In the last few decades, many researchers emphasized expanding or generalizing the Korovkin-type hypotheses from numerous points of view in light of a few distinct angles, containing (for instance) space of functions, Banach spaces summability theory, etc. Certainly, the change of Korovkin-type hypothesis is far from being finished till today. For additional points of interest and outcomes associated with the Korovkin-type hypothesis and other related advancements, we allude the reader to the current works [7–10, 22], and [17]. The main objective of this paper is to extend the notion of statistical convergence by the help of the deferred weighted regular technique and to show how this technique leads to a number of results based upon an approximation of functions of two variables over the Banach space \(C_{B}(\mathcal{D})\). Moreover, we establish some important approximation theorems related to the statistical deferred weighted \(\mathcal{B}\)-summability and deferred weighted \(\mathcal{B}\)-statistical convergence, which will effectively extend and improve most (if not all) of the existing results depending upon the choice of sequences of the deferred weighted \(\mathcal{B}\) means. Based upon the proposed methodology and techniques, we intend to estimate the rate of convergence and investigate the Korovkin-type approximation results. In fact, we extend here the result of Kadak et al. [1] by using the notion of statistical deferred weighted \(\mathcal{B}\)-summability and present the following theorem.
Let \(\mathcal{D}\) be any compact subset of the real two-dimensional space. We denote by \(C_{B}(\mathcal{D})\) the space of all continuous real-valued functions on \(\mathcal{D}=I\times I\) (\(I=[0,A]\)), \(A\leq\frac{1}{2}\) and equipped with the norm
$$\Vert f \Vert _{C_{B}(\mathcal{D})} =\sup\bigl\{ \bigl\vert f(x,y) \bigr\vert :(x,y)\in\mathcal{D}\bigr\} ,\quad f\in C_{B}(\mathcal{D}). $$
Let \(T:C_{B}(\mathcal{D})\rightarrow C_{B}(\mathcal{D})\) be a linear operator. Then we say that T is a positive linear operator provided
$$f\geqq0\quad \textrm{implies} \quad T(f)\geqq0. $$
Also, we use the notation \(T(f;x,y)\) for the values of \(T(f)\) at the point \((x,y)\in\mathcal{D}\).
Theorem 3
Let
\(\mathcal{B}\in \mathcal{R}^{+}_{D(w)}\), and let
\((a_{n})\)
and
\((b_{n})\)
be sequences of non-negative integers. Let
\(T_{n}\) (\(n\in \mathbb{N}\)) be a sequence of positive linear operators from
\(C_{B}(\mathcal{D})\)
into itself, and let
\(f\in C_{B}(\mathcal{D})\). Then
$$\begin{aligned} & \mathrm{stat}_{D(\bar{N})}\lim_{n} \bigl\Vert T_{n}\bigl(f(s,t);x,y\bigr)- f(x,y) \bigr\Vert _{C_{B}(\mathcal{D})}=0, \quad f\in C_{B}(\mathcal{D}) \end{aligned}$$
(3.1)
if and only if
$$\begin{aligned} & \mathrm{stat}_{D(\bar{N})}\lim_{n} \bigl\Vert T_{n}\bigl(f_{j}(s,t);x,y\bigr)- f(x,y) \bigr\Vert _{C_{B}(\mathcal{D})}=0, \quad (j=0,1,2,3), \end{aligned}$$
