Inequalities for Approximation of New Defined Fuzzy Post-Quantum Bernstein Polynomials via Interval-Valued Fuzzy Numbers

Abstract

In this study, we introduce new defined fuzzy post-quantum Bernstein polynomials and present examples illustrating their existence. We investigate their approximation properties via interval-valued fuzzy numbers. We obtain a fuzzy Korovkin-type approximation result, and we obtain inequalities estimating the rate of fuzzy convergence for these polynomials by means of the fuzzy modulus of continuity and Lipschitz-type fuzzy functions. Lastly, we present a Voronovskaja type asymptotic result for fuzzy post-quantum Bernstein polynomials. The methods in this paper are crucial and symmetric in terms of extending the approximation results of these polynomials from the real function space to the fuzzy function space and the applicability to the other operators.

1. Introduction

Fuzzy mathematics is a significant field in mathematics that has attracted attention and has been used in applied sciences such as engineering, medicine, finance, chemistery, biology, nuclear science, robotics, ecology, education and so forth.

Fuzzy sets theory was first presented by Zadeh in 1965 with his paper “Fuzzy Sets” [1]. Some basic studies related to fuzzy mathematics can be seen in references [2,3,4,5,6,7,8,9].

Recently, fuzzy mathematics has gained importance in many areas of mathematics such as in the approximation theory. For instance, Gal has presented the fuzzy version of the Weierstrass approximation theorem [10]. Congxin and Danghang have given an another version of the fuzzy Weierstrass approximation theorem [11]. Gal has estimated the degree of approximation of fuzzy mappings by fuzzy polynomials in [12], and Gal et al. have also studied some approximation results in references [4,13,14,15,16]. Moreover, Anastassiou has investigated a fuzzy Korovkin-type approximation theorem and its very important applications in operator theory [3,4,17,18].

Berstein polynomials were introduced to prove the Weierstrass approximation theorem constructively [19]. After the improvement of quantum calculus, Bernstein polynomials were generalized to quantum Bernstein polynomials [20,21].

Latterly, with the development of post-quantum calculus, Mursaleen et al. [22] applied post-quantum calculus to the approximation theory and introduced the post quantum Bernstein operators. Approximation results for post-quantum analogue of Bernstein-type rational functions and some other operators can also be seen in references [23,24,25,26,27].

We recall now certain notations of post-quantum calculus, called (p,q)-calculus.

Let $0<q<p\le 1$. For each non-negative integer n, k such that $n\ge k\ge 0$, (p,q)-integer ${\left[n\right]}_{p,q}$, (p,q)-factorial ${\left[n\right]}_{p,q}!$ and (p,q)-binomial coefficients are respectively defined by:

$${\left[n\right]}_{p,q}:=\frac{p^n-q^n}{p-q},$$

$${\left[n\right]}_{p,q}!:=\left\{\begin{array}{cc}{\left[n\right]}_{p,q}{\left[n-1\right]}_{p,q}...{\left[2\right]}_{p,q}{\left[1\right]}_{p,q},& n=1,2,3,...\\ 1,& n=0,\end{array}\right.$$

and

$${\left[\begin{array}{c}n\\ k\end{array}\right]}_{p,q}:=\frac{{\left[n\right]}_{p,q}!}{{\left[n-k\right]}_{p,q}!{\left[k\right]}_{p,q}!}.$$

Mursaalen et al. [22] have introduced post-quantum Bernstein polynomials, which are defined for any $x\in \left[0,1\right]$ and $f\in C\left[0,1\right]$ by:

$$B_n^{p,q}\left(f;x\right)=\sum _{i=0}^n{\tau }_{n,i}(x;p,q)f\left({\xi }_{n,i}^{p,q}\right),$$

(1)

where

$${\tau }_{n,i}(x;p,q)=\frac{p^{i(i-1)/2}}{p^{n(n-1)/2}}{\left[\begin{array}{c}n\\ i\end{array}\right]}_{p,q}{x}^i\prod _{s=0}^{n-i-1}\left(p^s-q^{s}x\right)$$

and

$${\xi }_{n,i}^{p,q}=\frac{{\left[i\right]}_{p,q}}{p^{i-n}{\left[n\right]}_{p,q}}.$$

It is clear that these polynomials are positive linear operators from $C\left[0,1\right]$ into itself. In [22], Mursaalen et al. calculated the following results at monomials:

$$B_n^{p,q}\left(1;x\right)=1,$$

(2)

$$B_n^{p,q}\left(t;x\right)=x,$$

(3)

$$B_n^{p,q}\left(t^2;x\right)=\frac{q{\left[n-1\right]}_{p,q}}{{\left[n\right]}_{p,q}}{x}^2+\frac{p^{n-1}}{{\left[n\right]}_{p,q}}x,$$

(4)

and investigated Korovkin-type approximation properties of the post-quantum Bernstein polynomials.

In this study, we construct new defined fuzzy post-quantum Bernstein polynomials and investigate their interval-valued approximation properties. We aim to extend their approximation results to the fuzzy function space.

In Section 2, we recall basic concepts of the fuzzy mathematics. In Section 3, we construct fuzzy post-quantum Bernstein polynomials and provide some auxiliary results. We also present examples illustrating their existence for certain basic fuzzy number-valued continuous functions. In Section 4, we demonstrate a fuzzy Korovkin-type approximation result for these polynomials, and we estimate the degree of approximation of the fuzzy post-quantum Bernstein polynomials by using the fuzzy modulus of continuity and Lipschitz-type fuzzy functions. In Section 5, we present a Voronovskaja type asymptotic estimation for the fuzzy post-quantum Bernstein polynomials. In Section 6, we discuss conclusions.

2. Background

Fuzzy numbers are a significant concept in fuzzy mathematics. We firstly recall some basic concepts of fuzzy mathematics, the details of which can be found in references [3,8,18,28,29,30].

Definition 1.

Let $z:\mathbb{R}\to \left[0,1\right]$ be a function. If z satisfies the following properties, then z is called a fuzzy number [2], and the set of all fuzzy numbers is denoted by ${\mathbb{R}}_{\mathcal{F}}:$

(i) 

z is normal, i.e., there exists at least an element $x_0\in \mathbb{R}$ satisfying $z\left(x_0\right)=1$;

(ii) 

z is a convex fuzzy subset, i.e., for each $\lambda \in \left[0,1\right]$ and $x,y\in \mathbb{R},$

$$z(\lambda x+(1-\lambda )y)\ge min\left\{z\left(x\right),z\left(y\right)\right\};$$

(iii) 

z is upper semi-continuous on $\mathbb{R},$ i.e., there exists at least a neighborhood $N\left(x_0\right)$ such that

$$z\left(x\right)\le z\left(x_0\right)+\epsilon ,$$

for each $x\in N\left(x_0\right)$;

(iv) 

The closure of the set $\left\{x\in \mathbb{R}:z\left(x\right)>0\right\}$ is compact in $\mathbb{R}$, where $\overline{A}$ denotes the closure of the set $A.$

A fuzzy number is represented with an r-level set ${\left[z\right]}^r$ defined as:

$${\left[z\right]}^r=\left\{\begin{array}{cc}\left\{x\in \mathbb{R}:z\left(x\right)\ge r\right\},& r\in \left(0,1\right]\\ \overline{\left\{x\in \mathbb{R}:z\left(x\right)>0\right\}},& r=0,\end{array}\right.$$

which is a bounded and closed interval of $\mathbb{R}$ [9] and is denoted by:

$${\left[z\right]}^r=\left[z_{-}^r,z_{+}^r\right]$$

for all $r\in \left[0,1\right]$, where $z_{-}^r$ and $z_{+}^r$ denote the left and right endpoints of the r-level interval ${\left[z\right]}^r$. It is clear that if $r_1,r_2\in \left[0,1\right]$ such that $r_1\le r_2$ then ${\left[z\right]}^{r_2}\subseteq {\left[z\right]}^{r_1}$.

For $z,w\in {\mathbb{R}}_{\mathcal{F}}$ and $k\in \mathbb{R}$, the summation and the product with scalar on ${\mathbb{R}}_{\mathcal{F}}$ are uniquely defined by

$${\left[z\oplus w\right]}^r={\left[z\right]}^r+{\left[w\right]}^r\mathrm{and}{\left[k\odot z\right]}^r=k{\left[z\right]}^r,r\in \left[0,1\right],$$

where ${\left[z\right]}^r+{\left[w\right]}^r$ means the usual sum of two intervals as subsets of $\mathbb{R}$ and $k{\left[z\right]}^r$ means the usual product between a real scalar and a subset of $\mathbb{R}$, i.e.,

$${\left(z\oplus w\right)}_{-}^r=z_{-}^r+w_{-}^r\mathrm{and}{\left(z\oplus w\right)}_{+}^r=z_{+}^r+w_{+}^r,$$

$${\left(k\odot z\right)}_{-}^r=k{z}_{-}^r\mathrm{and}{\left(k\odot z\right)}_{+}^r=k{z}_{+}^r.$$

By $z\oplus w$ and $k\odot z$, $z,w\in {\mathbb{R}}_{\mathcal{F}}$, we denote the summation and the scalar product on ${\mathbb{R}}_{\mathcal{F}}$, respectively. If there exists an element $e\in {\mathbb{R}}_{\mathcal{F}}$ such that $z\oplus w=e$, then we call z the difference e and w on ${\mathbb{R}}_{\mathcal{F}}$ denoted by $z:=e\ominus w$ [2].

