Bernstein Approximations for Fuzzy-Valued Functions

Abstract

Studying the Bernstein approximations for fuzzy functions is a new attempt. With the help of considering the support functions of fuzzy sets in which the weak* topology for the normed dual space is involved, we are able to approximate continuous fuzzy functions by considering the Bernstein polynomials for fuzzy functions. We first study the Bernstein approximations for the support functions of fuzzy sets. Using the concept of isometry between the metric spaces of fuzzy sets and the normed spaces of support functions of fuzzy sets, the Bernstein approximations for the support functions of fuzzy sets can naturally lead to the Bernstein approximations for continuous fuzzy functions.

1. Introduction

Bernstein polynomials and their approximations to continuous functions have been studied for a long time. Many different variants of Bernstein polynomials have been investigated by many researchers, such as multivariate Bernstein polynomials (ref. Ding and Cao [1]), the extension to the infinite-dimensional case (ref. D’Ambrosio [2]), and Bernstein polynomials with integer coefficients (ref. Draganov [3] and the references therein). Also, the behavior of Bernstein polynomials has been presented, like when studying the optimality of the generalized Bernstein operator (ref. Aldaz and Render [4]), studying the convergence of derivatives of Bernstein approximation (ref. Floater [5]), and considering a new family of generalized Bernstein operators (ref. Chen et al. [6]). In this paper, the Bernstein approximations for continuous fuzzy functions are going to be studied by using the support functions of fuzzy sets.

Let $(U,\Vert ·{\Vert }_U)$ be a normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$. This paper studies the Bernstein approximations for the following fuzzy functions:

$$\tilde{f}:[0,1]\to {\mathcal{F}}_k\left(U\right) \mathrm{and} \tilde{f}:[0,1]\to {\mathcal{F}}_k^C\left(U\right).$$

A detailed description of the families ${\mathcal{F}}_k\left(U\right)$ and ${\mathcal{F}}_k^C\left(U\right)$ of fuzzy sets will be introduced in the subsequent text. The separability of normed space $(U,\Vert ·{\Vert }_U)$ and the boundedness of fuzzy functions $\tilde{f}$ will be important issues for presenting the Bernstein approximations in which the concept of a weak* topology for the normed dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$ must be invoked.

The concepts of pointwise Bernstein approximation and uniform Bernstein approximation will be introduced by considering the continuity of fuzzy functions at a single point or on a compact set. Using the concept of isometry between the metric spaces of fuzzy sets and the normed spaces of support functions of fuzzy sets, we first study the Bernstein approximations for the support functions of fuzzy sets, which can naturally lead to the Bernstein approximations for continuous fuzzy functions.

We say that $\tilde{A}$ is a fuzzy set in a normed space $(U,\Vert ·{\Vert }_U)$ when a membership function ${\xi }_{\tilde{A}}:U\to [0,1]$ is associated with $\tilde{A}$. For $\alpha \in (0,1]$, the $\alpha$-level sets ${\tilde{A}}_{\alpha }$ of $\tilde{A}$ are defined by

$${\tilde{A}}_{\alpha }=\left\{x\in U:{\xi }_{\tilde{A}}\left(x\right)\ge \alpha \right\}.$$

For $\alpha \in [0,1)$, the strong $\alpha$-level sets ${\tilde{A}}_{\alpha +}$ of $\tilde{A}$ are defined by

$${\tilde{A}}_{\alpha +}=\left\{x\in U:{\xi }_{\tilde{A}}\left(x\right)>\alpha \right\}.$$

Then, we have

$${\tilde{A}}_{\alpha +}=\bigcup _{\beta \in (\alpha ,1]}{\tilde{A}}_{\beta }, \mathrm{f}\mathrm{o}\mathrm{r} \alpha \in [0,1).$$

The support ${\tilde{A}}_{0+}$ of $\tilde{A}$ is given by

$${\tilde{A}}_{0+}=\left\{x\in U:{\xi }_{\tilde{A}}\left(x\right)>0\right\}.$$

In this case, we can define the 0-level set of $\tilde{A}$ by ${\tilde{A}}_0=cl\left({\tilde{A}}_{0+}\right)$, which means that the 0-level set of $\tilde{A}$ is the closure of the support of $\tilde{A}$. In this paper, we assume that all the fuzzy sets are normal. In other words, the $\alpha$-level sets of each fuzzy set are nonempty for all $\alpha \in [0,1]$.

Let $\tilde{A}$ and $\tilde{B}$ be two fuzzy sets in a normed space $(U,\Vert ·{\Vert }_U)$. We consider the addition $\tilde{A}\oplus \tilde{B}$ and scalar multiplication $\lambda \tilde{A}$ with membership functions defined by

$${\xi }_{\tilde{A}\oplus \tilde{B}}\left(z\right)=\underset{\left\{\right(x,y)\in U\times U:x+y=z\}}{sup}min\left\{{\xi }_{\tilde{A}}\left(x\right),{\xi }_{\tilde{B}}\left(y\right)\right\}$$

(1)

and

$${\xi }_{\lambda \tilde{A}}\left(z\right)=\left\{\begin{array}{cc}{\xi }_{\tilde{A}}(x/\lambda )\hfill & \mathrm{if}\lambda \ne 0\hfill \\ \hfill & \\ {\displaystyle \underset{x\in U}{sup}{\xi }_{\tilde{A}}\left(x\right)}\hfill & \mathrm{if}\lambda =0 \mathrm{and} z=\theta ,\hfill \\ \hfill & \\ 0\hfill & \mathrm{if}\lambda =0 \mathrm{and} z\ne \theta ,\hfill \end{array}\right.$$

(2)

where $\theta$ is the zero element of U.

Let $\mathcal{P}\left(U\right)$ be the family of all nonempty subsets of a normed space $(U,\Vert ·\Vert )$. For any $A,B\in \mathcal{P}\left(U\right)$, the Pompeiu–Hausdorff metric of A and B is defined by

$$d_H(A,B)=max\left\{\underset{a\in A}{sup}\underset{b\in B}{inf}\Vert a-b\Vert ,\underset{b\in B}{sup}\underset{a\in A}{inf}\Vert a-b\Vert \right\}.$$

Given a fuzzy set $\tilde{A}$ in U, we consider the following function:

$${\eta }_{\tilde{A}}:[0,1]\to (\mathcal{P}\left(U\right),d_H)\mathrm{defined} \mathrm{by}\alpha \mapsto {\tilde{A}}_{\alpha },$$

(3)

The function ${\eta }_{\tilde{A}}$ is said to be continuous with respect to the Pompeiu–Hausdorff metric $d_H$ when, given any $ϵ>0$, there exists $\delta >0$ such that

$$|\alpha -\beta |<\delta \mathrm{implies} d_H({\tilde{A}}_{\alpha },{\tilde{A}}_{\beta })<ϵ.$$

Consider the following:

• Let ${\mathcal{F}}_b\left(U\right)$ be the family of all fuzzy sets in U with nonempty bounded $\alpha$-level sets for $\alpha \in [0,1]$.
• Let ${\mathcal{F}}_k\left(U\right)$ be the family of all fuzzy sets in U with nonempty compact $\alpha$-level sets for $\alpha \in [0,1]$.
• Let ${\mathcal{F}}_{kc}\left(U\right)$ be the family of all fuzzy sets in U with nonempty compact and convex $\alpha$-level sets for $\alpha \in [0,1]$.
• Let ${\mathcal{F}}_k^C\left(U\right)$ be the family of elements $\tilde{A}$ in ${\mathcal{F}}_k\left(U\right)$ such that the function ${\eta }_{\tilde{A}}$ is continuous.
• Let $\overline{\mathcal{F}}\left(U\right)$ be the family of fuzzy sets in U satisfying

  $$\mathrm{cl}\left({\tilde{A}}_{\alpha +}\right)=\mathrm{cl}\left({\tilde{A}}_{\alpha }\right), \mathrm{for} \mathrm{all}\alpha \in [0,1).$$
  More precisely, we have

  $$\overline{\mathcal{F}}\left(U\right)=\left\{\tilde{A}\in \mathcal{F}\left(U\right):\mathrm{cl}\left({\tilde{A}}_{\alpha +}\right)=\mathrm{cl}\left({\tilde{A}}_{\alpha }\right) \mathrm{for} \mathrm{all}\alpha \in [0,1)\right\}.$$

We define

$${\overline{\mathcal{F}}}_k\left(U\right)=\overline{\mathcal{F}}\left(U\right)\cap {\mathcal{F}}_k\left(U\right).$$

Then, we can obtain ${\mathcal{F}}_k^C\left(U\right)={\overline{\mathcal{F}}}_k\left(U\right)$. The families ${\mathcal{F}}_k\left(U\right)$ and ${\mathcal{F}}_k^C\left(U\right)$ are closed under addition and scalar multiplication. In other words, given any $\tilde{A},\tilde{B}\in {\mathcal{F}}_k\left(U\right)$, we have

$$\tilde{A}\oplus \tilde{B}\in {\mathcal{F}}_k\left(U\right) \mathrm{and} \lambda \tilde{A}\in {\mathcal{F}}_k\left(U\right)$$

(4)

for any $\lambda \in \mathbb{R}$. For any $\tilde{A},\tilde{B}\in {\mathcal{F}}_k^C\left(U\right)$, we also have the operations presented in (4). Moreover, the operations regarding the $\alpha$-level sets are given by

$${\left(\tilde{A}\oplus \tilde{B}\right)}_{\alpha }={\tilde{A}}_{\alpha }+{\tilde{B}}_{\alpha } \mathrm{and} {\left(\lambda \tilde{A}\right)}_{\alpha }=\lambda {\tilde{A}}_{\alpha }$$

(5)

for all $\alpha \in [0,1]$. For more detailed properties, the reader may refer to Diamond and Kloeden [7] and Wu [8].