(3.2)
where
$$\begin{aligned} f_{0}(s,t)&=1, \qquad f_{1}(s,t)=\frac{s}{1-s}, \qquad f_{2}(s,t)=\frac{t}{1-t}\quad \textit{and} \\ f_{3}(s,t)&= \biggl(\frac{s}{1-s} \biggr)^{2}+ \biggl( \frac{t}{1-t} \biggr)^{2}. \end{aligned}$$
Proof
Since each of the functions \(f_{j}(s,t)\in C_{B}(\mathcal{D})\), the following implication
$$(3.1)\quad \Longrightarrow\quad (3.2) $$
is fairly obvious. In order to complete the proof of Theorem 3, we first assume that (3.2) holds true. Let \(f\in C_{B}(\mathcal{D})\), \(\forall(x,y)\in\mathcal{D}\). Since \(f(x,y)\) is bounded on \(\mathcal{D}\), then there exists a constant \(\mathcal{M}>0\) such that
$$\bigl\vert f(x,y) \bigr\vert \leqq \mathcal{M}\quad (\forall x,y\in \mathcal{D}), $$
which implies that
$$ \bigl\vert f(s,t)-f(x,y) \bigr\vert \leqq 2\mathcal{M} \quad (s,t,x,y\in\mathcal{D}). $$
(3.3)
Clearly, f is a continuous function on \(\mathcal{D}\), for given \(\epsilon>0\), there exists \(\delta=\delta(\epsilon)>0\) such that
$$ \bigl\vert f(s,t)-f(x,y) \bigr\vert < \epsilon\quad \mbox{whenever } \biggl\vert \frac{s}{1-s}-\frac{x}{1-x} \biggr\vert < \delta \mbox{ and } \biggl\vert \frac{t}{1-t}-\frac{y}{1-y} \biggr\vert < \delta $$
(3.4)
for all \(s,t,x,y\in\mathcal{D}\).
From equations (3.3) and (3.4), we get
$$ \bigl\vert f(s,t)-f(x,y) \bigr\vert < \epsilon+ \frac{2\mathcal{M}}{\delta^{2}} \bigl(\bigl[\varphi(s,x)\bigr]^{2}+\bigl[\varphi(t,y) \bigr]^{2} \bigr), $$
(3.5)
where
$$\varphi(s,x)=\frac{s}{1-s}-\frac{x}{1-x}\quad \mbox{and}\quad \varphi(t,y)=\frac{t}{1-t}-\frac{y}{1-y}. $$
Since the function \(f\in C_{B}(\mathcal{D})\), inequality (3.5) holds for \(s,t,x,y\in\mathcal{D}\).
Now, since the operator \(T_{n}(f(s,t);x,y)\) is linear and monotone, so inequality (3.5) under this operator becomes
$$\begin{aligned} \bigl\vert T_{n}\bigl(f(s,t);x,y\bigr)-f(x,y) \bigr\vert =& \bigl\vert T_{n}\bigl(f(s,t)-f(x,y);x,y\bigr)+f(x,y) \bigl[T_{k}(f_{0};x,y)-f_{0}\bigr] \bigr\vert \\ \leqq& \bigl\vert T_{n}\bigl(f(s,t)-f(x,y);x,y\bigr)+\mathcal{M} \bigl[T_{k}(1;x,y)-1\bigr] \bigr\vert \\ \leqq& \biggl\vert T_{n} \biggl(\epsilon+\frac{2\mathcal{M}}{\delta^{2}} \bigl[ \varphi(s,x)^{2}+\varphi(t,y)^{2} \bigr];x,y \biggr) \biggr\vert \\ &{} +\mathcal{M} \bigl\vert T_{n}(1;x,y)-1 \bigr\vert \\ \leqq&\epsilon+(\epsilon+\mathcal{M}) \bigl\vert T_{n}(f_{0};x,y)-f_{0}(x,y) \bigr\vert \\ &{} +\frac{2\mathcal{M}}{\delta^{2}} \bigl\vert T_{n}(f_{3};x,y)-f_{3}(x,y) \bigr\vert \\ &{}-\frac{4\mathcal{M}}{\delta^{2}} \biggl(\frac{x}{1-x} \biggr) \bigl\vert T_{n}(f_{1};x,y)-f_{1}(x,y) \bigr\vert \\ &{}-\frac{4\mathcal{M}}{\delta^{2}} \biggl(\frac{y}{1-y} \biggr) \bigl\vert T_{n}(f_{2};x,y)-f_{2}(x,y) \bigr\vert \\ &{}+\frac{2\mathcal{M}}{\delta^{2}} \biggl( \biggl(\frac{x}{1-x} \biggr)^{2} + \biggl(\frac{y}{1-y}^{2} \biggr) \biggr) \bigl\vert T_{n}(f_{0};x,y)-f_{0}(x,y) \bigr\vert \\ \leqq &\epsilon+ \biggl(\epsilon+\mathcal{M}+\frac{4\mathcal{M}}{\delta^{2}} \biggr) \bigl\vert T_{n}(1;x,y)-1 \bigr\vert \\ &{}+\frac{4\mathcal{M}}{\delta^{2}} \bigl\vert T_{n}(f_{1};x,y)-f_{1}(x,y) \bigr\vert +\frac{4\mathcal{M}}{\delta^{2}} \bigl\vert T_{n}(f_{2};x,y)-f_{2}(x,y) \bigr\vert \\ &{}+\frac{2\mathcal{M}}{\delta^{2}} \bigl\vert T_{n}(f_{3};x,y)-f_{3}(x,y) \bigr\vert . \end{aligned}$$