Definition 2

([2]). Let D be a non-negative real-valued function defined on ${\mathbb{R}}_{\mathcal{F}}\times {\mathbb{R}}_{\mathcal{F}}$ by

$$D(z,w)=\underset{r\in \left[0,1\right]}{sup}max\left\{\left|z_{-}^r-w_{-}^r\right|,\left|z_{+}^r-w_{+}^r\right|\right\},$$

where ${\left[z\right]}^r=\left[z_{-}^r,z_{+}^r\right]$, ${\left[w\right]}^r=\left[w_{-}^r,w_{+}^r\right]\subset \mathbb{R}$.

D is a complete metric space on ${\mathbb{R}}_{\mathcal{F}}$ satisfying the following properties:

(i) 

$D(z\oplus v,w\oplus v)=D(z,w),\forall z,v,w\in {\mathbb{R}}_{\mathcal{F}}$;

(ii) 

$D(k\odot z,k\odot w)=\left|k\right|D(z,w)$, $\forall z,w\in {\mathbb{R}}_{\mathcal{F}},k\in \mathbb{R};$

(iii) 

$D(z\oplus w,v\oplus e)\le D(z,v)+D(w,e),\forall z,v,w,e\in {\mathbb{R}}_{\mathcal{F}}.$

Definition 3

([2]). A partial order on ${\mathbb{R}}_{\mathcal{F}}$ is defined by “≼”$:z,w\in {\mathbb{R}}_{\mathcal{F}}$, $z\preccurlyeq w$ iff $z_{-}^r\le w_{-}^r$ and $z_{+}^r\le w_{+}^r$ for each $r\in \left[0,1\right].$ Here, “≤” is partial order on $\mathbb{R}.$

Lemma 1

([4]). Let $k,l\in \mathbb{R}$, $w,z\in {\mathbb{R}}_{\mathcal{F}}$ and let $\tilde{o}={\chi }_{\left\{0\right\}}$ be the characteristic function of $\left\{0\right\}$.

Then:

(i) 

A neutral element with regard to ⊕ is $\tilde{o}\in {\mathbb{R}}_{\mathcal{F}}$;

(ii) 

For each $z\ne \tilde{o}$, $z\in {\mathbb{R}}_{\mathcal{F}}$ does not have an opposite in ${\mathbb{R}}_{\mathcal{F}}$;

(iii) 

If $k,l\ge 0$ or $k,l\le 0$, we have

$$(k+l)\odot z=k\odot z\oplus l\odot z,$$

(For general $k,l\in \mathbb{R}$ (iii) is not true);

(iv) 

$$k\odot (w\oplus z)=k\odot w\oplus k\odot z;$$

(v) 

$$k\odot (l\odot z)=(k.l)\odot z.$$

Definition 4

([3]). A function h from $\left[a,b\right]$ to ${\mathbb{R}}_{\mathcal{F}}$ is called a fuzzy number-valued function, which has the following representation:

$${\left[h\left(x\right)\right]}^r=\left[h_{-}^r\left(x\right),h_{+}^r\left(x\right)\right],$$

for each $x\in \left[a,b\right]$ and $r\in \left[0,1\right]$, where $h_{-}^r\left(x\right)$ and $h_{+}^r\left(x\right)$ denote the left and right endpoints of the interval ${\left[h\left(x\right)\right]}^r$. Also, $h_{-}^r$ and $h_{+}^r$ are real-valued functions defined on $\left[a,b\right].$

Similarly, a fuzzy algebraic polynomial of degree n has the following form:

$$P_n\left(x\right)=\underset{k=0}{\overset{n}{⨁}}{x}^k\odot c_k,$$

where $c_k\in {\mathbb{R}}_{\mathcal{F}}$, $k=0,1,...,n.$ Here, ${\displaystyle \underset{k=0}{\overset{n}{⨁}}}$ denotes the finite fuzzy n-summation.

Definition 5

([3]). Let l and h be fuzzy number-valued functions defined on $\left[a,b\right]$. The distance between l and h is defined by:

$$\begin{array}{ccc}\hfill D^{*}(l,h)& =& \underset{x\in \left[a,b\right]}{sup}D(l\left(x\right),h\left(x\right))\hfill \\ & =& \underset{x\in \left[a,b\right]}{sup}\underset{r\in \left[0,1\right]}{sup}max\left\{\left|l_{-}^r\left(x\right)-h_{-}^r\left(x\right)\right|,\left|l_{+}^r\left(x\right)-h_{+}^r\left(x\right)\right|\right\}.\hfill \end{array}$$

Definition 6

([8,28]). A fuzzy number-valued function h defined on $\left[a,b\right]$ is called a fuzzy continuous at $x_0\in \left[a,b\right]$ if and only if h is sequential continuous at $x_0\in \left[a,b\right]$. If h is continuous for each $x_0\in \left[a,b\right]$, then h is called a fuzzy continuous function on $\left[a,b\right]$, and by $C_{\mathcal{F}}\left[a,b\right]$ be denoted the space of all fuzzy continuous functions on $\left[a,b\right].$$\left(C_{\mathcal{F}}\left[a,b\right],D^{*}\right)$ is a complete metric space.

Any fuzzy continuous function h has the following representation of r-level interval:

$${\left[h\left(x\right)\right]}^r=\left[h_{-}^r\left(x\right),h_{+}^r\left(x\right)\right],$$

for each $x\in \left[a,b\right]$ and $r\in \left[0,1\right],$ where $h_{-}^r$ and $h_{+}^r$ are real-valued continuous functions defined on $\left[a,b\right]$.

The summation and the scalar product in $C_{\mathcal{F}}\left[a,b\right]$ are defined by:

$$\left(h\oplus l\right)\left(x\right)=h\left(x\right)\oplus l\left(x\right),$$

$$\left(k\odot h\right)\left(x\right)=k\odot h\left(x\right),$$

for each $x\in \left[a,b\right],$$k\in \mathbb{R},$$h,l\in C_{\mathcal{F}}\left[a,b\right].$ Additionally, $\tilde{0}$ is a fuzzy number-valued function defined on $\left[a,b\right]$ such that $\tilde{0}\left(x\right)=\tilde{o}$ for each $x\in \left[a,b\right]$, where $\tilde{o}$ is the neutral element with regard to ⊕ in ${\mathbb{R}}_{\mathcal{F}}$.

Considering the representation of ${\left[h\left(x\right)\right]}^r=\left[h_{-}^r\left(x\right),h_{+}^r\left(x\right)\right]$ and ${\left[l\left(x\right)\right]}^r=\left[l_{-}^r\left(x\right),l_{+}^r\left(x\right)\right]$ for each $x\in \left[a,b\right]$ and $r\in \left[0,1\right],$ we have

$${\left(h\oplus l\right)}_{+}^r=h_{+}^r+l_{+}^r\mathit{and}{\left(h\oplus l\right)}_{-}^r=h_{-}^r+l_{-}^r,$$

(5)

$${\left(k\odot h\right)}_{-}^r=k{h}_{-}^r\mathit{and}{\left(k\odot h\right)}_{+}^r=k{h}_{+}^r,\mathit{for}k\ge 0,$$

(6)

$${\left(k\odot h\right)}_{-}^r=k{h}_{+}^r\mathit{and}{\left(k\odot h\right)}_{+}^r=k{h}_{-}^r,\mathit{for}k<0.$$

(7)

Lemma 2

([8,28]). For $k,m\in \mathbb{R}$ and $h,l,f,g\in C_{\mathcal{F}}\left[a,b\right]$, we have the following properties:

(i) 

$$h\oplus l=l\oplus h,$$

$$\left(h\oplus l\right)\oplus g=h\oplus \left(l\oplus g\right),$$

$$h\oplus \tilde{0}=\tilde{0}\oplus h;$$

(ii) 

With respect to $\tilde{0}$ in $C_{\mathcal{F}}\left[a,b\right]$ for any $h\in C_{\mathcal{F}}\left[a,b\right]$ with $h\left(\left[a,b\right]\right)\cap {\mathbb{R}}_{\mathcal{F}}\ne \varnothing$ has no opposite member regarding ⊕ in $C_{\mathcal{F}}\left[a,b\right];$

(iii) 

For $k,m\in \mathbb{R}$ with $k,m\ge 0$ or $k,m\le 0$

$$\left(k+m\right)\odot h=k\odot h\oplus m\odot h,$$

(For general $k,m\in \mathbb{R}$, this property does not hold);

(iv) 

$$k\odot \left(h\oplus l\right)=k\odot h\oplus k\odot l,$$

$$k\odot \left(m\odot h\right)=\left(km\right)\odot h;$$

(v) 

$$D^{*}(k\odot h,m\odot h)=\left|k-m\right|D^{*}(\tilde{0},h),$$

$$D^{*}(h\oplus g,l\oplus g)=D^{*}(h,l),$$

$$D^{*}(h\oplus l,g\oplus f)=D^{*}(h,g)+D^{*}(l,f).$$

Definition 7

([2]). A fuzzy number-valued function h defined on $\left[a,b\right]$ is called differentiable at $x\in \left[0,1\right]$ if there exists $h^{{}^{\prime }}\left(x\right)\in {\mathbb{R}}_{\mathcal{F}}$ such that

$$h^{{}^{\prime }}\left(x\right)=\underset{t\to 0}{lim}\frac{h\left(x+t\right)\ominus h\left(x\right)}{t},$$

where ⊖ denotes the difference on ${\mathbb{R}}_{\mathcal{F}}$. The $k$-times derivatives $h^{{}^{\left(k\right)}}\left(x\right)$, $k\in \mathbb{N}$ of h can be defined similarly.