In Section 2, we introduce some useful properties of support functions of fuzzy sets in which the concept of a weak* topology for the normed dual space is included. The important issue is to present the weak*-compactness and weak*-metrizability of the closed unit ball ${\overline{S}}^{*}$. In Section 3, we study the Bernstein approximations of support functions of fuzzy sets by using the properties presented in Section 2, in which the concept of compatibility for the metric space of fuzzy sets is introduced. In Section 4, using the convergence obtained in Section 3, we are able to present the Bernstein approximations for fuzzy functions in which the function values of the concerned fuzzy functions are taken with the properties of compactness, convexity, and continuity. In Section 5, some practical examples are provided based on the Euclidean space such that the metric between any two fuzzy sets can be more clearly realized using the $\alpha$-level sets of fuzzy sets in which the Pompeiu–Hausdorff metric is considered.

2. Support Functions of Fuzzy Sets

Let $(U,\Vert ·{\Vert }_U)$ be a normed space. The closed unit ball in U is defined by

$$\overline{S}=\left\{{x:\Vert x\Vert }_U\le 1\right\}.$$

The topology ${\tau }_U^{\left(s\right)}$ generated by the norm of U is called the norm topology or strong topology. In this case, we have a topological space $(U,{\tau }_U^{\left(s\right)})$. Sometimes, we also write

$${(U,\Vert ·\Vert }_U)=\left(U,{\tau }_U^{\left(s\right)}\right).$$

The family $U^{*}$ of all continuous linear functionals defined on U with respect to the norm topology is called the dual space of U. The weakest topology for U such that all linear functionals in $U^{*}$ are continuous is called the weak topology ${\tau }_U^{\left(w\right)}$ for U.

For any $x^{*}\in U^{*}$, we write $x^{*}\left(x\right)=\langle x^{*},x\rangle \in \mathbb{R}$ for $x\in U$ and define

$$\Vert x^{*}{\Vert }_{U^{*}}=\underset{\Vert x\Vert =1}{sup}|\langle x^{*},x\rangle |=\underset{\Vert x\Vert \le 1}{sup}\left|\langle x^{*},x\rangle \right|.$$

Then, $(U^{*},\Vert ·{\Vert }_{U^{*}})$ is also a normed space.

The norm ${\Vert ·\Vert }_{U^{*}}$ can also generate a norm topology (strong topology) ${\tau }_{U^{*}}^{\left(s\right)}$ for $U^{*}$. Also, the weakest topology for $U^{*}$ such that all linear functionals in $U^{**}$ are continuous is called the weak topology ${\tau }_{U^{*}}^{\left(w\right)}$ for $U^{*}$. We consider a natural embedding of $\varphi :U\to U^{**}$. It is clear to see that $\varphi \left(U\right)\subset U^{**}$. The normed space U is said to be reflexive when we have the equality $\varphi \left(U\right)=U^{**}$. Now, the weakest topology for $U^{*}$ such that all linear functionals in $\varphi \left(U\right)$ are continuous is called the weak* topology ${\tau }_{U^{*}}^{(w*)}$ for $U^{*}$. It is clear to see that the weak* topology for $U^{*}$ is weaker than the weak topology for $U^{*}$. In other words, when U is reflexive, the weak and weak* topologies for $U^{*}$ will coincide.

Although the closed unit ball

$${\overline{S}}^{*}=\left\{x^{*}\in U^{*}:{\Vert x^{*}\Vert }_{U^{*}}\le 1\right\}$$

is not a compact subset of $U^{*}$ with respect to the norm topology for $U^{*}$, Alaoglu’s theorem says that ${\overline{S}}^{*}$ is a weak* compact subset of $U^{*}$. In other words, the closed unit ball ${\overline{S}}^{*}$ is a compact set with respect to the weak* topology ${\tau }_{U^{*}}^{(w*)}$ for $U^{*}$. In this case, we can generate a topology ${\tau }_{{\overline{S}}^{*}}^{(w*)}$ for ${\overline{S}}^{*}$ such that $({\overline{S}}^{*},{\tau }_{{\overline{S}}^{*}}^{(w*)})$ is a compact topological space. More precisely, given any ${\overline{O}}^{*}\in {\tau }_{{\overline{S}}^{*}}^{(w*)}$, there exists $O^{*}\in {\tau }_{U^{*}}^{(w*)}$ satisfying ${\overline{O}}^{*}=O^{*}\cap {\overline{S}}^{*}$.

We say that a topological space $(X,\tau )$ is metrizable when its topology $\tau$ can be generated by some metric. Regarding the closed unit ball ${\overline{S}}^{*}$, we define a metric $d_{{\overline{S}}^{*}}^{\left(s\right)}$ by

$$d_{{\overline{S}}^{*}}^{\left(s\right)}(x^{*},y^{*})={\Vert x^{*}-y^{*}\Vert }_{U^{*}}.$$

(6)

such that $({\overline{S}}^{*},d_{{\overline{S}}^{*}}^{\left(s\right)})$ forms a metric space, which can also generate a metric topology ${\tau }_{{\overline{S}}^{*}}^{\left(s\right)}$, Let ${\tau }_{U^{*}}^{\left(s\right)}$ be the norm topology for $U^{*}$. Then, we can also generate a topological subspace $({\overline{S}}^{*},{\tau }_{{\overline{S}}^{*}})$ of $U^{*}$ such that each ${\overline{O}}^{*}\in {\tau }_{{\overline{S}}^{*}}$ means ${\overline{O}}^{*}=O^{*}\cap {\overline{S}}^{*}$ for some $O^{*}\in {\tau }_{U^{*}}^{\left(s\right)}$. It is not difficult to show ${\tau }_{{\overline{S}}^{*}}^{\left(s\right)}={\tau }_{{\overline{S}}^{*}}$. In other words, the topology ${\tau }_{{\overline{S}}^{*}}$ is metrizable with the metric $d_{{\overline{S}}^{*}}^{\left(s\right)}$ defined in the way of (6).

Proposition 1

(Aliprantis and Border [9], p. 254). Let $(U,\Vert ·{\Vert }_U)$ be a normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$. The closed unit ball ${\overline{S}}^{*}$ in $U^{*}$ is weak*-metrizable if and only if U is separable.

When the normed space $(U,\Vert ·\Vert )$ is separable, from Proposition 1, we see that there exists a metric $d_{{\overline{S}}^{*}}^{(w*)}$ defined on ${\overline{S}}^{*}$ such that the compact space $({\overline{S}}^{*},{\tau }_{{\overline{S}}^{*}}^{(w*)})$ can be generated by the metric $d_{{\overline{S}}^{*}}^{(w*)}$. In other words, the metric topology generated by the metric $d_{{\overline{S}}^{*}}^{(w*)}$ coincides with the topology ${\tau }_{{\overline{S}}^{*}}^{(w*)}$. More precisely, the metric can be taken by

$$d_{{\overline{S}}^{*}}^{(w*)}(x^{*},y^{*})=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{2^n}}·\left|\langle x^{*},x_n\rangle -\langle y^{*},x_n\rangle \right|=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{2^n}}·\left|\langle x^{*}-y^{*},x_n\rangle \right|,$$

(7)

where $\{x_1,x_2,\dots \}$ is a countable dense subset of the closed unit ball in U and $x^{*},y^{*}\in U^{*}$. Since each $x_n$ is in the closed unit ball of U, it follows that $\Vert x_n\Vert \le 1$ for all n. Therefore, we have

$$\left|\langle x^{*}-y^{*},x_n\rangle \right|\le \underset{\Vert x\Vert \le 1}{sup}\left|\langle x^{*}-y^{*},x\rangle \right|=\Vert x^{*}-y^{*}\Vert ,$$

which also implies

$$d_{{\overline{S}}^{*}}^{(w*)}(x^{*},y^{*})=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{2^n}}·\left|\langle x^{*}-y^{*},x_n\rangle \right|\le \sum _{n=1}^{\infty }{\displaystyle \frac{1}{2^n}}·\Vert x^{*}-y^{*}\Vert =\Vert x^{*}-y^{*}\Vert =d_{{\overline{S}}^{*}}^{\left(s\right)}(x^{*},y^{*}).$$

Let $C(X,Y)$ be the family of all continuous functions from the topological space X into the normed space $(Y,\Vert ·{\Vert }_Y)$. For each $f\in C(X,Y)$, we define

$$\Vert f\Vert =\underset{x\in X}{sup}{\Vert f\left(x\right)\Vert }_Y.$$

Let $C_b(X,Y)$ be the space of all continuous functions $f:X\to Y$ satisfying $\Vert f\Vert <+\infty$. We can show that the spaces $C(X,Y)$ and $C_b(X,Y)$ are also normed spaces.

Proposition 2.

Assume that Y is a Banach space. Then, $C(X,Y)$ and $C_b(X,Y)$ are Banach spaces.