(3.6)
Next, taking \(\sup_{x,y\in\mathcal{D}}\) on both sides of (3.6), we get
$$ \bigl\Vert T_{n}\bigl(f(s,t);x,y\bigr)-f(x,y) \bigr\Vert _{C_{B}(\mathcal{D})} \leqq\epsilon+ N\sum_{j=0}^{3} \bigl\Vert T_{n}\bigl(f_{j}(s,t);x,y\bigr)-f_{j}(x,y) \bigr\Vert _{C_{B}(\mathcal{D})}, $$
(3.7)
where
$$N= \biggl\{ \epsilon+\mathcal{M}+\frac{4\mathcal{M}}{\delta^{2}} \biggr\} . $$
We now replace \(T_{n}(f(s,t);x,y)\) by
$$\mathfrak{L}_{n}\bigl(f(s,t);x,y\bigr)=\frac{1}{P_{n}}\sum _{m={a_{n}+1}}^{b_{n}} \sum_{k=0}^{\infty}p_{m}b_{m,k}(i)T_{k} \bigl(f(s,t);x,y\bigr)\quad (\forall i,m\in\mathbb{N}) $$
in equation (3.7).
Now, for given \(r>0\), we choose \(0<\epsilon'<r\), and by setting
$$\mathcal{K}_{n}= \bigl\vert \bigl\{ n:n\leqq \mathbb{N}\mbox{ and } \bigl\vert \mathfrak{L}_{n}\bigl(f(s,t);x,y\bigr)-f(x,y) \bigr\vert \geqq r\bigr\} \bigr\vert $$
and
$$\mathcal{K}_{j,n}= \biggl\vert \biggl\{ n:n\leqq \mathbb{N}\mbox{ and } \bigl\vert \mathfrak{L}_{n}\bigl(f_{j}(s,t);x,y \bigr)-f_{j}(x,y) \bigr\vert \geqq \frac{r-\epsilon'}{4N} \biggr\} \biggr\vert \quad (j=0,1,2,3), $$
we easily find from (3.7) that
$$\mathcal{K}_{n}\leqq \sum_{j=0}^{3} \mathcal{K}_{j,n}. $$
Thus, we have
$$ \frac{ \Vert \mathcal{K}_{n} \Vert _{C_{B}(\mathcal{D})}}{n}\leqq \sum_{j=0}^{3} \frac{ \Vert \mathcal{K}_{j,n} \Vert _{C_{B}(\mathcal{D})}}{n}. $$
(3.8)
Clearly, from the above supposition for the implication in (3.2) and Definition 4, the right-hand side of (3.8) tends to zero \((n\rightarrow\infty)\). Subsequently, we obtain
$$\mathrm{stat}_{D(\bar{N})}\lim_{n\rightarrow\infty} \bigl\Vert T_{n} \bigl(f_{j}(s,t);x,y\bigr)-f_{j}(x,y) \bigr\Vert _{C_{B}(\mathcal{D})}=0\quad (j=0,1,2,3). $$
Hence, implication (3.1) is fairly true, which completes the proof of Theorem 3. □
Remark 2
If
$$\mathcal{B}=\mathcal{A},\qquad a_{n}=0,\quad \mbox{and}\quad b_{n}=n \quad (\forall n) $$
in our Theorem 3, then we obtain a statistical weighted \(\mathcal{A}\)-summability version of Korovkin-type approximation theorem (see [28]). Furthermore, if we substitute
$$a_{n}+1=\alpha(n)\quad \textrm{and}\quad b_{n}=\beta(n)\quad (\forall n) $$
in our Theorem 3, then we obtain a statistical weighted \(\mathcal{B}\)-summability version of Korovkin-type approximation theorem (see [1]). Finally,
$$\mathcal{B}=I\quad (\textrm{identity matrix}),\quad a_{n}=0,\quad \textrm{and}\quad b_{n}=n \quad (\forall n) $$
in our Theorem 3, then we obtain a statistical weighted convergence version of Korovkin-type approximation theorem (see [19]).