Definition 8

([8,28]). Let T be an operator from $C_{\mathcal{F}}\left[a,b\right]$ into itself, such that

$$T\left(l\oplus h\right)=T\left(l\right)\oplus T\left(h\right),$$

$$T\left(k\odot h\right)=k\odot T\left(h\right),$$

for each $k\in \mathbb{R},$$h,l\in C_{\mathcal{F}}\left[a,b\right]$, then T is called a fuzzy linear operator. We say that a fuzzy linear operator T from $C_{\mathcal{F}}\left[a,b\right]$ into itself is positive, if and only if whenever $h,l\in C_{\mathcal{F}}\left[a,b\right]$ are such that $h⪯l$ then $T\left(h\right)⪯T\left(l\right)$, if and only if ${\left(T\left(h\right)\right)}_{+}^r\le {\left(T\left(l\right)\right)}_{+}^r$ and ${\left(T\left(h\right)\right)}_{-}^r\le {\left(T\left(l\right)\right)}_{-}^r$ for all $r\in \left[0,1\right]$. Here, we have the following representation of r-level interval:

$${\left[T\left(h\right)\left(x\right)\right]}^r=\left[{\left(T\left(h\right)\left(x\right)\right)}_{-}^r,{\left(T\left(h\right)\left(x\right)\right)}_{+}^r\right],$$

for all $x\in \left[a,b\right]$, $r\in \left[0,1\right]$, and also “⪯” and “≤” are partial orders on $C_{\mathcal{F}}\left[a,b\right]$ and $C\left[a,b\right]$, respectively.

Now, we recall the following fuzzy Korovkin-type approximation theorem.

Theorem 1

([18], p. 103). Assume that there exists a corresponding sequence of linear positive operators ${\left\{{\tilde{T}}_n\right\}}_{n\in \mathbb{N}}$ from $C\left[a,b\right]$ into itself for any sequence of fuzzy linear positive operators ${\left\{T_n\right\}}_{n\in \mathbb{N}}$ from $C_{\mathcal{F}}\left[a,b\right]$ into itself for $\left[a,b\right]\subset \mathbb{R}$ satisfying

$${\left(T_n\left(h\right)\right)}_{\pm }^r={\tilde{T}}_n\left(h_{\pm }^r\right)$$

(8)

for all $r\in \left[0,1\right]$ and $h\in C_{\mathcal{F}}\left[a,b\right]$, respectively. Moreover, suppose that ${\tilde{T}}_n\left(t^i\right)$ converge to $x^i$ for $i=0,1,2$ uniformly, then $D^{*}\left(T_n\left(h\right),h\right)$ converges to zero uniformly for any $h\in C_{\mathcal{F}}\left[a,b\right]$, i.e., $T_n\left(h\right)$ converges to h with respect to the metric $D^{*}$.

Now, we recall the concept of fuzzy modulus continuity.

Definition 9

([3], pp. 128–130). Fuzzy modulus of continuity for any fuzzy number-valued continuous function h defined on $\left[a,b\right]$ is defined by:

$${\omega }_1^{\mathcal{F}}\left(h;\mu \right):=\underset{\begin{array}{c}t,s\in \left[a,b\right]\\ \left|t-s\right|\le \mu \end{array}}{sup}D\left(h\left(t\right),h\left(s\right)\right),\mu >0.$$

Suppose that ${\omega }_1^{\mathcal{F}}\left(h;\mu \right)$, ${\omega }_1\left(h_{-}^r;\mu \right)$ and ${\omega }_1\left(h_{+}^r;\mu \right)$ are all finite for each $\mu >0$ and $r\in \left[0,1\right]$. Here, ${\omega }_1$ is the usual modulus of continuity defined by:

$${\omega }_1\left(f;\mu \right)=\underset{\begin{array}{c}t,x\in \left[a,b\right]\\ \left|t-x\right|\le \mu \end{array}}{sup}\left|f\left(t\right)-f\left(x\right)\right|,\mu >0,f\in C\left[0,1\right].$$

Then,

$${\omega }_1^{\mathcal{F}}\left(h;\mu \right)=\underset{r\in \left[0,1\right]}{sup}max\left\{{\omega }_1\left(h_{-}^r;\mu \right),{\omega }_1\left(h_{+}^r;\mu \right)\right\},\mu >0.$$

Additionally, ${\omega }_1^{\mathcal{F}}\left(h;\mu \right)$ has the following properties:

$${\omega }_1^{\mathcal{F}}\left(h;{\mu }_1+{\mu }_2\right)\le {\omega }_1^{\mathcal{F}}\left(h;{\mu }_1\right)+{\omega }_1^{\mathcal{F}}\left(h;{\mu }_2\right),{\mu }_1,{\mu }_2>0,$$

$${\omega }_1^{\mathcal{F}}\left(h;n\mu \right)\le n{\omega }_1^{\mathcal{F}}\left(h;\mu \right),\mu >0,n\in \mathbb{N},$$

$${\omega }_1^{\mathcal{F}}\left(h;\kappa \mu \right)\le \left(\kappa +1\right){\omega }_1^{\mathcal{F}}\left(h;\mu \right),\mu >0,\kappa >0,$$

$$h\in C_{\mathcal{F}}\left[a,b\right]\mathit{implies}\underset{\mu \to 0}{lim}{\omega }_1^{\mathcal{F}}\left(h;\mu \right)=0.$$

Definition 10

([3], p. 144). Let h be a fuzzy number-valued continuous function defined on $\left[a,b\right]$. If h satisfies the following property:

$$D\left(h\left(t\right),h\left(s\right)\right)\le M{\left|t-s\right|}^{\gamma },\gamma \in \left(0,1\right],M>0,s,t\in \left[0,1\right],$$

then h is called a Lipschitz-type fuzzy function denoted by $Li{p}^{\mathcal{F}}\left(M\right).$

3. Construction of Fuzzy Post-Quantum Bernstein Polynomials

In this part, we construct fuzzy post-quantum Bernstein polynomials and give some auxiliary results.

Definition 11.

Let h be a fuzzy continuous function defined on $\left[0,1\right]$. We define fuzzy post-quantum Bernstein polynomials by:

$$B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)=\underset{k=0}{\overset{n}{⨁}}{\tau }_{n,k}\left(x;p,q\right)\odot h\left({\xi }_{n,k}^{p,q}\right),x\in \left[0,1\right],$$

where ${\displaystyle \underset{k=0}{\overset{n}{⨁}}}$ denotes the finite fuzzy n-summation on ${\mathbb{R}}_{\mathcal{F}}$ and ⊙ denotes the fuzzy scalar product on ${\mathbb{R}}_{\mathcal{F}}$, and

$${\tau }_{n,k}\left(x;p,q\right):=\frac{p^{k(k-1)/2}}{p^{n(n-1)/2}}{\left[\begin{array}{c}n\\ k\end{array}\right]}_{p,q}{x}^k\prod _{s=0}^{n-k-1}\left(p^s-q^{s}x\right),$$

$${\xi }_{n,k}^{p,q}=\frac{{\left[k\right]}_{p,q}}{p^{k-n}{\left[n\right]}_{p,q}}.$$

It is clear that ${\tau }_{n,k}\left(x;p,q\right)\ge 0$ for all $x\in \left[0,1\right]$ and ${\tau }_{n,0}\left(x;p,q\right)$, ${\tau }_{n,1}\left(x;p,q\right)$, …,${\tau }_{n,n}\left(x;p,q\right)$ are linearly independent real-valued algebraic polynomials of degree ≤n. When $p=q=1,$ the fuzzy post-quantum Bernstein polynomials are reduced to the fuzzy Bernstein polynomials defined in [28].

We give the following auxiliary results, which are prominent in all the proofs of the results.

By the definition of fuzzy continuous functions, we have the following relation between the fuzzy post-quantum Bernstein polynomials $B_{n,p,q}^{\mathcal{F}}$ and the post-quantum Bernstein polynomials $B_n^{p,q}.$

Lemma 3.

For the fuzzy post-quantum Bernstein polynomials $B_{n,p,q}^{\mathcal{F}}$, we have

$${\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{\pm }^r=B_n^{p,q}(h_{\pm }^r;x),$$

where $x\in \left[0,1\right],$$h\in C_{\mathcal{F}}\left[0,1\right],$$h_{-}^r,h_{+}^r\in C\left[0,1\right]$ for each $r\in \left[0,1\right]$ and $B_n^{p,q}$ are the post-quantum Bernstein polynomials defined by (1).

Proof. 

We know that $B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)$ has the following representation of the r-level interval:

$${\left[B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right]}^r=\left[{\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{-}^r,{\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{+}^r\right].$$

We directly can see:

$$\begin{array}{ccc}\hfill {\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{-}^r& =& {\left(\underset{k=0}{\overset{n}{⨁}}{\tau }_{n,k}\left(x;p,q\right)\odot h\left({\xi }_{n,k}^{p,q}\right)\right)}_{-}^r\hfill \\ & =& \sum _{k=0}^n{h}_{-}^r\left({\xi }_{n,k}^{p,q}\right){\tau }_{n,k}\left(x;p,q\right)\hfill \\ & =& B_n^{p,q}(h_{-}^r;x).\hfill \end{array}$$

Similarly, we get:

$${\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{+}^r=B_n^{p,q}(h_{+}^r;x).$$

□

We have the following basic property of fuzzy post-quantum Bernstein polynomials $B_{n,p,q}^{\mathcal{F}}.$

Lemma 4.

The fuzzy post-quantum Bernstein polynomials $B_{n,p,q}^{\mathcal{F}}$ are fuzzy linear positive operators.

Proof. 