Diamond and Kloeden [10] and Puri and Ralescu [11] considered the concept of the support function of fuzzy sets in finite-dimensional Euclidean space ${\mathbb{R}}^n$. Now, the support function based on the infinite-dimensional normed space is considered in this paper.

Definition 1.

Let $(U,\Vert ·{\Vert }_U)$ be a normed space with a normed dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$ of U, and let $\tilde{A}$ be a fuzzy set in U. The support function

$$s_{\tilde{A}}:[0,1]\times U^{*}\to \mathbb{R}$$

of $\tilde{A}$ is defined by

$$s_{\tilde{A}}(\alpha ,x^{*})=\underset{x\in {\tilde{A}}_{\alpha }}{sup}\langle x^{*},x\rangle .$$

(8)

The norm of support function $s_{\tilde{A}}$ is defined by

$$\Vert s_{\tilde{A}}\Vert =\underset{(\alpha ,x^{*})\in [0,1]\times U^{*}}{sup}\left|s_{\tilde{A}}(\alpha ,x^{*})\right|.$$

From (8), it is clear to see that

$$\underset{(\alpha ,x^{*})\in [0,1]\times U^{*}}{sup}\left|s_{\tilde{A}}(\alpha ,x^{*})\right|=\underset{\alpha \in [0,1]}{sup}\underset{x^{*}\in U^{*}}{sup}\left|s_{\tilde{A}}(\alpha ,x^{*})\right|.$$

Since $x^{*}$ is a continuous linear functional, it is also bounded. When each $\alpha$-level set ${\tilde{A}}_{\alpha }$ is bounded for $\alpha \in [0,1]$, it follows that the support function $s_{\tilde{A}}$ is bounded. In other words, for any $\tilde{A}\in {\mathcal{F}}_b\left(U\right)$, we have $s_{\tilde{A}}(\alpha ,x^{*})<+\infty$ for any $(\alpha ,x^{*})\in [0,1]\times U^{*}$. For $\tilde{A}\in {\mathcal{F}}_k\left(U\right)$, this means that each $\alpha$-level set ${\tilde{A}}_{\alpha }$ is a compact subset of U for all $\alpha \in [0,1]$. The continuity of $x^{*}$ says that the supremum (8) is attained, i.e.,

$$s_{\tilde{A}}(\alpha ,x^{*})=\underset{x\in {\tilde{A}}_{\alpha }}{sup}\langle x^{*},x\rangle =\langle x^{*},x_0\rangle <+\infty$$

for some $x_0\in {\tilde{A}}_{\alpha }$.

The support function $s_{\tilde{A}}$ is frequently restricted on $[0,1]\times {\overline{S}}^{*}$, where ${\overline{S}}^{*}$ is the closed unit ball in $U^{*}$. In this case, the norm of $s_{\tilde{A}}$ is given by

$$\Vert s_{\tilde{A}}\Vert =\underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{sup}\left|s_{\tilde{A}}(\alpha ,x)\right|.$$

Example 1.

For $U=\mathbb{R}$, it follows that $U^{*}=\mathbb{R}$. In this case, we have ${\overline{S}}^{*}=[-1,1]$. Therefore, the support function $s_{\tilde{A}}$ is defined on $[0,1]\times [-1,1]$ and is given by

$$s_{\tilde{A}}(\alpha ,x)=\underset{y\in {\tilde{A}}_{\alpha }}{sup}\langle x,y\rangle =\underset{y\in {\tilde{A}}_{\alpha }}{sup}xy.$$

Let $\tilde{A}$ be a fuzzy interval. Then, the α-level set $\tilde{A}$ is a bounded closed interval given by

$${\tilde{A}}_{\alpha }=\left[{\tilde{A}}_{\alpha }^L,{\tilde{A}}_{\alpha }^U\right], for all \alpha \in [0,1].$$

In this case, we obtain

$$s_{\tilde{A}}(\alpha ,x)=\underset{y\in {\tilde{A}}_{\alpha }}{sup}xy=\underset{y\in [{\tilde{A}}_{\alpha }^L,{\tilde{A}}_{\alpha }^U]}{sup}xy=\left\{\begin{array}{cc}x·{\tilde{A}}_{\alpha }^U\hfill & if x\in [0,1]\hfill \\ & \\ x·{\tilde{A}}_{\alpha }^L\hfill & if x\in [-1,0].\hfill \end{array}\right.$$

The norm of $s_{\tilde{A}}$ is given by

$$\begin{array}{cc}\hfill \Vert s_{\tilde{A}}\Vert & =\underset{(\alpha ,x)\in [0,1]\times [-1,1]}{sup}\left|s_{\tilde{A}}(\alpha ,x)\right|\hfill \\ \hfill & =max\left\{\underset{(\alpha ,x)\in [0,1]\times [-1,0]}{sup}\left|x·{\tilde{A}}_{\alpha }^L\right|,\underset{(\alpha ,x)\in [0,1]\times [0,1]}{sup}\left|x·{\tilde{A}}_{\alpha }^U\right|\right\}.\hfill \end{array}$$

Puri and Ralescu [12] considered the function ${\widehat{d}}_{\mathcal{F}}$ defined on $\mathcal{F}\left(U\right)\times \mathcal{F}\left(U\right)$ by

$${\widehat{d}}_{\mathcal{F}}\left(\tilde{A},\tilde{B}\right)=\underset{\alpha \in (0,1]}{sup}{d}_H\left({\tilde{A}}_{\alpha },{\tilde{B}}_{\alpha }\right).$$

In this paper, we define

$$d_{\mathcal{F}}\left(\tilde{A},\tilde{B}\right)d_{\mathcal{F}}=\underset{\alpha \in [0,1]}{sup}{d}_H\left({\tilde{A}}_{\alpha },{\tilde{B}}_{\alpha }\right),$$

(9)

where $\alpha =0$ is considered. For any $\tilde{A},\tilde{B}\in \mathcal{F}\left(U\right)$, the metrics ${\widehat{d}}_{\mathcal{F}}(\tilde{A},\tilde{B})$ and $d_{\mathcal{F}}(\tilde{A},\tilde{B})$ are not necessarily identical. However, given any $\tilde{A},\tilde{B}\in \overline{\mathcal{F}}\left(U\right)$, we have

$${\widehat{d}}_{\mathcal{F}}(\tilde{A},\tilde{B})=d_{\mathcal{F}}\left(\tilde{A},\tilde{B}\right)=\underset{\alpha \in [0,1]}{sup}{d}_H\left({\tilde{A}}_{\alpha },{\tilde{B}}_{\alpha }\right)=\underset{\alpha \in [0,1]}{max}{d}_H\left({\tilde{A}}_{\alpha },{\tilde{B}}_{\alpha }\right).$$

When $(U,\Vert ·\Vert )$ is a Banach space, the spaces $({\mathcal{F}}_k^C\left(U\right),d_{\mathcal{F}})$ and $({\mathcal{F}}_k\left(U\right),d_{\mathcal{F}})$ are complete metric spaces. The following interesting results will be used in the subsequent study.

Proposition 3

(Wu [13]). Let $(U,\Vert ·{\Vert }_U)$ be a normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$, and let $\tilde{A}$ be a fuzzy set in U. We have the following properties:

(i) 

Given any $\tilde{A},\tilde{B}\in {\mathcal{F}}_k\left(U\right)$, for $\lambda \ge 0$, we have

$$s_{\tilde{A}\oplus \tilde{B}}=s_{\tilde{A}}+s_{\tilde{B}} and s_{\lambda \tilde{A}}=\lambda ·s_{\tilde{A}}.$$

(ii) 

When the support function $s_{\tilde{A}}$ is restricted on $[0,1]\times {\overline{S}}^{*}$, we have

$$\left|s_{\tilde{A}}(\alpha ,x^{*})\right|\le \Vert {\tilde{A}}_{\alpha }\Vert \le \Vert {\tilde{A}}_0\Vert ,$$

where

$$\Vert {\tilde{A}}_{\alpha }\Vert =\underset{x\in {\tilde{A}}_{\alpha }}{sup}{\Vert x\Vert }_U, for \alpha \in [0,1].$$

This also says that $s_{\tilde{A}}$ is uniformly bounded on $[0,1]\times {\overline{S}}^{*}$.

(iii) 

Given any $\tilde{A},\tilde{B}\in {\mathcal{F}}_{kc}\left(U\right)$, we have

$$\begin{array}{cc}\hfill d_{\mathcal{F}}(\tilde{A},\tilde{B})& =\underset{\{(\alpha ,x^{*}):\alpha \in [0,1],\Vert x^{*}{\Vert }_{U^{*}}\le 1\}}{sup}\left|s_{\tilde{A}}(\alpha ,x^{*})-s_{\tilde{B}}(\alpha ,x^{*})\right|\hfill \\ \hfill & =\underset{\{(\alpha ,x^{*}):\alpha \in [0,1],\Vert x^{*}{\Vert }_{U^{*}}=1\}}{sup}\left|s_{\tilde{A}}(\alpha ,x^{*})-s_{\tilde{B}}(\alpha ,x^{*})\right|.\hfill \end{array}$$

Proposition 4

(Wu [13]). Let $(U,\Vert ·{\Vert }_U)$ be a normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$. We have the following properties:

(i) 

For $\tilde{A}\in \mathcal{F}\left(U\right)$ with the support function $s_{\tilde{A}}$, given any fixed $\alpha \in [0,1]$, the function $s_{\tilde{A}}(\alpha ,·)$ is lower semi-continuous and weak*-lower semi-continuous on $U^{*}$. We also have the following properties:

(a) 

For $\tilde{A}\in {\mathcal{F}}_b\left(U\right)$, the function $s_{\tilde{A}}(\alpha ,·)$ is continuous on $U^{*}$.