Now we recall the generating function type Meyer–König and Zeller operators of two variables (see [30] and [31]).
Let us take the following sequence of generalized linear positive operators:
$$\begin{aligned} L_{n,m}\bigl(f(s,t);x,y\bigr) =&\frac{1}{h_{n}(x,s)h_{m}(y,t)}\sum _{k=0}^{\infty}\sum_{l=0}^{\infty} f \biggl(\frac{a_{k,n}}{a_{k,n}+q_{n}},\frac{c_{l,m}}{c_{l,m}+r_{m}} \biggr) \\ &{}\times\Gamma_{k,n}(s) \Gamma_{l,m}(t)x^{k}y^{l}, \end{aligned}$$
(3.9)
where
$$0\leqq\frac{a_{k,n}}{a_{k,n}+q_{n}}\leqq A\quad \textrm{and}\quad 0\leqq\frac{c_{l,m}}{c_{l,m}+r_{m}} \leqq B\quad \bigl(\forall A,B\in(0,1)\bigr). $$
For the sequences of functions, \((\Gamma_{k,n}(s) )_{n\in\mathbb{N}}\) and \((\Gamma_{l,m}(t) )_{n\in\mathbb{N}}\) are the generating functions, \(h_{n}(x,s)\) and \(h_{m}(y,t)\) are defined by
$$h_{n}(x,s)=\sum_{k=0}^{\infty} \Gamma_{k,n}(s)x^{k}\quad \textrm{and}\quad h_{m}(y,t) =\sum_{l=0}^{\infty} \Gamma_{l,m}(t)y^{l}\quad \bigl(s,t\in I\times I\subset \mathbb{R}^{2}\bigr). $$
Because the nodes are given by
$$s=\frac{a_{k,n}}{a_{k,n}+q_{n}}\quad \textrm{and}\quad t=\frac{c_{l,m}}{c_{l,m}+r_{m}}, $$
the denominators of
$$\frac{s}{1-s}=\frac{a_{n,k}}{q_{n}}\quad \textrm{and}\quad \frac{t}{1-t}= \frac{c_{l,m}}{r_{m}} $$
are independent of k and l, respectively.