For the fuzzy linearity of $B_{n,p,q}^{\mathcal{F}},$ let l and h be fuzzy continuous functions defined on $\left[0,1\right]$ and $k\in \mathbb{R}.$

By Lemma 3, we have:

$${\left(B_{n,p,q}^{\mathcal{F}}\left(l\oplus h\right)\left(x\right)\right)}_{\pm }^r=B_n^{p,q}\left({\left(l\oplus h\right)}_{\pm }^r;x\right),$$

(9)

for each $x,r\in \left[0,1\right]$ respect to − and $+,$ respectively.

Since $B_{n,p,q}^{\mathcal{F}}\left(l\oplus h\right)\in C_{\mathcal{F}}\left[0,1\right],$ considering the representation of $B_{n,p,q}^{\mathcal{F}},$ we have ${\left(B_{n,p,q}^{\mathcal{F}}\left(l\oplus h\right)\right)}_{-}^r,$${\left(B_{n,p,q}^{\mathcal{F}}\left(l\oplus h\right)\right)}_{+}^r\in C\left[0,1\right].$

Considering (5) and the linearity of $B_n^{p,q},$ we can write

$$\begin{array}{ccc}\hfill B_n^{p,q}\left({\left(l\oplus h\right)}_{\pm }^r;x\right)& =& B_n^{p,q}\left(l_{\pm }^r+h_{\pm }^r;x\right)\hfill \\ & =& B_n^{p,q}\left(l_{\pm }^r;x\right)+B_n^{p,q}\left(h_{\pm }^r;x\right)\hfill \end{array}$$

(10)

Applying (10) to (9) and considering Lemma 3, we obtain

$${\left(B_{n,p,q}^{\mathcal{F}}\left(l\oplus h\right)\left(x\right)\right)}_{\pm }^r={\left(B_{n,p,q}^{\mathcal{F}}\left(l\right)\left(x\right)\right)}_{\pm }^r+{\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{\pm }^r,$$

(11)

for each $x,r\in \left[0,1\right]$ respect to − and $+,$ respectively.

Using (11) and considering the summation of the interval, we have

$$\begin{array}{ccc}\hfill {\left[B_{n,p,q}^{\mathcal{F}}\left(l\oplus h\right)\left(x\right)\right]}^r& =& \left[{\left(B_{n,p,q}^{\mathcal{F}}\left(l\oplus h\right)\left(x\right)\right)}_{-}^r,{\left(B_{n,p,q}^{\mathcal{F}}\left(l\oplus h\right)\left(x\right)\right)}_{+}^r\right]\hfill \\ & =& \left[{\left(B_{n,p,q}^{\mathcal{F}}\left(l\right)\left(x\right)\right)}_{-}^r+{\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{-}^r,{\left(B_{n,p,q}^{\mathcal{F}}\left(l\right)\left(x\right)\right)}_{+}^r+{\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{+}^r\right]\hfill \\ & =& \left[{\left(B_{n,p,q}^{\mathcal{F}}\left(l\right)\left(x\right)\right)}_{-}^r,{\left(B_{n,p,q}^{\mathcal{F}}\left(l\right)\right)}_{+}^r\right]\hfill \\ & & +\left[{\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{-}^r,{\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{+}^r\right]\hfill \\ & =& {\left[B_{n,p,q}^{\mathcal{F}}\left(l\right)\left(x\right)\right]}^r+{\left[B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right]}^r\hfill \\ & =& {\left[B_{n,p,q}^{\mathcal{F}}\left(l\right)\left(x\right)\oplus B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right]}^r\hfill \\ & =& {\left[\left(B_{n,p,q}^{\mathcal{F}}\left(l\right)\oplus B_{n,p,q}^{\mathcal{F}}\left(h\right)\right)\left(x\right)\right]}^r,\hfill \end{array}$$

for each $x\in \left[0,1\right].$ Thus,

$$B_{n,p,q}^{\mathcal{F}}\left(l\oplus h\right)=B_{n,p,q}^{\mathcal{F}}\left(l\right)\oplus B_{n,p,q}^{\mathcal{F}}\left(h\right),l,h\in C_{\mathcal{F}}\left[0,1\right].$$

Suppose that $k\ge 0$.

By Lemma 3,

$${\left(B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\left(x\right)\right)}_{\pm }^r=B_n^{p,q}\left({\left(k\odot h\right)}_{\pm }^r;x\right),$$

(12)

for each $x,r\in \left[0,1\right]$ with respect to − and $+,$ respectively. Since $B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\in C_{\mathcal{F}}\left[0,1\right],$ considering the r-level interval of $B_{n,p,q}^{\mathcal{F}},$ we have ${\left(B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\right)}_{-}^r,$${\left(B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\right)}_{+}^r\in C\left[0,1\right].$

Considering (6) and the linearity of $B_n^{p,q},$ we can write

$$B_n^{p,q}\left({\left(k\odot h\right)}_{\pm }^r;x\right)=B_n^{p,q}\left(k{h}_{\pm }^r;x\right)=k{B}_n^{p,q}\left(h_{\pm }^r;x\right).$$

(13)

Applying (13) to (12) and considering Lemma 3, we obtain

$${\left(B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\left(x\right)\right)}_{\pm }^r=k{\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{\pm }^r={\left(\left(k\odot B_{n,p,q}^{\mathcal{F}}\right)\left(h\right)\left(x\right)\right)}_{\pm }^r,$$

(14)

for each $x,r\in \left[0,1\right]$ with respect to − and $+,$ respectively. Using (14), we have:

$$\begin{array}{ccc}\hfill {\left[B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\left(x\right)\right]}^r& =& \left[{\left(B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\left(x\right)\right)}_{-}^r,{\left(B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\left(x\right)\right)}_{+}^r\right]\hfill \\ & =& \left[{\left(\left(k\odot B_{n,p,q}^{\mathcal{F}}\right)\left(h\right)\left(x\right)\right)}_{-}^r,{\left(\left(k\odot B_{n,p,q}^{\mathcal{F}}\right)\left(h\right)\left(x\right)\right)}_{+}^r\right]\hfill \\ & =& {\left[\left(k\odot B_{n,p,q}^{\mathcal{F}}\right)\left(h\right)\left(x\right)\right]}^r.\hfill \end{array}$$

Therefore,

$$B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)=\left(k\odot B_{n,p,q}^{\mathcal{F}}\right)\left(h\right),k\ge 0,h\in C_{\mathcal{F}}\left[0,1\right].$$

Assume that $k<0$. Considering the linearity of $B_n^{p,q}$ and applying (7) to (12), we can write:

$$B_n^{p,q}\left({\left(k\odot h\right)}_{-}^r;x\right)=B_n^{p,q}\left(k{h}_{+}^r;x\right)=k{B}_n^{p,q}\left(h_{+}^r;x\right),$$

(15)

$$B_n^{p,q}\left({\left(k\odot h\right)}_{+}^r;x\right)=B_n^{p,q}\left(k{h}_{-}^r;x\right)=k{B}_n^{p,q}\left(h_{-}^r;x\right).$$

(16)

Using (12), (15), (16) and (7), considering Lemma 3, we have:

$${\left(B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\left(x\right)\right)}_{-}^r=k{\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{+}^r={\left(\left(k\odot B_{n,p,q}^{\mathcal{F}}\right)\left(h\right)\left(x\right)\right)}_{-}^r$$

(17)

$${\left(B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\left(x\right)\right)}_{+}^r=k{\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{-}^r={\left(\left(k\odot B_{n,p,q}^{\mathcal{F}}\right)\left(h\right)\left(x\right)\right)}_{+}^r.$$

(18)

Using (17) and (18), we can write:

$$\begin{array}{ccc}\hfill {\left[B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\left(x\right)\right]}^r& =& \left[{\left(B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\left(x\right)\right)}_{-}^r,{\left(B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)\left(x\right)\right)}_{+}^r\right]\hfill \\ & =& \left[{\left(\left(k\odot B_{n,p,q}^{\mathcal{F}}\right)\left(h\right)\left(x\right)\right)}_{-}^r,{\left(\left(k\odot B_{n,p,q}^{\mathcal{F}}\right)\left(h\right)\left(x\right)\right)}_{+}^r\right]\hfill \\ & =& {\left[\left(k\odot B_{n,p,q}^{\mathcal{F}}\right)\left(h\right)\left(x\right)\right]}^r.\hfill \end{array}$$

Therefore, we obtain:

$$B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)=\left(k\odot B_{n,p,q}^{\mathcal{F}}\right)\left(h\right),k<0,h\in C_{\mathcal{F}}\left[0,1\right].$$

Thus, we get:

$$B_{n,p,q}^{\mathcal{F}}\left(k\odot h\right)=\left(k\odot B_{n,p,q}^{\mathcal{F}}\right)\left(h\right),\forall k\in \mathbb{R},h\in C_{\mathcal{F}}\left[0,1\right].$$

For the fuzzy positivity of $B_{n,p,q}^{\mathcal{F}},$ let h and l be fuzzy continuous functions defined on $\left[0,1\right]$ with $h⪯l,$ where ”⪯ “ is the partial order on $C_{\mathcal{F}}\left[0,1\right]$. Then $h_{-}^r\le l_{-}^r$ and $h_{+}^r\le l_{+}^r,$ where ”≤ “ is the partial order on $C\left[0,1\right].$

Since $h_{-}^r,h_{+}^r,l_{-}^r$ and $l_{+}^r\in C\left[0,1\right]$ and the positivity of $B_n^{p,q},$ we have:

$$B_n^{p,q}\left(h_{\pm }^r\right)\le B_n^{p,q}\left(l_{\pm }^r\right),$$

(19)

with respect to − and +, respectively.