(b) 

Suppose that the normed space U is separable. For $\tilde{A}\in {\mathcal{F}}_k\left(U\right)$, the function $s_{\tilde{A}}(\alpha ,·)$ is weak*-continuous on ${\overline{S}}^{*}$.

(ii) 

For $\tilde{A}\in \overline{\mathcal{F}}\left(U\right)$ with the support function $s_{\tilde{A}}$, given any fixed $x^{*}\in U^{*}$, the function $s_{\tilde{A}}(·,x^{*})$ is continuous on $[0,1]$. In fact, it is right-continuous at 0 and left-continuous at 1.

Remark 1.

Let $(U,\Vert ·{\Vert }_U)$ be a separable normed space. From Proposition 1, we see that the closed unit ball ${\overline{S}}^{*}$ is weak*-compact and weak*-metrizable with the metric $d_{{\overline{S}}^{*}}^{(w*)}$ given in (7). This metric $d_{{\overline{S}}^{*}}^{(w*)}$ can generate the topology ${\tau }_{{\overline{S}}^{*}}^{(w*)}$ such that $({\overline{S}}^{*},{\tau }_{{\overline{S}}^{*}}^{(w*)})$ turns into a compact topological space. It is clear to see that $[0,1]$ is a compact subset of $\mathbb{R}$ with respect to the usual topology ${\tau }_{\mathbb{R}}$. Therefore, we can endow a topology ${\tau }_{[0,1]}$ to $[0,1]$ such that $([0,1],{\tau }_{[0,1]})$ is a compact topological subspace of $(\mathbb{R},{\tau }_{\mathbb{R}})$. Using the topologies ${\tau }_{{\overline{S}}^{*}}^{(w*)}$ and ${\tau }_{[0,1]}$, we can generate a product topology ${\tau }_{[0,1]\times {\overline{S}}^{*}}^{(w*)}$ for the product space $[0,1]\times {\overline{S}}^{*}$. Tychonoff’s theorem (ref. Royden [14]) says that the product space

$$\left([0,1]\times {\overline{S}}^{*},{\tau }_{[0,1]\times {\overline{S}}^{*}}^{(w*)}\right).$$

is also a compact topological space.

Proposition 5

(Wu [13]). Let $(U,\Vert ·{\Vert }_U)$ be a separable normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$. Given any $\tilde{A}\in {\mathcal{F}}_k^C\left(U\right)$, the support function $s_{\tilde{A}}$ is ${\tau }_{[0,1]\times {\overline{S}}^{*}}^{(w*)}$-continuous on $[0,1]\times {\overline{S}}^{*}$.

3. Bernstein Approximation for Support Functions

Let $(U,\Vert ·{\Vert }_U)$ be a normed space. We consider the function

$$f:[0,1]\to \left({U,\Vert ·\Vert }_U\right)$$

defined on $[0,1]$. The n-th Bernstein polynomial of the function f is a function

$$B_{n}f:[0,1]\to \left({U,\Vert ·\Vert }_U\right) \mathrm{defined} \mathrm{by} \left(B_{n}f\right)\left(t\right)=\sum _{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)t^j{(1-t)}^{n-j}f\left({\displaystyle \frac{j}{n}}\right).$$

In this case, we have a sequence ${\left\{B_{n}f\right\}}_{n=1}^{\infty }$ of functions that consist of Bernstein polynomials of f. Two concepts of Bernstein approximation are defined below:

• We say that the sequence ${\left\{B_{n}f\right\}}_{n=1}^{\infty }$ is the pointwise Bernstein approximation of f at $t_0\in [0,1]$ when we have

  $$\underset{n\to \infty }{lim}{\Vert \left(B_{n}f\right)\left(t_0\right)-f\left(t_0\right)\Vert }_U=0.$$
• We say that the sequence ${\left\{B_{n}f\right\}}_{n=1}^{\infty }$ is the uniform Bernstein approximation of f when the sequence ${\left\{B_{n}f\right\}}_{n=1}^{\infty }$ of functions converges to function f uniformly on $[0,1]$. More precisely, given any $ϵ>0$, there exists an integer N such that

  $$n\ge N \mathrm{implies} \Vert \left(B_{n}f\right)\left(t\right)-f\left(t\right){\Vert }_U<ϵ, \mathrm{for} \mathrm{all} t\in [0,1],$$

  where N is independent of t.

Let $(U,\Vert ·{\Vert }_U)$ be a normed space, and let $\left(\mathcal{F}\right(U),d)$ be a metric space, where d can take $d_{\mathcal{F}}$ defined in (9). We consider the fuzzy function

$$\tilde{f}:[0,1]\to (\mathcal{F}\left(U\right),d)$$

defined on $[0,1]$, which means that the function value $\tilde{f}\left(t\right)$ is a fuzzy set in the normed space $(U,\Vert ·{\Vert }_U)$ for all $t\in [0,1]$. We say that a fuzzy function $\tilde{f}$ is continuous at $t_0\in [0,1]$ when, given any $ϵ>0$, there exists $\delta >0$ such that

$$|t-t_0|<\delta \mathrm{implies} d\left(\tilde{f}\left(t\right),\tilde{f}\left(t_0\right)\right)<ϵ.$$

Recall that addition and scalar multiplication in $\mathcal{F}\left(U\right)$ are defined in (1) and (2), respectively. The n-th Bernstein polynomial of the fuzzy function $\tilde{f}$ is a fuzzy function

$$\left(B_n\tilde{f}\right):[0,1]\to \mathcal{F}\left(U\right)$$

defined by

$$\left(B_n\tilde{f}\right)\left(t\right)=\underset{j=0}{\overset{n}{⨁}}{b}_{nj}\left(t\right)\tilde{f}\left({\displaystyle \frac{j}{n}}\right) \mathrm{with} b_{nj}\left(t\right)=\left(\begin{array}{c}n\\ j\end{array}\right)·t^j·{(1-t)}^{n-j}$$

(10)

Since the family ${\mathcal{F}}_k\left(U\right)$ is closed under addition and scalar multiplication, we consider the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k\left(U\right),d\right).$$

According to (4) and (5), the n-th Bernstein polynomial of the fuzzy function $\tilde{f}$ is also a fuzzy function

$$B_n\tilde{f}:[0,1]\to \left({\mathcal{F}}_k\left(U\right),d\right)$$

defined in the way of (10). Therefore, the support functions $s_{\tilde{f}(j/n)}$ and $s_{\left(B_n\tilde{f}\right)\left(t\right)}$ are bounded. In this case, we can study the Bernstein approximation for the support functions of fuzzy sets.

We further assume that the normed space $(U,\Vert ·{\Vert }_U)$ is separable. Proposition 5 says

$$s_{\tilde{f}\left(t\right)}\in C\left([0,1]\times {\overline{S}}^{*},{\tau }_{[0,1]\times {\overline{S}}^{*}}^{(w*)}\right),foreacht\in [0,1].$$

In this case, we can define a function

$$\zeta :[0,1]\to \left(C\left([0,1]\times {\overline{S}}^{*},{\tau }_{[0,1]\times {\overline{S}}^{*}}^{(w*)}\right),\Vert ·\Vert \right) \mathrm{by} \zeta \left(t\right)=s_{\tilde{f}\left(t\right)}.$$

Without any ambiguity, we also treat $s_{\tilde{f}}$ as a function given by

$$s_{\tilde{f}}:[0,1]\to \left(C\left([0,1]\times {\overline{S}}^{*},{\tau }_{[0,1]\times {\overline{S}}^{*}}^{(w*)}\right),\Vert ·\Vert \right).$$

(11)

Given any fixed $t\in [0,1]$, according to part (i) of Proposition 4, the support function of fuzzy set $\left(B_n\tilde{f}\right)\left(t\right)$ is given by

$$s_{\left(B_n\tilde{f}\right)\left(t\right)}(\alpha ,x^{*})=\sum _{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)t^j{(1-t)}^{n-j}{s}_{\tilde{f}\left({\displaystyle \frac{j}{n}}\right)}(\alpha ,x^{*})=\sum _{j=0}^n{b}_{nj}\left(t\right)s_{\tilde{f}\left({\displaystyle \frac{j}{n}}\right)}(\alpha ,x^{*})$$

(12)

for $(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}$. We can simply write

$$s_{\left(B_n\tilde{f}\right)\left(t\right)}=\sum _{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)t^j{(1-t)}^{n-j}{s}_{\tilde{f}\left({\displaystyle \frac{j}{n}}\right)}=\sum _{j=0}^n{b}_{nj}\left(t\right)s_{\tilde{f}\left({\displaystyle \frac{j}{n}}\right)}.$$

(13)

In this case, we can also treat $s_{\left(B_n\tilde{f}\right)}$ as a function given by

$$s_{\left(B_n\tilde{f}\right)}:[0,1]\to \left(C\left([0,1]\times {\overline{S}}^{*},{\tau }_{[0,1]\times {\overline{S}}^{*}}^{(w*)}\right),\Vert ·\Vert \right).$$

Under some suitable conditions, we are going to show that the function $s_{\left(B_n\tilde{f}\right)}$ is the n-th Bernstein polynomial of function $s_{\tilde{f}}$ such that the sequence ${\left\{s_{\left(B_n\tilde{f}\right)}\right\}}_{n=1}^{\infty }$ of functions is the Bernstein approximation of function $s_{\tilde{f}}$.