We also suppose that the following conditions hold true:
-
(i)
\(h_{n}(x,s)=(1-x)h_{n+1}(x,s)\) and \(h_{m}(y,t)=(1-y)h_{m+1}(y,t)\);
-
(ii)
\(q_{n}\Gamma_{k,n+1}(s)=a_{k+1,n}\Gamma_{k+1,n}(s)\) and \(r_{m}\Gamma_{l,m+1}(t)=c_{l+1,m}\Gamma_{l+1,m}(t)\);
-
(iii)
\(q_{n}\rightarrow\infty\), \(\frac{q_{n+1}}{q_{n}}\rightarrow\infty\) (\(r_{m}\rightarrow\infty\)), \(\frac{r_{m+1}}{r_{m}}\rightarrow\infty\) and \(q_{n},r_{m}\neq0\) (\(\forall m,n\));
-
(iv)
\(a_{k+1,n}-a_{k,n+1}=\phi_{n}\) and \(c_{l+1,m}-a_{l,m+1}=\psi_{m}\),
where
$$\phi_{n}\leqq n\leqq\infty,\qquad \psi_{m}\leqq m\leqq \infty\quad \mbox{and}\quad a_{0,n}=c_{m,0}=0. $$
It is easy to see that \(L_{n}(f(s,t);x,y)\) is positive linear operators. We also observe that
$$L_{n}(1;x,y)=1,\qquad L_{n} \biggl(\frac{s}{1-s};x,y \biggr)=\frac{x}{1-x},\qquad L_{n} \biggl(\frac{t}{1-t};x,y \biggr)=\frac{t}{1-t} $$
and
$$L_{n} \biggl( \biggl(\frac{s}{1-s} \biggr)^{2}+ \biggl( \frac{t}{1-t} \biggr)^{2};x,y \biggr) =\frac{x^{2}}{(1-x)^{2}} \frac{q_{n+1}}{q_{n}}+\frac{y^{2}}{(1-y)^{2}}\frac{r_{n+1}}{r_{n}} +\frac{x}{1-x} \frac{\phi_{n}}{q_{n}}+\frac{y}{1-y}\frac{\psi_{n}}{r_{n}}. $$
Example 3
Let \(T_{n}:C_{B}(\mathcal{D})\rightarrow C_{B}(\mathcal{D})\), \(\mathcal{D}=[0,A]\times [0,A]\), \(A\leq\frac{1}{2}\) be defined by
$$ T_{n}(f;x,y)=(1+x_{n})L_{n}(f;x,y), $$
(3.10)
where \((x_{n})\) is a sequence defined as in Example 2. It is clear that the sequence \((T_{n})\) satisfies the conditions (3.2) of our Theorem 3, thus we obtain
$$\begin{aligned} &\mathrm{stat}_{D(\bar{N})}\lim_{n} \bigl\Vert T_{n}(1;x,y)- 1 \bigr\Vert _{C_{B}(\mathcal{D})}=0 \\ &\mathrm{stat}_{D(\bar{N})}\lim_{n} \biggl\Vert T_{n} \biggl(\frac{s}{1-s};x,y \biggr)-\frac{x}{1-x} \biggr\Vert _{C_{B}(\mathcal{D})}=0 \\ &\mathrm{stat}_{D(\bar{N})}\lim_{n} \biggl\Vert T_{n} \biggl(\frac{t}{1-t};x,y \biggr)-\frac{y}{1-y} \biggr\Vert _{C_{B}(\mathcal{D})}=0 \end{aligned}$$
and
$$\mathrm{stat}_{D(\bar{N})}\lim_{n} \biggl\Vert T_{n} \biggl[ \biggl(\frac{s}{1-s} \biggr)^{2}+ \biggl( \frac{t}{1-t} \biggr)^{2};x,y \biggr]- \biggl[ \biggl( \frac{s}{1-s} \biggr)^{2}+ \biggl(\frac{t}{1-t} \biggr)^{2} \biggr] \biggr\Vert _{C_{B}(\mathcal{D})}=0. $$
Therefore, from Theorem 3, we have
$$\mathrm{stat}_{D(\bar{N})}\lim_{n} \bigl\Vert T_{n}\bigl(f(s,t);x,y\bigr)- f(x,y) \bigr\Vert _{C_{B}(\mathcal{D})}=0, \quad f\in C_{B}(\mathcal{D}). $$
However, since \((x_{n})\) is not statistically weighted \(\mathcal{B}\)-summable, so the result of Kadak et al. [1], p. 85, Theorem 3, certainly does not hold for the operators defined by us in (3.10). Moreover, as \((x_{n})\) is statistically deferred weighted \(\mathcal{B}\)-summable, therefore we conclude that our Theorem 3 works for the operators which we consider here.