Considering (19) and Lemma 3, we obtain:

$${\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\right)}_{\pm }^r\le {\left(B_{n,p,q}^{\mathcal{F}}\left(l\right)\right)}_{\pm }^r,$$

with respect to − and +, respectively, that indicates

$$B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)⪯B_{n,p,q}^{\mathcal{F}}\left(l\right)\left(x\right),$$

which gives the fuzzy positivity of $B_{n,p,q}^{\mathcal{F}}.$ □

Now, we present the following examples, illustrating the existence of fuzzy post-quantum Bernstein polynomials.

Let $A\subset \mathbb{R}$. The membership function of the subset A is defined by:

$${\chi }_A=\left\{\begin{array}{cc}1,& s\in A\\ 0,& s\notin A,\end{array}\right.$$

which is a fuzzy number in due of satisfying (i)–(iv) of Definition 1, i.e., ${\chi }_A\in {\mathbb{R}}_{\mathcal{F}}.$

Example 1.

Let us choose $A=\left[0,1\right]$ and define a function $h_0:\mathbb{R}\to$${\mathbb{R}}_{\mathcal{F}}$ by

$$h_0\left(s\right)={\chi }_{\left[0,1\right]}=\left\{\begin{array}{cc}1,& s\in \left[0,1\right]\\ 0,& s\notin \left[0,1\right].\end{array}\right.$$

It is clear that $h_0$ is a basic fuzzy number-valued function defined on: $\mathbb{R}.$ For each $s\in \mathbb{R}$ and $r\in \left[0,1\right]$, we have

$${\left[h_0\left(s\right)\right]}^r=\left\{\begin{array}{cc}\left[0,1\right],& s\in \left[0,1\right]\\ \varnothing ,& s\notin \left[0,1\right].\end{array}\right.$$

Indeed, let $s\in \left[0,1\right].$

Case 1. For $r\in \left(0,1\right]$, we obtain:

$$\begin{array}{ccc}\hfill {\left[h_0\left(s\right)\right]}^r& =& \left\{s\in \mathbb{R}:h_0\left(s\right)\ge r\right\}\cap \left\{s\in \mathbb{R}:s\in \left[0,1\right]\right\}\hfill \\ & =& \left\{s\in \mathbb{R}:1\ge r\right\}\cap \left[0,1\right]\hfill \\ & =& \mathbb{R}\cap \left[0,1\right]=\left[0,1\right].\hfill \end{array}$$

Case 2. For $r=0$, we get:

$$\begin{array}{ccc}\hfill {\left[h_0\left(s\right)\right]}^r& =& \overline{\left\{s\in \mathbb{R}:h_0\left(s\right)>r\right\}}\cap \left\{s\in \mathbb{R}:s\in \left[0,1\right]\right\}\hfill \\ & =& \overline{\left\{s\in \mathbb{R}:1>0\right\}}\cap \left[0,1\right]\hfill \\ & =& \overline{\mathbb{R}}\cap \left[0,1\right]=\mathbb{R}\cap \left[0,1\right]=\left[0,1\right].\hfill \end{array}$$

By Case 1 and Case 2, we obtain that ${\left[h_0\left(s\right)\right]}^r=\left[0,1\right]$ for each $r\in \left[0,1\right].$

Let be $s\in \mathbb{R}\backslash \left[0,1\right].$

Case 3. For each $r\in \left(0,1\right]$, we obtain:

$$\begin{array}{ccc}\hfill {\left[h_0\left(s\right)\right]}^r& =& \left\{s\in \mathbb{R}:h_0\left(s\right)\ge r\right\}\cap \left\{s\in \mathbb{R}:s\notin \left[0,1\right]\right\}\hfill \\ & =& \left\{s\in \mathbb{R}:0\ge r\right\}\cap \left(\mathbb{R}\backslash \left[0,1\right]\right)\hfill \\ & =& \varnothing \cap \left(\mathbb{R}\backslash \left[0,1\right]\right)=\varnothing .\hfill \end{array}$$

Case 4. For $r=0$, we get:

$$\begin{array}{ccc}\hfill {\left[h_0\left(s\right)\right]}^r& =& \overline{\left\{s\in \mathbb{R}:h_0\left(s\right)>r\right\}}\cap \left\{s\in \mathbb{R}:s\notin \left[0,1\right]\right\}\hfill \\ & =& \overline{\left\{s\in \mathbb{R}:0>0\right\}}\cap \left(\mathbb{R}\backslash \left[0,1\right]\right)\hfill \\ & =& \overline{\varnothing }\cap \left[0,1\right]=\varnothing \cap \left[0,1\right]=\varnothing .\hfill \end{array}$$

By Case 3 and Case 4, we get that ${\left[h_0\left(s\right)\right]}^r=\varnothing$ for each $r\in \left[0,1\right]$.

If we define ${\tilde{h}}_0:\left[0,1\right]\to {\mathbb{R}}_{\mathcal{F}}$ such that ${\tilde{h}}_0\left(s\right)=h_0\left(s\right)$ for each $s\in \left[0,1\right],$ then

$${\left[{\tilde{h}}_0\left(s\right)\right]}^r=\left[{\left({\tilde{h}}_0\left(s\right)\right)}_{-}^r,{\left({\tilde{h}}_0\left(s\right)\right)}_{+}^r\right]=\left[0,1\right],$$

for each $s,r\in \left[0,1\right]$, which indicates

$${\left({\tilde{h}}_0\left(s\right)\right)}_{-}^r=0and{\left({\tilde{h}}_0\left(s\right)\right)}_{+}^r=1,$$

for each $s,r\in \left[0,1\right]$ and ${\left({\tilde{h}}_0\right)}_{-}^r,{\left({\tilde{h}}_0\right)}_{+}^r\in C\left[0,1\right].$

Let $x_0$ be any fixed point in $\left[0,1\right]$ and $\left(x_n\right)$ be any sequence in $\left[0,1\right]$ satisfying ${lim}_{n\to \infty }{x}_n=x_0.$ Then,

$$\begin{array}{ccc}\hfill D\left({\tilde{h}}_0\left(x_n\right),{\tilde{h}}_0\left(x_0\right)\right)& =& \underset{r\in \left[0,1\right]}{sup}max\left\{\left|{\left({\tilde{h}}_0\left(x_n\right)\right)}_{-}^r-{\left({\tilde{h}}_0\left(x_0\right)\right)}_{-}^r\right|,\left|{\left({\tilde{h}}_0\left(x_n\right)\right)}_{+}^r-{\left({\tilde{h}}_0\left(x_0\right)\right)}_{+}^r\right|\right\}\hfill \\ & =& 0.\hfill \end{array}$$

Since

$$\underset{n\to \infty }{lim}D\left({\tilde{h}}_0\left(x_n\right),{\tilde{h}}_0\left(x_0\right)\right)=0,$$

${\tilde{h}}_0$ is fuzzy continuous in any point $x_0\in \left[0,1\right],$ i.e., ${\tilde{h}}_0$ is fuzzy continuous on $\left[0,1\right]$. Thus ${\tilde{h}}_0\in C_{\mathcal{F}}\left[0,1\right].$ Considering Lemma 3 and (2), we obtain:

$$\begin{array}{ccc}\hfill {\left[B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_0\left(s\right)\right)\left(x\right)\right]}^r& =& \left[{\left(B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_0\left(s\right)\right)\left(x\right)\right)}_{-}^r,{\left(B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_0\left(s\right)\right)\left(x\right)\right)}_{+}^r\right]\hfill \\ & =& \left[B_n^{p,q}\left({\left({\tilde{h}}_0\left(s\right)\right)}_{-}^r;x\right),B_n^{p,q}\left({\left({\tilde{h}}_0\left(s\right)\right)}_{+}^r;x\right)\right]\hfill \\ & =& \left[B_n^{p,q}(0;x),B_n^{p,q}(1;x)\right]\hfill \\ & =& \left[0,1\right],\hfill \end{array}$$

for each $s,r\in \left[0,1\right]$. Thus, $B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_0\right)\in C_{\mathcal{F}}\left[0,1\right].$

Let us choose sequences $p=p_n$ and $q=q_n$ such that $0<q_n<p_n\le 1$, ${lim}_{n\to \infty }{p}_n=1$ and ${lim}_{n\to \infty }{q}_n=1$. Since $B_n^{p,q}\left({\left({\tilde{h}}_0\left(s\right)\right)}_{-}^r;x\right)=B_n^{p,q}(0;x)$ and $B_n^{p,q}\left({\left({\tilde{h}}_0\left(s\right)\right)}_{+}^r;x\right)=B_n^{p,q}(1;x)$ converge uniformly to 0 and 1, respectively. Therefore, $B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_0\left(s\right)\right)\left(x\right)$ converges uniformly to ${\tilde{h}}_0\left(x\right)$ for each $x\in \left[0,1\right]$ with respect to the fuzzy metric D, i.e., $B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_0\right)$ converges uniformly to ${\tilde{h}}_0$ on $\left[0,1\right]$ with regard to the metric $D^{*}.$

Example 2.