Definition 2.

Let $(U,\Vert ·{\Vert }_U)$ be a normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$, and let $\left(\mathcal{F}\right(U),d)$ be a metric space. We have the following:

• We say that the metric space $\left(\mathcal{F}\right(U),d)$ is compatible with distance when, for any $\tilde{A},\tilde{B}\in \mathcal{F}\left(U\right)$, we have

  $$d(\tilde{A},\tilde{B})=\Vert s_{\tilde{A}}-s_{\tilde{B}}\Vert =\underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{sup}\left|s_{\tilde{A}}(\alpha ,x^{*})-s_{\tilde{B}}(\alpha ,x^{*})\right|.$$
• We say that the metric space $\left(\mathcal{F}\right(U),d)$ is sub-compatible with distance when, for any $\tilde{A},\tilde{B}\in \mathcal{F}\left(U\right)$, we have

  $$d(\tilde{A},\tilde{B})\le \Vert s_{\tilde{A}}-s_{\tilde{B}}\Vert =\underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{sup}\left|s_{\tilde{A}}(\alpha ,x^{*})-s_{\tilde{B}}(\alpha ,x^{*})\right|.$$
• We say that the metric space $\left(\mathcal{F}\right(U),d)$ is sup-compatible with distance when, for any $\tilde{A},\tilde{B}\in \mathcal{F}\left(U\right)$, we have

  $$d(\tilde{A},\tilde{B})\ge \Vert s_{\tilde{A}}-s_{\tilde{B}}\Vert =\underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{sup}\left|s_{\tilde{A}}(\alpha ,x^{*})-s_{\tilde{B}}(\alpha ,x^{*})\right|.$$

Remark 2.

Part (iii) of Proposition 3 says that the metric space $({\mathcal{F}}_{kc}\left(U\right),d_{\mathcal{F}})$ is compatible with distance.

Since the family ${\mathcal{F}}_k^C\left(U\right)$ is also closed under addition and scalar multiplication, we first study the Bernstein approximations for the fuzzy function

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k^C\left(U\right),d\right),$$

which are presented below.

Theorem 1.

Let $(U,\Vert ·{\Vert }_U)$ be a separable normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$, and let $({\mathcal{F}}_k^C\left(U\right),d)$ be a metric space. Consider the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k^C\left(U\right),d\right).$$

Then, the function $s_{\left(B_n\tilde{f}\right)}$ is the n-th Bernstein polynomial of function $s_{\tilde{f}}$. Suppose that the metric space $({\mathcal{F}}_k^C\left(U\right),d)$ is compatible with distance. Then, we also have the following properties:

(i) 

Suppose that the fuzzy function $\tilde{f}$ is continuous at $t_0\in [0,1]$. Then, the function $s_{\tilde{f}}$ given in $\left(11\right)$ is continuous at $t_0$. Also, the sequence ${\left\{s_{B_n\tilde{f}}\right\}}_{n=1}^{\infty }$ of functions given in $\left(13\right)$ is the pointwise Bernstein approximation of function $s_{\tilde{f}}$ at $t_0$.

(ii) 

Suppose that the fuzzy function $\tilde{f}$ is continuous on $[0,1]$. Then, the function $s_{\tilde{f}}$ is uniformly continuous on $[0,1]$. Also, the sequence ${\left\{s_{B_n\tilde{f}}\right\}}_{n=1}^{\infty }$ of functions is the uniform Bernstein approximation of function $s_{\tilde{f}}$.

Proof .

Since $(U,\Vert ·{\Vert }_U)$ is separable, Proposition 5 says that we can define a function

$$\zeta :[0,1]\to \left(C\left([0,1]\times {\overline{S}}^{*},{\tau }_{[0,1]\times {\overline{S}}^{*}}^{(w*)}\right),\Vert ·\Vert \right) \mathrm{by} \zeta \left(t\right)=s_{\tilde{f}\left(t\right)}.$$

The continuity of the norm $\Vert ·\Vert$ and the compact set $[0,1]$ say that the following supremum is attained:

$$\underset{t\in [0,1]}{sup}\Vert \zeta \left(t\right)\Vert =\underset{t\in [0,1]}{max}\Vert \zeta \left(t\right)\Vert =\Vert \zeta \left(t_0\right)\Vert =\Vert s_{\tilde{f}\left(t_0\right)}\Vert$$

for some $t_0\in [0,1]$. Since the product set

$$\left([0,1]\times {\overline{S}}^{*},{\tau }_{[0,1]\times {\overline{S}}^{*}}^{(w*)}\right)$$

is a compact space, referring to Remark 1, and the support function $s_{\tilde{f}\left(t_0\right)}$ is ${\tau }_{[0,1]\times {\overline{S}}^{*}}^{(w*)}$-continuous on $[0,1]\times {\overline{S}}^{*}$ by Proposition 5, the following supremum is also attained:

$$\begin{array}{cc}\hfill \Vert s_{\tilde{f}\left(t_0\right)}\Vert & =\underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{sup}\left|s_{\tilde{f}\left(t_0\right)}(\alpha ,x^{*})\right|\hfill \\ \hfill & =\underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{max}\left|s_{\tilde{f}\left(t_0\right)}(\alpha ,x^{*})\right|=\left|s_{\tilde{f}\left(t_0\right)}({\alpha }_0,x_0^{*})\right|<+\infty \hfill \end{array}$$

for some $({\alpha }_0,x_0^{*})\in [0,1]\times {\overline{S}}^{*}$. This shows that the function $\zeta$ is bounded in the sense of

$$\Vert \zeta \left(t\right)\Vert \le \underset{t\in [0,1]}{sup}\Vert \zeta \left(t\right)\Vert =\Vert s_{\tilde{f}\left(t_0\right)}\Vert <+\infty , \mathrm{for} \mathrm{all} t\in [0,1].$$

(14)

Since $\zeta (j/n)=s_{\tilde{f}(j/n)}$, using (12), we have

$$\begin{array}{cc}\hfill s_{\left(B_n\tilde{f}\right)\left(t\right)}& =\sum _{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)t^j{(1-t)}^{n-j}{s}_{\tilde{f}\left({\displaystyle \frac{j}{n}}\right)}\hfill \\ \hfill & =\sum _{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)t^j{(1-t)}^{n-j}\zeta \left({\displaystyle \frac{j}{n}}\right)\equiv \left(B_n\zeta \right)\left(t\right),\hfill \end{array}$$

(15)

which says that $s_{\left(B_n\tilde{f}\right)}$ is the n-th Bernstein polynomial of function $\zeta =s_{\tilde{f}}$.

Given any $\tilde{A},\tilde{B}\in {\mathcal{F}}_k^C\left(U\right)$, the assumption of compatibility with distance says

$$d(\tilde{A},\tilde{B})=\Vert s_{\tilde{A}}-s_{\tilde{B}}\Vert .$$

(16)

Suppose that $\tilde{f}$ is continuous at $t_0\in [0,1]$. Using (16), given any $ϵ>0$, there exists $\delta >0$ such that

$$|t-t_0|<\delta \mathrm{implies} \Vert \zeta \left(t\right)-\zeta \left(t_0\right)\Vert =\Vert s_{\tilde{f}\left(t\right)}-s_{\tilde{f}\left(t_0\right)}\Vert =d(\tilde{f}\left(t\right),\tilde{f}\left(t_0\right))<ϵ,$$

which shows that the function $\zeta$ is continuous at $t_0$.

We write

$$b_{nj}\left(t\right)=\left(\begin{array}{c}n\\ j\end{array}\right)t^j{(1-t)}^{n-j}.$$

Then, we have

$$\sum _{j=0}^n{b}_{nj}\left(t\right)=\sum _{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)t^j{(1-t)}^{n-j}={(t+(1-t))}^n=1.$$

Moreover, we also have

$$\sum _{j=0}^{n}j{b}_{nj}\left(t\right)=nt\sum _{i=0}^{n-1}\left(\begin{array}{c}n-1\\ i\end{array}\right)t^i{(1-t)}^{n-1-i}=nt{(t+(1-t))}^{n-1}=nt$$

and

$$\sum _{j=0}^{n}j(j-1)b_{nj}\left(t\right)=n(n-1)t^2\sum _{i=0}^{n-2}\left(\begin{array}{c}n-2\\ i\end{array}\right)t^i{(1-t)}^{n-2-i}=n(n-1)t^2.$$

Therefore, we obtain

$$\begin{array}{cc}\hfill T& =\sum _{j=0}^n{(j-nt)}^2{b}_{nj}\left(t\right)\hfill \\ \hfill & =\sum _{j=0}^n\left[j(j-1)-j(2nt-1)+n^2{t}^2\right]b_{nj}\left(t\right)\hfill \\ \hfill & =n^2{t}^2-(2nt-1)nt+n(n-1)t^2=nt(1-t).\hfill \end{array}$$

