(x+h)+f(x)|.$$

Also, for $f\in C_B[0,1]$, the usual modulus of continuity is given by

$$w(f,\delta )=\underset{0\le h\le \delta }{sup}\underset{0\le x\le 1}{sup}|f(x+h)-f(x)|.$$

Theorem 4.1 Let $f\in C_B[0,1]$ and $0\le {\omega }_n^{(i,j)}<1$. Then, for all $x\in [0,1]$ and $n\in \mathbb{N}$, there exists an absolute constant $C>0$ s.t.

$$|T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f;x)-f(x)|\le C{w}_2(f,{\delta }_n(x)),$$

where

$${\delta }_n^2(x)=x^2[{\left({\omega }_n^{(1,1)}\right)}^2-2{\omega }_n^{(1,1)}+1]+\frac{x{\omega }_n^{(1,1)}}{n}.$$

Proof Let $g\in W^2$. From Taylor’s expansion

$$g(t)=g(x)+g^{\prime }(x)(t-x)+{\int }_x^t(t-x)g^{″}udu,t\in [0,1],$$

and from Lemmas (2.1), (2.2) and (2.3), we get

$$T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(g,x)=g(x)+T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}({\int }_x^t(t-x)g^{″}(u)du,x),$$

hence

$$\begin{array}{rl}|T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(g,x)-g(x)|& \le \left|T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}({\int }_x^t(t-x)g^{″}(u)du,x)\right|\\ \le \left|L_n^{u^{(1)},u^{(2)}}({(t-x)}^2,x)\right|\parallel g^{″}\parallel .\end{array}$$

Using Remark 2.2, we obtain

$$|T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(g,x)-g(x)|\le [x^2{\left({\omega }^{(1,1)}\right)}^2+\frac{2x{\omega }^{(1,1)}}{n}-2x^2{\omega }^{(1,1)}+x^2+\frac{1}{3n^2}-\frac{x}{n}]\parallel g^{″}\parallel .$$

On the other hand, by the definition of $T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f,x)$, we have

$$|T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f;x)|\le \parallel f\parallel .$$

Next

$$\begin{array}{r}|T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f,x)-f(x)|\\ \le |T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}((f-g);x)-(f-g)(x)|+|T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(g,x)-g(x)|\\ \le \parallel f-g\parallel +[x^2{\left({\omega }^{(1,1)}\right)}^2+\frac{2x{\omega }^{(1,1)}}{n}-2x^2{\omega }^{(1,1)}+x^2+\frac{1}{3n^2}-\frac{x}{n}]\parallel g^{″}\parallel .\end{array}$$

Hence, taking infimum on the right-hand side over all $g\in W^2$, we get

$$|T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f,x)-f(x)|\le C{K}_2(f,{\delta }_n^2(x)).$$

In view of the property of K-functional, for every $0<{\omega }_n^{(i,j)}<1$, we get

$$|T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f,x)-f(x)|\le C{w}_2(f,{\delta }_n(x)).$$

This completes the proof of the theorem. □

Theorem 4.2 Let $f\in C_B[0,1]$ be such that $f^{\prime },f^{″}\in C_B[0,1]$ and $0<{\omega }_n^{(i,j)}<1$, $j=1,2,3,\dots ,n$, such that ${\omega }_n^{(i,j)}\to 1$ as $n\to \mathrm{\infty }$. Then the following equality holds:

$$\underset{n\to \mathrm{\infty }}{lim}n(T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f,x)-f(x))=\frac{x}{2}{f}^{″}(x)$$

uniformly on $[0,1]$.

Proof By the Taylor’s formula, we may write

$$f(t)=f(x)+f^{\prime }(x)(t-x)+\frac{1}{2}{f}^{″}(x){(t-x)}^2+r(t,x){(t-x)}^2,$$

(4.1)

where $r(t,x)$ is the remaining term and ${lim}_{t\to x}r(t,x)=0$. Applying $T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f;x)$ to (4.1), we obtain

$$\begin{array}{r}n(T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f,x)-f(x))\\ =n{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(t-x;x)f^{\prime }(x)\\ +n{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}({(t-x)}^2;x)\frac{f^{″}(x)}{2}+n{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(r(t,x){(t-x)}^2;x).\end{array}$$

By the Cauchy-Schwartz inequality, we have

$$T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(r(t,x){(t-x)}^2;x)\le \sqrt{T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(r^2{(t,x)}^2;x)}\sqrt{T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}({(t-x)}^4;x)}.$$

(4.2)

Observe that $r^2(x,x)=0$ and $r^2(\cdot ,x)\in C[0,1]$. Then it follows from Theorem 4.1 that

$$\underset{n\to \mathrm{\infty }}{lim}{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(r^2(t,x);x)=r^2(x,x)=0$$

(4.3)

uniformly with respect to $x\in [0,1]$.

Now, from (4.2), (4.3) and Remark 2.2, we get

$$\underset{n\to \mathrm{\infty }}{lim}n{T}_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(r(t,x){(t-x)}^2;x)=0.$$

Finally, using Remark 2.1, we get the following:

$$\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim}n(T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(f,x)-f(x))=& \underset{n\to \mathrm{\infty }}{lim}n(f^{\prime }(x)T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}((t-x);x))\\ +\frac{1}{2}{f}^{″}(x)T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}({(t-x)}^2;x)\\ +\frac{1}{2}{f}^{″}(x)T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}({(t-x)}^2;x)\\ +T_n^{{\omega }^{(1,1)},{\omega }^{(1,2)}}(r(t,x){(t-x)}^2;x)\\ =\frac{x}{2}{f}^{″}(x).\end{array}$$

 □