Let us choose $A=\left[0,t\right]$ for any $t\in \left[0,1\right],$ and define a function $h_1:\mathbb{R}\to$${\mathbb{R}}_{\mathcal{F}}$ by

$$h_1\left(s\right)={\chi }_{\left[0,t\right]}=\left\{\begin{array}{cc}1,& s\in \left[0,t\right]\\ 0,& s\notin \left[0,t\right].\end{array}\right.$$

It is clear that $h_1$ is a fuzzy number-valued function defined on $\mathbb{R}.$ For each $s\in \mathbb{R}$ and $r\in \left[0,1\right]$, we obtain

$${\left[h_1\left(s\right)\right]}^r=\left\{\begin{array}{cc}\left[0,t\right],& s\in \left[0,t\right]\\ \varnothing ,& s\notin \left[0,t\right].\end{array}\right.$$

If we define ${\tilde{h}}_1:\left[0,t\right]\to {\mathbb{R}}_{\mathcal{F}}$ such that ${\tilde{h}}_1\left(s\right)=h_1\left(s\right)$ for each $s\in \left[0,t\right],$ then

$${\left[{\tilde{h}}_1\left(s\right)\right]}^r=\left[{\left({\tilde{h}}_1\left(s\right)\right)}_{-}^r,{\left({\tilde{h}}_1\left(s\right)\right)}_{+}^r\right]=\left[0,t\right],$$

for each $s,r\in \left[0,1\right]$, which implies

$${\left({\tilde{h}}_1\left(s\right)\right)}_{-}^r=0and{\left({\tilde{h}}_1\left(s\right)\right)}_{+}^r=t,$$

for each $r\in \left[0,1\right]$ and ${\left({\tilde{h}}_1\right)}_{-}^r,{\left({\tilde{h}}_1\right)}_{+}^r\in C\left[0,t\right].$

Let $y_0$ be any fixed point in $\left[0,t\right]$ and $\left(y_n\right)$ be any sequence in $\left[0,t\right]$ satisfying ${lim}_{n\to \infty }{y}_n=y_0.$

Then,

$$\begin{array}{ccc}\hfill D\left({\tilde{h}}_1\left(y_n\right),{\tilde{h}}_1\left(y_0\right)\right)& =& \underset{r\in \left[0,1\right]}{sup}max\left\{\left|{\left({\tilde{h}}_1\left(y_n\right)\right)}_{-}^r-{\left({\tilde{h}}_1\left(y_0\right)\right)}_{-}^r\right|,\left|{\left({\tilde{h}}_1\left(y_n\right)\right)}_{+}^r-{\left({\tilde{h}}_1\left(y_0\right)\right)}_{+}^r\right|\right\}\hfill \\ & =& 0.\hfill \end{array}$$

Since

$$\underset{n\to \infty }{lim}D\left({\tilde{h}}_1\left(y_n\right),{\tilde{h}}_1\left(y_0\right)\right)=0,$$

${\tilde{h}}_1$ is fuzzy continuous at any point $y_0\in \left[0,t\right],$ i.e., ${\tilde{h}}_1$ is fuzzy continuous on $\left[0,t\right]$. Therefore, ${\tilde{h}}_1\in C_{\mathcal{F}}\left[0,t\right].$ Considering Lemma 3 and (3), we obtain

$$\begin{array}{ccc}\hfill {\left[B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_1\left(s\right)\right)\left(x\right)\right]}^r& =& \left[{\left(B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_1\left(s\right)\right)\left(x\right)\right)}_{-}^r,{\left(B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_1\left(s\right)\right)\left(x\right)\right)}_{+}^r\right]\hfill \\ & =& \left[B_n^{p,q}\left({\left({\tilde{h}}_1\left(s\right)\right)}_{-}^r;x\right),B_n^{p,q}\left({\left({\tilde{h}}_1\left(s\right)\right)}_{+}^r;x\right)\right]\hfill \\ & =& \left[B_n^{p,q}(0;x),B_n^{p,q}(t;x)\right]\hfill \\ & =& \left[0,x\right],\hfill \end{array}$$

for each $r\in \left[0,1\right]$. Thus $B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_1\right)\in C_{\mathcal{F}}\left[0,t\right].$

Let us choose sequences $p=p_n$ and $q=q_n$ such that $0<q_n<p_n\le 1$, ${lim}_{n\to \infty }{p}_n=1$ and ${lim}_{n\to \infty }{q}_n=1$. Since $B_n^{p,q}\left({\left({\tilde{h}}_1\left(s\right)\right)}_{-}^r;x\right)=B_n^{p,q}(0;x)$ and $B_n^{p,q}\left({\left({\tilde{h}}_1\left(s\right)\right)}_{+}^r;x\right)=B_n^{p,q}(t;x)$ converge uniformly to 0 and x, respectively. Therefore, $B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_1\left(s\right)\right)\left(x\right)$ converges uniformly to ${\tilde{h}}_1\left(x\right)$ for each $x\in \left[0,t\right]$ with respect to the fuzzy metric D, i.e., $B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_1\right)$ converges uniformly to ${\tilde{h}}_1$ on $\left[0,t\right]$ with regard to the metric $D^{*}.$

Example 3.

Let us choose $A=\left[0,t^2\right]$ for any $t\in \left[0,1\right],$ and define a function $h_2:\mathbb{R}\to$${\mathbb{R}}_{\mathcal{F}}$ by

$$h_2\left(s\right)={\chi }_{\left[0,t^2\right]}=\left\{\begin{array}{cc}1,& s\in \left[0,t^2\right]\\ 0,& s\notin \left[0,t^2\right].\end{array}\right.$$

It is clear that $h_2$ is a fuzzy number-valued function defined on $\mathbb{R}.$ For each $s\in \mathbb{R}$ and $r\in \left[0,1\right]$, with a similar way as in (i) and (ii), we obtain

$${\left[h_2\left(s\right)\right]}^r=\left\{\begin{array}{cc}\left[0,t^2\right],& s\in \left[0,t^2\right]\\ \varnothing ,& s\notin \left[0,t^2\right].\end{array}\right.$$

If we define ${\tilde{h}}_2:\left[0,t^2\right]\to {\mathbb{R}}_{\mathcal{F}}$ such that ${\tilde{h}}_2\left(s\right)=h_2\left(s\right)$ for each $s\in \left[0,t^2\right].$ Then,

$${\left[{\tilde{h}}_2\left(s\right)\right]}^r=\left[{\left({\tilde{h}}_2\left(s\right)\right)}_{-}^r,{\left({\tilde{h}}_2\left(s\right)\right)}_{+}^r\right]=\left[0,t^2\right],$$

for each $s,r\in \left[0,1\right]$, which indicates

$${\left({\tilde{h}}_2\left(s\right)\right)}_{-}^r=0and{\left({\tilde{h}}_2\left(s\right)\right)}_{+}^r=t^2,$$

for each $r\in \left[0,1\right]$ and ${\left({\tilde{h}}_2\right)}_{-}^r,{\left({\tilde{h}}_2\right)}_{+}^r\in C\left[0,t^2\right].$

Let $z_0$ be any fixed point in $\left[0,t^2\right]$ and $\left(z_n\right)$ be any sequence in $\left[0,t^2\right]$ satisfying ${lim}_{n\to \infty }{z}_n=z_0.$ Then,

$$\begin{array}{ccc}\hfill D\left({\tilde{h}}_2\left(z_n\right),{\tilde{h}}_2\left(z_0\right)\right)& =& \underset{r\in \left[0,1\right]}{sup}max\left\{\left|{\left({\tilde{h}}_2\left(z_n\right)\right)}_{-}^r-{\left({\tilde{h}}_2\left(z_0\right)\right)}_{-}^r\right|,\left|{\left({\tilde{h}}_2\left(z_n\right)\right)}_{+}^r-{\left({\tilde{h}}_2\left(y_0\right)\right)}_{+}^r\right|\right\}\hfill \\ & =& 0.\hfill \end{array}$$

Since

$$\underset{n\to \infty }{lim}D\left({\tilde{h}}_2\left(z_n\right),{\tilde{h}}_2\left(z_0\right)\right)=0,$$

${\tilde{h}}_2$ is fuzzy continuous at any point $z_0\in \left[0,t^2\right],$ i.e., ${\tilde{h}}_2$ is fuzzy continuous on $\left[0,t^2\right]$. Thus ${\tilde{h}}_2\in C_{\mathcal{F}}\left[0,t^2\right].$ Considering Lemma 3 and (4), we obtain:

$$\begin{array}{ccc}\hfill {\left[B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_2\left(s\right)\right)\left(x\right)\right]}^r& =& \left[{\left(B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_2\left(s\right)\right)\left(x\right)\right)}_{-}^r,{\left(B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_2\left(s\right)\right)\left(x\right)\right)}_{+}^r\right]\hfill \\ & =& \left[B_n^{p,q}\left({\left({\tilde{h}}_2\left(s\right)\right)}_{-}^r;x\right),B_n^{p,q}\left({\left({\tilde{h}}_2\left(s\right)\right)}_{+}^r;x\right)\right]\hfill \\ & =& \left[B_{n,}^{p,q}(0;x),B_{n,}^{p,q}(t^2;x)\right]\hfill \\ & =& \left[0,\frac{q{\left[n-1\right]}_{p,q}}{{\left[n\right]}_{p,q}}{x}^2+\frac{p^{n-1}}{{\left[n\right]}_{p,q}}x\right],\hfill \end{array}$$

for each $r\in \left[0,1\right]$. Thus, $B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_2\right)\in C_{\mathcal{F}}\left[0,t^2\right].$

Let us choose sequences $p=p_n$ and $q=q_n$ such that $0<q_n<p_n\le 1$, ${lim}_{n\to \infty }{p}_n=1$ and ${lim}_{n\to \infty }{q}_n=1$.Since $B_n^{p,q}\left({\left({\tilde{h}}_2\left(s\right)\right)}_{-}^r;x\right)=B_n^{p,q}(0;x)$ and $B_n^{p,q}\left({\left({\tilde{h}}_2\left(s\right)\right)}_{+}^r;x\right)=B_n^{p,q}(t^2;x)$ converge uniformly to 0 and x, respectively. Therefore, $B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_2\left(s\right)\right)\left(x\right)$ converges uniformly to ${\tilde{h}}_2\left(x\right)$ for each $x\in \left[0,t^2\right]$ with respect to the fuzzy metric D, i.e., $B_{n,p,q}^{\mathcal{F}}\left({\tilde{h}}_2\right)$ converges uniformly to ${\tilde{h}}_2$ on $\left[0,t^2\right]$ with regard to the metric $D^{*}.$

4. Fuzzy Korovkin-Type Approximation Results

In this section, we present a fuzzy Korovkin-type approximation result for the fuzzy post-quantum Bernstein polynomials, and we obtain estimates by using the fuzzy modulus of continuity and Lipshitz-type fuzzy functions.