Since $t(1-t)\le 1/4$ for $t\in [0,1]$, we have the following inequality:

$$\sum _{|{\displaystyle \frac{j}{n}}-t|\ge \delta }{b}_{nj}\left(t\right)\le {\displaystyle \frac{1}{{\delta }^2}}\sum _{|{\displaystyle \frac{j}{n}}-t|\ge \delta }{\left({\displaystyle \frac{j}{n}}-t\right)}^2{b}_{nj}\left(t\right)={\displaystyle \frac{T}{n^2{\delta }^2}}={\displaystyle \frac{t(1-t)}{n{\delta }^2}}\le {\displaystyle \frac{1}{4n{\delta }^2}}.$$

(17)

Since the function $\zeta$ is bounded by referring to (14), we have

$$\Vert \zeta \left(t\right)\Vert \le M, \mathrm{for} \mathrm{any} t\in [0,1].$$

(18)

Since $\zeta$ is continuous at $t_0$, given any $ϵ>0$, there exists $\delta >0$ such that

$$|t-t_0|<\delta \mathrm{implies} \Vert \zeta \left(t\right)-\zeta \left(t_0\right)\Vert <ϵ/2,$$

(19)

where $\delta$ depends on $t_0$. Now, we have

$$\begin{array}{cc}\hfill & \Vert \zeta \left(t_0\right)-\left(B_n\zeta \right)\left(t_0\right)\Vert =\left|\left|\sum _{j=0}^n\zeta \left(t_0\right)b_{nj}\left(t_0\right)-\left(B_n\zeta \right)\left(t_0\right)\right|\right|(\mathrm{since} {\displaystyle \sum _{j=0}^n{b}_{nj}\left(t_0\right)=1})\hfill \\ \hfill & =\left|\left|\sum _{j=0}^n\left[\zeta \left(t_0\right)-\zeta (j/n)\right]b_{nj}\left(t_0\right)\right|\right|(\mathrm{using} (15\left)\right)\hfill \\ \hfill & \le \sum _{|{\displaystyle \frac{j}{n}}-t_0|<\delta }\Vert \zeta \left(t_0\right)-\zeta (j/n)\Vert b_{nj}\left(t_0\right)+\sum _{|{\displaystyle \frac{j}{n}}-t_0|\ge \delta }\Vert \zeta \left(t_0\right)-\zeta (j/n)\Vert b_{nj}\left(t_0\right)\hfill \\ \hfill & \le {\displaystyle \frac{ϵ}{2}}\sum _{|{\displaystyle \frac{j}{n}}-t_0|<\delta }{b}_{nj}\left(t_0\right)+2M\sum _{|{\displaystyle \frac{j}{n}}-t_0|\ge \delta }{b}_{nj}\left(t_0\right)(\mathrm{using} (19) \mathrm{and} (18\left)\right)\hfill \\ \hfill & \le {\displaystyle \frac{ϵ}{2}}\sum _{j=0}^n{b}_{nj}\left(t_0\right)+2M\sum _{|{\displaystyle \frac{j}{n}}-t_0|\ge \delta }{b}_{nj}\left(t_0\right)\hfill \\ \hfill & \le {\displaystyle \frac{ϵ}{2}}+{\displaystyle \frac{2M}{4n{\delta }^2}}={\displaystyle \frac{ϵ}{2}}+{\displaystyle \frac{M}{2n{\delta }^2}}(\mathrm{using} {\displaystyle \sum _{j=0}^n{b}_{nj}\left(t_0\right)=1}\mathrm{and} \left(17\right)).\hfill \end{array}$$

When an integer n is sufficiently large satisfying $n>M/ϵ{\delta }^2$, we have

$$\Vert \zeta \left(t_0\right)-\left(B_n\zeta \right)\left(t_0\right)\Vert <{\displaystyle \frac{ϵ}{2}}+{\displaystyle \frac{ϵ}{2}}=ϵ.$$

(20)

Therefore, we obtain

$$\underset{n\to \infty }{lim}\Vert \left(B_n\zeta \right)\left(t_0\right)-\zeta \left(t_0\right)\Vert =0.$$

(21)

Using (15), we also have

$$\underset{n\to \infty }{lim}\Vert s_{\tilde{f}\left(t_0\right)}-s_{\left(B_n\tilde{f}\right)\left(t_0\right)}\Vert =\underset{n\to \infty }{lim}\Vert \zeta \left(t_0\right)-\left(B_n\zeta \right)\left(t_0\right)\Vert =0.$$

which proves part (i).

To prove part (ii), since the fuzzy function $\tilde{f}$ is continuous on $[0,1]$, part (i) says that the function $\zeta$ is continuous on the compact set $[0,1]$. It follows that the function $\zeta =s_{\tilde{f}}$ is also uniformly continuous on $[0,1]$. In other words, given any $ϵ>0$, there exists $\delta >0$ such that

$$|t-s|<\delta \mathrm{implies} \Vert \zeta \left(t\right)-\zeta \left(s\right)\Vert <ϵ/2,$$

where $\delta$ is independent of s and t. Then, we can similarly obtain

$$\Vert \zeta \left(t\right)-\left(B_n\zeta \right)\left(t\right)\Vert =\left|\left|\sum _{j=0}^n\zeta \left(t\right)b_{nj}\left(t\right)-\left(B_n\zeta \right)\left(t\right)\right|\right|\le {\displaystyle \frac{ϵ}{2}}+{\displaystyle \frac{M}{2n{\delta }^2}}, \mathrm{for} \mathrm{all} t\in [0,1],$$

where $\delta$ is independent of t. When an integer n is sufficiently large satisfying $n>M/ϵ{\delta }^2$, we have

$$\Vert \zeta \left(t\right)-\left(B_n\zeta \right)\left(t\right)\Vert <{\displaystyle \frac{ϵ}{2}}+{\displaystyle \frac{ϵ}{2}}=ϵ, \mathrm{for} \mathrm{all} t\in [0,1],$$

which shows $\Vert \zeta \left(t\right)-\left(B_n\zeta \right)\left(t\right)\Vert \to 0$ uniformly on $[0,1]$ as $n\to \infty$. Using (15), we also have

$$\Vert s_{\tilde{f}\left(t\right)}-s_{\left(B_n\tilde{f}\right)\left(t\right)}\Vert =\Vert \zeta \left(t\right)-\left(B_n\zeta \right)\left(t\right)\Vert \to 0$$

uniformly on $[0,1]$ as $n\to \infty$. This completes the proof. □

When the normed space $(U,\Vert ·{\Vert }_U)$ is not separable, we cannot have the Bernstein approximation like the one given in Theorem 1. However, the different types of Bernstein approximation can be obtained. In this case, we can consider the metric space $({\mathcal{F}}_k\left(U\right),d)$ using sup-compatibility with distance instead of the metric space $({\mathcal{F}}_k^C\left(U\right),d)$ adopted in Theorem 1 using compatibility with distance.

Consider the fuzzy function

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k\left(U\right),d\right),$$

which means that the $\alpha$-level sets ${\left(\tilde{f}\left(t\right)\right)}_{\alpha }$ of $\tilde{f}\left(t\right)$ are bounded for $\alpha \in [0,1]$. We say that the fuzzy function $\tilde{f}$ is bounded when there exists a positive constant M satisfying

$$\Vert {\left(\tilde{f}\left(t\right)\right)}_0\Vert \le M, \mathrm{for} \mathrm{all} t\in [0,1],$$

(22)

where $\Vert {\left(\tilde{f}\left(t\right)\right)}_0\Vert$ denotes the norm of subset of U given by

$$\Vert A\Vert =\underset{a\in A}{sup}{\Vert a\Vert }_U, \mathrm{f}\mathrm{o}\mathrm{r} A\subset U.$$

Given any fixed $({\alpha }_0,x_0^{*})\in [0,1]\times {\overline{S}}^{*}$, we see that $s_{\tilde{f}\left(t\right)}({\alpha }_0,x_0^{*})$ and $s_{\left(B_n\tilde{f}\right)\left(t\right)}({\alpha }_0,x_0^{*})$ are real-valued functions of variable t in the sense of

$$\eta \left(t\right)\equiv s_{\tilde{f}\left(t\right)}({\alpha }_0,x_0^{*}) \mathrm{and} {\eta }_n\left(t\right)\equiv s_{\left(B_n\tilde{f}\right)\left(t\right)}({\alpha }_0,x_0^{*})$$

defined on $[0,1]$. The different types of Bernstein approximation are provided below.

Theorem 2.

Let $(U,\Vert ·{\Vert }_U)$ be a normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$, and let $({\mathcal{F}}_k\left(U\right),d)$ be a metric space. Consider the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k\left(U\right),d\right).$$

Then, for any fixed $(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}$, the real-valued function $s_{B_n\tilde{f}\left(t\right)}(\alpha ,x^{*})$ of variable t on $[0,1]$ is the n-th Bernstein polynomial of the real-valued function $s_{\tilde{f}\left(t\right)}(\alpha ,x^{*})$ of variable t on $[0,1]$. Suppose that the fuzzy function $\tilde{f}$ is bounded and that the metric space $({\mathcal{F}}_k\left(U\right),d)$ is sup-compatible with distance. Then, we have the following properties:

(i) 

Suppose that the fuzzy function $\tilde{f}$ is continuous at $t_0\in [0,1]$. Then, the real-valued function $s_{\tilde{f}\left(t\right)}(\alpha ,x^{*})$ of variable t is continuous at $t_0$ for any $(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}$, and the sequence ${\left\{s_{B_n\tilde{f}\left(t_0\right)}\right\}}_{n=1}^{\infty }$ of real-valued functions converges to the real-valued function $s_{\tilde{f}\left(t_0\right)}$ uniformly on $[0,1]\times {\overline{S}}^{*}$. In other words, the sequence ${\left\{s_{B_n\tilde{f}\left(t\right)}(\alpha ,x^{*})\right\}}_{n=1}^{\infty }$ of real-valued functions of variable t is the pointwise Bernstein approximation of real-valued function $s_{\tilde{f}\left(t\right)}(\alpha ,x^{*})$ of variable t at $t_0$ for any $(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}$.

(ii) 

Suppose that the fuzzy function $\tilde{f}$ is continuous on $[0,1]$. Then, the real-valued function $s_{\tilde{f}\left(t\right)}(\alpha ,x^{*})$ of variable t is uniformly continuous on $[0,1]$ for any $(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}$, and the sequence ${\left\{s_{B_n\tilde{f}\left(t\right)}(\alpha ,x^{*})\right\}}_{n=1}^{\infty }$ of real-valued functions of variable t is the uniform Bernstein approximation of real-valued function $s_{\tilde{f}\left(t\right)}(\alpha ,x^{*})$ of variable t for any $(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}$.

Proof .

By referring to the support function of fuzzy set $\left(B_n\tilde{f}\right)\left(t\right)$ given in (12), for any fixed $({\alpha }_0,x_0^{*})\in [0,1]\times {\overline{S}}^{*}$, we write $\eta \left(t\right)=s_{\tilde{f}\left(t\right)}({\alpha }_0,x_0^{*})$. It is clear to see that the real-valued function $s_{\left(B_n\tilde{f}\right)\left(t\right)}({\alpha }_0,x_0^{*})$ of variable t is the n-th Bernstein polynomial of the real-valued function $\eta \left(t\right)=s_{\tilde{f}\left(t\right)}({\alpha }_0,x_0^{*})$.

Since the fuzzy function $\tilde{f}$ is bounded, by referring to (22) and using part (ii) of Proposition 3, we have

$$\left|\eta \left(t\right)\right|=\left|s_{\tilde{f}\left(t\right)}({\alpha }_0,x_0^{*})\right|\le \Vert {\left(\tilde{f}\left(t\right)\right)}_0\Vert \le M, \mathrm{for} \mathrm{all} t\in [0,1],$$

which shows that the real-valued function $\eta$ is bounded on $[0,1]$.

Given any $\tilde{A},\tilde{B}\in {\mathcal{F}}_k\left(U\right)$, the assumption of sup-compatibility with distance says

$$d(\tilde{A},\tilde{B})\ge \Vert s_{\tilde{A}}-s_{\tilde{B}}\Vert .$$

(23)

Suppose that the fuzzy function $\tilde{f}$ is continuous at $t_0\in [0,1]$. Given any $ϵ>0$, there exists $\delta >0$ such that

$$|t-t_0|<\delta \mathrm{implies} d\left(s_{\tilde{f}\left(t\right)},s_{\tilde{f}\left(t_0\right)}\right)<ϵ.$$

Now, using (23), we have

$$\begin{array}{cc}\hfill \left|\eta \left(t\right)-\eta \left(t_0\right)\right|& =\left|s_{\tilde{f}\left(t\right)}({\alpha }_0,x_0^{*})-s_{\tilde{f}\left(t_0\right)}({\alpha }_0,x_0^{*})\right|\hfill \\ \hfill & \le \underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{sup}\left|s_{\tilde{f}\left(t\right)}\left(x^{*}\right)-s_{\tilde{f}\left(t_0\right)}\left(x^{*}\right)\right|=\left|\left|s_{\tilde{f}\left(t\right)}-s_{\tilde{f}\left(t_0\right)}\right|\right|\le d\left(s_{\tilde{f}\left(t\right)},s_{\tilde{f}\left(t_0\right)}\right)<ϵ,\hfill \end{array}$$

which says that the real-valued function $\eta$ is continuous at $t_0$. We also see that $({\alpha }_0,x_0^{*})$ is independent of $\delta$ and $ϵ$. In other words, the argument of proving the continuity of $\eta$ at $t_0$ is independent of $({\alpha }_0,x_0^{*})$.

The real-valued function $s_{\left(B_n\tilde{f}\right)\left(t\right)}(\alpha ,x^{*})$ of variable t is the n-th Bernstein polynomial of the real-valued function $\eta \left(t\right)=s_{\tilde{f}\left(t\right)}(\alpha ,x^{*})$ for any $(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}$, which also means that this situation is independent of $(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}$. Therefore, given any $ϵ>0$, using the boundedness of real-valued function $\eta$ and the proof of part (i) of Theorem 1 by referring to (15) and (20), in which $\zeta$ is replaced by $\eta$ and the argument is independent of $(\alpha ,x^{*})$, we conclude that there exists an integer N such that

$$n\ge N \mathrm{implies} \left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,x^{*})-s_{\tilde{f}\left(t_0\right)}(\alpha ,x^{*})\right|<{\displaystyle \frac{ϵ}{2}}, \mathrm{for} \mathrm{all} (\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*},$$

(24)

which shows that the sequence ${\left\{s_{B_n\tilde{f}\left(t_0\right)}\right\}}_{n=1}^{\infty }$ of real-valued functions converges to the real-valued function $s_{\tilde{f}\left(t_0\right)}$ uniformly on $[0,1]\times {\overline{S}}^{*}$. In other words, the sequence ${\left\{s_{B_n\tilde{f}\left(t\right)}(\alpha ,x^{*})\right\}}_{n=1}^{\infty }$ of real-valued functions of variable t is the pointwise Bernstein approximation of the real-valued function $s_{\tilde{f}\left(t\right)}(\alpha ,x^{*})$ of variable t at $t_0$ for any $(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}$, which proves part (i).

To prove part (ii), since the fuzzy function $\tilde{f}$ is continuous on $[0,1]$, part (i) says that the real-valued function $\eta$ is continuous on the compact set $[0,1]$. It follows that the real-valued function $\eta \left(t\right)=s_{\tilde{f}\left(t\right)}(\alpha ,x^{*})$ is also uniformly continuous on $[0,1]$ for any $(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}$. From the proof of part (ii) of Theorem 1, we can also obtain

$$\left|s_{\left(B_n\tilde{f}\right)\left(t\right)}(\alpha ,x^{*})-s_{\tilde{f}\left(t\right)}(\alpha ,x^{*})\right|\to 0 \mathrm{as} n\to \infty$$

(25)

uniformly on $[0,1]$ with respect to the variable t for any $(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}$. This completes the proof. □

We denote by $C\left(X\right)$ the family of all continuous real-valued functions defined on the topological space. Proposition 2 says that $C\left(X\right)$ is a Banach space. Consider the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k\left(U\right),d\right).$$

Then, we have $\tilde{f}\left(t\right),\left(B_n\tilde{f}\right)\left(t\right)\in {\mathcal{F}}_k\left(U\right)$ for all $t\in [0,1]$. Given any fixed $\alpha \in [0,1]$, we define

$$\gamma \left(t\right)\equiv s_{\tilde{f}\left(t\right)}(\alpha ,·) \mathrm{and} {\gamma }_n\left(t\right)\equiv s_{\left(B_n\tilde{f}\right)\left(t\right)}(\alpha ,·)$$

on $[0,1]$. We further assume that the normed space $(U,\Vert ·{\Vert }_U)$ is separable. Part (i) of Proposition 4 says that $\gamma \left(t\right)$ and ${\gamma }_n\left(t\right)$ are weak*-continuous real-valued functions on ${\overline{S}}^{*}$, i.e.,

$$\gamma \left(t\right),{\gamma }_n\left(t\right)\in C\left({\overline{S}}^{*},d_{{\overline{S}}^{*}}^{(w*)}\right), \mathrm{for} \mathrm{any} t\in [0,1].$$

This also means that $\gamma$ is a function given by

$$\gamma :[0,1]\to \left(C\left({\overline{S}}^{*},d_{{\overline{S}}^{*}}^{(w*)}\right),\Vert ·\Vert \right).$$

The different kinds of Bernstein approximations are provided below.

Theorem 3.

Let $(U,\Vert ·{\Vert }_U)$ be a separable normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$, and let $({\mathcal{F}}_k\left(U\right),d)$ be a metric space. Consider the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k\left(U\right),d\right).$$

Then, for any fixed $\alpha \in [0,1]$, the function $s_{\left(B_n\tilde{f}\left(t\right)\right)}(\alpha ,·)$ of variable t on $[0,1]$ is the n-th Bernstein polynomial of function $s_{\tilde{f}\left(t\right)}(\alpha ,·)$ of variable t on $[0,1]$. Suppose that the metric space $({\mathcal{F}}_k\left(U\right),d)$ is sup-compatible with distance. Then, we also have the following properties:

(i) 

Suppose that the fuzzy function $\tilde{f}$ is continuous at $t_0\in [0,1]$. Then, the function $s_{\tilde{f}\left(t\right)}(\alpha ,·)$ of variable t is continuous at $t_0$ for any $\alpha \in [0,1]$. Also, the sequence ${\left\{s_{B_n\tilde{f}\left(t_0\right)}(\alpha ,·)\right\}}_{n=1}^{\infty }$ of functions of variable α converges to the function $s_{\tilde{f}\left(t_0\right)(\alpha ,·)}$ of variable α uniformly on $[0,1]$. In other words, the sequence ${\left\{s_{B_n\tilde{f}\left(t\right)}(\alpha ,·)\right\}}_{n=1}^{\infty }$ of functions of variable t is the pointwise Bernstein approximation of function $s_{\tilde{f}\left(t\right)}(\alpha ,·)$ of variable t at $t_0$ for any $\alpha \in [0,1]$.