Theorem 2.

Let $\left(p_n\right)$ and $\left(q_n\right)$ be any sequences such that $q_n\in \left(0,1\right]$ and $p_n\in \left(q_n,1\right]$ fulfilling the following conditions:

$$\underset{n\to \infty }{lim}{q}_n=\underset{n\to \infty }{lim}{p}_n=1\mathit{and}\underset{n\to \infty }{lim}{\left(p_n\right)}^n<\infty ,$$

(20)

and let ${\left\{B_{n,p_n,q_n}^{\mathcal{F}}\left(h\right)\left(x\right)\right\}}_{n\in N}$ be a sequence of fuzzy post-quantum Bernstein polynomials from $C_{\mathcal{F}}\left[0,1\right]$ into itself. Then $D^{*}\left(B_{n,p_n,q_n}^{\mathcal{F}}\left(h\right),h\right)$ converges to zero uniformly for any $h\in C_{\mathcal{F}}\left[0,1\right]$, i.e., $B_{n,p_n,q_n}^{\mathcal{F}}\left(h\right)$ converges to h with respect to $D^{*}$.

Proof. 

For the proof, we consider Theorem 1. By Lemma 4, $B_{n,p_n,q_n}^{\mathcal{F}}$ is a fuzzy linear positive operator. By Definition 6, as the fuzzy post-quantum Bernstein polynomial $B_{n,p_n,q_n}^{\mathcal{F}}$ maps $C_{\mathcal{F}}\left[0,1\right]$ into itself, the corresponding classical real post-quantum Bernstein operator $B_n^{p_n,q_n}$ maps $C\left[0,1\right]$ into itself, respectively. Considering Lemma 3, the assumption (8) of Theorem 1 is fulfilled. On the other hand, by Theorem 3.1 in [22], $B_n^{p_n,q_n}\left(t^i\right)$ converge uniformly to $x^i$ for $i=0,1,2$. Thus, the hypotheses of Theorem 1 are verified, which completes the proof. □

We can provide the fuzzy rate of convergence of fuzzy post-quantum Bernstein polynomials with the help of the fuzzy modulus of continuity.

Theorem 3.

If any $h\in C_{\mathcal{F}}\left[0,1\right]$, then

$$D^{*}\left(B_{n,p,q}^{\mathcal{F}}\left(h\right),h\right)\le 2{\omega }_1^{\mathcal{F}}\left(h;{\mu }_n\right),n\in \mathbb{N},$$

where ${\mu }_n=\frac{1}{2}\sqrt{\frac{p^{n-1}}{{\left[n\right]}_{p,q}}}.$

Proof. 

Since ${\displaystyle \sum _{k=0}^n}{\tau }_{n,k}\left(x;p,q\right)=1$, we can write

$$\begin{array}{ccc}\hfill h\left(x\right)& =& \left[\sum _{k=0}^n{\tau }_{n,k}\left(x;p,q\right)\right]\odot h\left(x\right)\hfill \\ & & (\mathrm{by}\mathrm{Lemma}1\left(\mathrm{iii}\right))\hfill \\ & =& \underset{k=0}{\overset{n}{⨁}}\left[{\tau }_{n,k}\left(x;p,q\right)\odot h\left(x\right)\right].\hfill \end{array}$$

Considering the last equality, Lemma 1 (iii)–(iv) and the properties of the metric D on ${\mathbb{R}}_{\mathcal{F}}$, we obtain:

$$\begin{array}{ccc}& & D\left(B_{n,p,q}^{\mathcal{F}}\left(h\left(t\right)\right)\left(x\right),h\left(x\right)\right)\hfill \\ & =& D\left(\underset{k=0}{\overset{n}{⨁}}\left[{\tau }_{n,k}\left(x;p,q\right)\odot h\left({\xi }_{n,k}^{p,q}\right)\right],\underset{k=0}{\overset{n}{⨁}}\left[{\tau }_{n,k}\left(x;p,q\right)\odot h\left(x\right)\right]\right)\hfill \\ & \le & \sum _{k=0}^n{\tau }_{n,k}\left(x;p,q\right)D\left(h\left({\xi }_{n,k}^{p,q}\right),h\left(x\right)\right).\hfill \end{array}$$

(21)

Additionally, we have

$$\begin{array}{ccc}\hfill D\left(h\left({\xi }_{n,k}^{p,q}\right),h\left(x\right)\right)& \le & {\omega }_1^{\mathcal{F}}\left(h;\left|{\xi }_{n,k}^{p,q}-x\right|\right)\hfill \\ & \le & \left(1+\frac{1}{\mu }\left|{\xi }_{n,k}^{p,q}-x\right|\right){\omega }_1^{\mathcal{F}}\left(h;\mu \right).\hfill \end{array}$$

(22)

Applying (22) to (21), we get

$$D\left(B_{n,p,q}^{\mathcal{F}}\left(h\left(t\right)\right)\left(x\right),h\left(x\right)\right)\le {\omega }_1^{\mathcal{F}}\left(h;\mu \right)+\frac{1}{\mu }{\omega }_1^{\mathcal{F}}\left(h;\mu \right)\sum _{k=0}^n{\tau }_{n,k}\left(x;p,q\right)\left|{\xi }_{n,k}^{p,q}-x\right|,$$

(23)

for all $n\in \mathbb{N}$, $\mu >0$, and $x\in \left[0,1\right]$.

Using Cauchy–Schwarz inequality and by (2)–(4), for each $x\in \left[0,1\right]$, we obtain:

$$\begin{array}{ccc}& & \sum _{k=0}^n{\tau }_{n,k}\left(x;p,q\right)\left|{\xi }_{n,k}^{p,q}-x\right|\hfill \\ & \le & {\left(\sum _{k=0}^n{\tau }_{n,k}\left(x;p,q\right){\left({\xi }_{n,k}^{p,q}-x\right)}^2\right)}^{1/2}{\left(\sum _{k=0}^n{\tau }_{n,k}\left(x;p,q\right)\right)}^{1/2}\hfill \\ & =& {\left(B_n^{p,q}\left(t^2;x\right)-2x{B}_n^{p,q}\left(t;x\right)+x^2{B}_n^{p,q}\left(1;x\right)\right)}^{1/2}\hfill \\ & =& {\left(\left(\frac{q{\left[n-1\right]}_{p,q}}{{\left[n\right]}_{p,q}}\right)x^2+\frac{p^{n-1}}{{\left[n\right]}_{p,q}}x\right)}^{1/2}\hfill \\ & =& {\left(-x^2+x\right)}^{1/2}\sqrt{\frac{p^{n-1}}{{\left[n\right]}_{p,q}}}\hfill \\ & \le & \frac{1}{2}\sqrt{\frac{p^{n-1}}{{\left[n\right]}_{p,q}}}={\mu }_n.\hfill \end{array}$$

(24)

Applying (24) to (23) and taking supremum the right hand side of (23) for each $x\in \left[0,1\right],$ we complete the proof. □

Remark 1.

When $p=1$ and $q=1$, the rate of convergence of Theorem 3 is a different result from Theorem 2.3 of [28].

Now, we present the rate of convergence of fuzzy post-quantum Bernstein polynomials with the help of Lipschitz-type fuzzy functions.

Theorem 4.

If $h\in Li{p}^{\mathcal{F}}\left(M\right)$, then the following inequality is valid:

$$D^{*}\left(B_{n,p,q}^{\mathcal{F}}\left(h\right),h\right)\le M{\left({\mu }_n\right)}^{\gamma },M>0,$$

where ${\mu }_n=\frac{1}{2}\sqrt{\frac{p^{n-1}}{{\left[n\right]}_{p,q}}}.$

Proof. 

Since $h\in Li{p}^{\mathcal{F}}\left(M\right)$, we have:

$$D\left(h\left(t\right),h\left(s\right)\right)\le M{\left|t-s\right|}^{\gamma },\gamma \in \left(0,1\right],M>0,s,t\in \left[0,1\right].$$

(25)

Considering (21) and (25), we can write:

$$\begin{array}{ccc}\hfill D\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right),h\left(x\right)\right)& \le & \sum _{k=0}^n{\tau }_{n,k}\left(x;p,q\right)D\left(h\left({\xi }_{n,k}^{p,q}\right),h\left(x\right)\right)\hfill \\ & \le & M\sum _{k=0}^n{\tau }_{n,k}\left(x;p,q\right){\left|{\xi }_{n,k}^{p,q}-x\right|}^{\gamma }.\hfill \end{array}$$

(26)

Applying Hölder inequality to (26), we get:

$$\begin{array}{ccc}\hfill D\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right),h\left(x\right)\right)& \le & M{\left(\sum _{k=0}^n{\tau }_{n,k}\left(x;p,q\right){\left({\xi }_{n,k}^{p,q}-x\right)}^2\right)}^{\gamma /2}\hfill \\ & =& M{\left[{\left(B_n^{p,q}\left({\left(t-x\right)}^2;x\right)\right)}^{1/2}\right]}^{\gamma }.\hfill \end{array}$$

In the last inequality, by using (2)–(4), with a similar calculation to the proof of Theorem 3 and taking supremum the right hand side for each $x\in \left[0,1\right]$, we obtain:

$$D^{*}\left(B_{n,p,q}^{\mathcal{F}}\left(h\right),h\right)\le M{\left({\mu }_n\right)}^{\gamma },M>0.$$

□

Remark 2.

In Theorems 3 and 4, if we choose sequences $\left(p_n\right)$ and $\left(q_n\right)$ satisfying (20) instead of p and q then ${lim}_{n\to \infty }{\mu }_n=0,$ i.e.,

$$\underset{n\to \infty }{lim}{\omega }_1^{\mathcal{F}}\left(h;{\mu }_n\right)=0,$$

therefore

$$\underset{n\to \infty }{lim}{D}^{*}\left(B_{n,p_n,q_n}^{\mathcal{F}}\left(h\right),h\right)=0.$$

Thus, Theorems 3 and 4 indicate that fuzzy post-quantum Bernstein polynomials $B_{n,p_n,q_n}^{\mathcal{F}}\left(h\right)$ converge to h with respect to the metric $D^{*}$ uniformly for any $h\in C_{\mathcal{F}}\left[0,1\right]$ in a fuzzy sense.

5. Asymptotic Approximation Result

In 1932, Voronovskaja [31] provided an asymptotic approximation result for the Bernstein polynomials. A Voronovskaja type result on a compact disk for the post-quantum Bernstein polynomials was studied in [32].

Now, we present a Voronovskaja type asymptotic approximation result for the fuzzy post-quantum Bernstein polynomials.

Let $C_{\mathcal{F}}^2\left[0,1\right]$ denote the space of all fuzzy continuous functions defined on $\left[0,1\right],$ that is, k-times differentiable continuously.

Theorem 5.

If any $h\in$$C_{\mathcal{F}}^2\left[0,1\right]$ then

$$\begin{array}{ccc}\hfill D^{*}\left(B_{n,p,q}^{\mathcal{F}}\left(h\right),h\oplus \frac{1}{2}{B}_n^{p,q}\left(\psi ;x\right)\odot h^{″}\right)& \le & D\left({\omega }_1^{\mathcal{F}}\left(h;{\mu }_n\right),\tilde{o}\right)\hfill \\ & & +\frac{1}{2}{\mu }_n^2{D}^{*}\left(h^{″},\tilde{0}\right),\hfill \end{array}$$

where ${\omega }_1^{\mathcal{F}}$ is the first order fuzzy modulus of continuity, $\tilde{o}$ is the neutral element with regard to ⊕ in ${\mathbb{R}}_{\mathcal{F}}$, $\tilde{0}$ is a fuzzy number-valued function defined on $\left[0,1\right]$ such that $\tilde{0}\left(x\right)=\tilde{o}$ for all $x\in \left[0,1\right],$$\psi \left(t\right)={\left(t-x\right)}^2$ and ${\mu }_n=\frac{1}{2}\sqrt{\frac{p^{n-1}}{{\left[n\right]}_{p,q}}}.$

Proof. 

Let $h\in$$C_{\mathcal{F}}^2\left[0,1\right]$ with the representation ${\left[h\left(x\right)\right]}^r=\left[h_{-}^r\left(x\right),h_{+}^r\left(x\right)\right]$ for each $x,r\in \left[0,1\right]$. By Definition 6, $h_{-}^r$ and $h_{+}^r$ are in $C^2\left[0,1\right]$. Additionally, $∥{\left(h^{″}\right)}_{-}^r∥$ and $∥{\left(h^{″}\right)}_{+}^r∥$ are bound in $\mathbb{R}.$

By the proof of Theorem 3, the asymptotic expansion of $B_n^{p,q}$ is obtained by:

$$\begin{array}{ccc}& & \left|B_n^{p,q}\left(h_{\pm }^r\right)\left(x\right)-h_{\pm }^r\left(x\right)-\frac{1}{2}\left({\left(h^{″}\right)}_{\pm }^r\right)B_n^{p,q}\left({\left(t-x\right)}^2;x\right)\right|\hfill \\ & \le & \left|B_n^{p,q}\left(h_{\pm }^r\right)\left(x\right)-h_{\pm }^r\left(x\right)\right|+\frac{1}{2}∥{\left(h^{″}\right)}_{\pm }^r∥\left|B_n^{p,q}\left({\left(t-x\right)}^2;x\right)\right|\hfill \\ & \le & 2{\omega }_1\left(h_{\pm }^r;{\mu }_n\right)+\frac{1}{2}{\mu }_n^2∥{\left(h^{″}\right)}_{\pm }^r∥,\hfill \end{array}$$

(27)

with regard to ±, respectively.

Consequently, by (27), considering Lemma 3, we get:

$$\begin{array}{ccc}& & D\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right),h\left(x\right)\oplus \frac{1}{2}{B}_n^{p,q}\left({\left(t-x\right)}^2;x\right)\odot h^{″}\left(x\right)\right)\hfill \\ & =& D\left(\left[{\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{-}^r,{\left(B_{n,p,q}^{\mathcal{F}}\left(h\right)\left(x\right)\right)}_{+}^r\right],\left[h_{-}^r\left(x\right),h_{+}^r\left(x\right)\right]\right.\hfill \\ & & \left.+\frac{1}{2}{B}_n^{p,q}\left({\left(t-x\right)}^2;x\right)\left[{\left(h^{″}\left(x\right)\right)}_{-}^r,{\left(h^{″}\left(x\right)\right)}_{+}^r\right]\right)\hfill \\ & =& \underset{r\in \left[0,1\right]}{sup}max\left\{\left|B_n^{p,q}(h_{-}^r;x)-h_{-}^r\left(x\right)-\frac{1}{2}\left({\left(h^{″}\left(x\right)\right)}_{-}^r\right)B_n^{p,q}\left({\left(t-x\right)}^2;x\right)\right|,\right.\hfill \\ & & \left.\left|B_n^{p,q}(h_{+}^r;x)-h_{+}^r\left(x\right)-\frac{1}{2}\left({\left(h^{″}\left(x\right)\right)}_{+}^r\right)B_n^{p,q}\left({\left(t-x\right)}^2;x\right)\right|\right\}\hfill \\ & \le & \underset{r\in \left[0,1\right]}{sup}max\left\{{\omega }_1\left(h_{-}^r;{\mu }_n\right)+\frac{1}{2}{\mu }_n^2∥{\left(h^{″}\right)}_{-}^r∥,{\omega }_1\left(h_{+}^r;{\mu }_n\right)+\frac{1}{2}{\mu }_n^2∥{\left(h^{″}\right)}_{+}^r∥\right\}\hfill \\ & \le & \underset{r\in \left[0,1\right]}{sup}max\left\{{\omega }_1\left(h_{-}^r;{\mu }_n\right),{\omega }_1\left(h_{+}^r;{\mu }_n\right)\right\}\hfill \\ & & +\underset{r\in \left[0,1\right]}{sup}max\left\{\frac{1}{2}{\mu }_n^2∥{\left(h^{″}\right)}_{-}^r∥,\frac{1}{2}{\mu }_n^2∥{\left(h^{″}\right)}_{+}^r∥\right\}\hfill \\ & \le & D\left({\omega }_1^{\mathcal{F}}\left(h;{\mu }_n\right),\tilde{o}\right)+\frac{1}{2}{\mu }_n^2{D}^{*}\left(h^{″},\tilde{0}\right).\hfill \end{array}$$

In the last inequality, taking supremum as the right hand side for each $x\in \left[0,1\right]$, we complete the proof. □

Remark 3.

In Theorem 5, if we choose sequences $\left(p_n\right)$ and $\left(q_n\right),we$ satisfy (20) instead of p and q. By Remark 2, we obtain:

$$\underset{n\to \infty }{lim}D\left({\omega }_1^{\mathcal{F}}\left(h;{\mu }_n\right),\tilde{o}\right)=0;$$

therefore,

$$\underset{n\to \infty }{lim}{D}^{*}\left(B_{n,p_n,q_n}^{\mathcal{F}}\left(h\right),h\oplus \frac{1}{2}{B}_n^{p_n,q_n}\left(\psi \right)\odot h^{″}\right)=0.$$

6. Conclusions

In this paper, we have introduced the fuzzy post-quantum Bernstein polynomials with the help of fuzzy number-valued functions defined on $\left[0,1\right]$ and we have investigated their fuzzy approximation properties.

Theorem 2 demonstrates that the sequence of fuzzy post-quantum Bernstein polynomials $D^{*}\left(B_{n,p_n,q_n}^{\mathcal{F}}\left(h\right),h\right)$ converges to zero uniformly for any $h\in C_{\mathcal{F}}\left[0,1\right]$, i.e., $B_{n,p_n,q_n}^{\mathcal{F}}\left(h\right)$ converges to h with respect to $D^{*}$. Theorems 3 and 4 estimate the degree of fuzzy approximation of fuzzy post-quantum Bernstein polynomials and Theorem 5 implies that the asymptotic expansion of fuzzy post-quantum Bernstein polynomials $B_{n,p_n,q_n}^{\mathcal{F}}\left(h\right)$ converges to h with respect to the metric $D^{*}$ for all $h\in C_{\mathcal{F}}^2\left[0,1\right]$ in a fuzzy sense.

Any real number $x_0\in \mathbb{R}$ can be identified with the membership function ${\chi }_{\left\{x_0\right\}},$ which satisfies the properties (i)–(iv) of Definition 1. Therefore, it is clear that $\mathbb{R}\subset {\mathbb{R}}_{\mathcal{F}}$ [8]. In this sense, this study is crucial in terms of investigating the approximation properties of the fuzzy post-quantum Bernstein polynomials in the fuzzy function space.