(ii) 

Suppose that the fuzzy function $\tilde{f}$ is continuous on $[0,1]$. Then, the function $s_{\tilde{f}\left(t\right)}(\alpha ,·)$ of variable t is uniformly continuous on $[0,1]$ for any $\alpha \in [0,1]$. Also, the sequence ${\left\{s_{B_n\tilde{f}\left(t\right)}(\alpha ,·)\right\}}_{n=1}^{\infty }$ of functions of variable t is the uniform Bernstein approximation of function $s_{\tilde{f}\left(t\right)}(\alpha ,·)$ of variable t for any $\alpha \in [0,1]$.

Proof .

By referring to the support function of fuzzy set $\left(B_n\tilde{f}\right)\left(t\right)$ given in (12), for any fixed ${\alpha }_0\in [0,1]$, we write $\gamma \left(t\right)=s_{\tilde{f}\left(t\right)}({\alpha }_0,·)$. Part (i) of Proposition 4 says that we can define a function

$$\gamma :[0,1]\to \left(C\left({\overline{S}}^{*},d_{{\overline{S}}^{*}}^{(w*)}\right),\Vert ·\Vert \right) \mathrm{by} \gamma \left(t\right)=s_{\tilde{f}\left(t\right)}({\alpha }_0,·).$$

The continuity of the norm $\Vert ·\Vert$ and the compact set $[0,1]$ say that the following supremum is attained:

$$\underset{t\in [0,1]}{sup}\Vert \gamma \left(t\right)\Vert =\underset{t\in [0,1]}{max}\Vert \gamma \left(t\right)\Vert =\Vert \gamma \left(t_0\right)\Vert =\Vert s_{\tilde{f}\left(t_0\right)}({\alpha }_0,·)\Vert$$

for some $t_0\in [0,1]$. Since the closed unit ball ${\overline{S}}^{*}$ is weak*-compact by Alaoglu’s theorem and the support function $s_{\tilde{f}\left(t_0\right)}({\alpha }_0,·)$ is weak*-continuous by part (i) of Proposition 4, we attain the supremum

$$\begin{array}{cc}\hfill \Vert s_{\tilde{f}\left(t_0\right)}({\alpha }_0,·)\Vert & =\underset{x^{*}\in {\overline{S}}^{*}}{sup}\left|s_{\tilde{f}\left(t_0\right)}({\alpha }_0,x^{*})\right|\hfill \\ \hfill & =\underset{x^{*}\in {\overline{S}}^{*}}{max}\left|s_{\tilde{f}\left(t_0\right)}({\alpha }_0,x^{*})\right|=\left|s_{\tilde{f}\left(t_0\right)}({\alpha }_0,x_0^{*})\right|<+\infty \hfill \end{array}$$

for some $x_0^{*}\in {\overline{S}}^{*}$, which shows that the function $\gamma$ is bounded in the sense of

$$\Vert \gamma \left(t\right)\Vert \le \underset{t\in [0,1]}{sup}\Vert \gamma \left(t\right)\Vert =\Vert s_{\tilde{f}\left(t_0\right)}({\alpha }_0,·)\Vert <+\infty , \mathrm{for} \mathrm{all} t\in [0,1].$$

Since $\gamma (j/n)=s_{\tilde{f}(j/n)}({\alpha }_0,·)$, using part (i) of Proposition 3, we have

$$\begin{array}{cc}\hfill s_{\left(B_n\tilde{f}\right)\left(t\right)}({\alpha }_0,·)& =\sum _{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)t^j{(1-t)}^{n-j}{s}_{\tilde{f}(j/n)}({\alpha }_0,·)\hfill \\ \hfill & =\sum _{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)t^j{(1-t)}^{n-j}\gamma (j/n)\equiv \left(B_n\gamma \right)\left(t\right),\hfill \end{array}$$

which says that the function $s_{\left(B_n\tilde{f}\right)\left(t\right)}({\alpha }_0,·)$ of variable t is the n-th Bernstein polynomial of the function $\gamma \left(t\right)=s_{\tilde{f}\left(t\right)}({\alpha }_0,·)$.

Given any $\tilde{A},\tilde{B}\in {\mathcal{F}}_k\left(U\right)$, the assumption of sup-compatibility with distance says

$$d(\tilde{A},\tilde{B})\ge \Vert s_{\tilde{A}}-s_{\tilde{B}}\Vert .$$

(26)

Suppose that $\tilde{f}$ is continuous at $t_0\in [0,1]$. Using (26), given any $ϵ>0$, there exists $\delta >0$ such that $|t-t_0|<\delta$ implies

$$\begin{array}{cc}\hfill \Vert \gamma \left(t\right)-\gamma \left(t_0\right)\Vert & =\Vert s_{\tilde{f}\left(t\right)}({\alpha }_0,·)-s_{\tilde{f}\left(t_0\right)}({\alpha }_0,·)\Vert \hfill \\ \hfill & =\underset{x^{*}\in {\overline{S}}^{*}}{sup}\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}({\alpha }_0,x^{*})-s_{\tilde{f}\left(t_0\right)}({\alpha }_0,x^{*})\right|\hfill \\ \hfill & \le \underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{sup}\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,x^{*})-s_{\tilde{f}\left(t_0\right)}(\alpha ,x^{*})\right|\hfill \\ \hfill & =\Vert s_{\tilde{f}\left(t\right)}-s_{\tilde{f}\left(t_0\right)}\Vert \le d(\tilde{f}\left(t\right),\tilde{f}\left(t_0\right))<{\displaystyle \frac{ϵ}{2}},\hfill \end{array}$$

which shows that the function $\gamma$ is continuous at $t_0$. We also see that the argument of proving the continuity of $\gamma$ at $t_0$ is independent of ${\alpha }_0$.

Given any $ϵ>0$, using the boundedness of function $\gamma$ and the proof of part (i) of Theorem 1 by referring to (15) and (20), in which $\zeta$ is replaced by $\gamma$ and the argument is independent of $\alpha$, we conclude that there exists an integer N such that

$$n\ge N \mathrm{implies} \left|\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,·)-s_{\tilde{f}\left(t_0\right)}(\alpha ,·)\right|\right|<{\displaystyle \frac{ϵ}{2}}, \mathrm{for} \mathrm{all} \alpha \in [0,1],$$

(27)

which shows that the sequence ${\left\{s_{B_n\tilde{f}\left(t_0\right)}(\alpha ,·)\right\}}_{n=1}^{\infty }$ of functions of variable $\alpha$ converges to the function $s_{\tilde{f}\left(t_0\right)(\alpha ,·)}$ of variable $\alpha$ uniformly on $[0,1]$. In other words, the sequence ${\left\{s_{B_n\tilde{f}\left(t\right)}(\alpha ,·)\right\}}_{n=1}^{\infty }$ of functions of variable t is the pointwise Bernstein approximation of the function $s_{\tilde{f}\left(t\right)}(\alpha ,·)$ of variable t at $t_0$ for any $\alpha \in [0,1]$, which proves part (i).

To prove part (ii), since the fuzzy function $\tilde{f}$ is continuous on $[0,1]$, part (i) says that the function $\gamma$ is continuous on the compact set $[0,1]$. It follows that the function $\gamma \left(t\right)=s_{\tilde{f}\left(t\right)}(\alpha ,·)$ is also uniformly continuous on $[0,1]$ for any $\alpha \in [0,1]$. From the proof of part (ii) of Theorem 1, we can also obtain

$$\left|\left|s_{\left(B_n\tilde{f}\right)\left(t\right)}(\alpha ,·)-s_{\tilde{f}\left(t\right)}(\alpha ,·)\right|\right|\to 0 \mathrm{as} n\to \infty$$

(28)

uniformly on $[0,1]$ with respect to the variable t for any $\alpha \in [0,1]$. This completes the proof. □

We denote by $C[0,1]$ the family of all continuous real-valued functions defined on $[0,1]$. Since the family ${\overline{\mathcal{F}}}_k\left(U\right)={\mathcal{F}}_k^C\left(U\right)$ is closed under addition and scalar multiplication, we can study the Bernstein approximations by considering the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\overline{\mathcal{F}}}_k\left(U\right),d\right).$$

This means that $\tilde{f}\left(t\right),\left(B_n\tilde{f}\right)\left(t\right)\in {\overline{\mathcal{F}}}_k\left(U\right)$ for all $t\in [0,1]$. Given any fixed $x^{*}\in {\overline{S}}^{*}$, we define

$$\phi \left(t\right)\equiv s_{\tilde{f}\left(t\right)}(·,x^{*}) \mathrm{and} {\phi }_n\left(t\right)\equiv s_{\left(B_n\tilde{f}\right)\left(t\right)}(·,x^{*})$$

on $[0,1]$. Part (ii) of Proposition 4 says that $\phi \left(t\right)$ and ${\phi }_n\left(t\right)$ are continuous real-valued functions on $[0,1]$, i.e.,

$$\phi \left(t\right),{\phi }_n\left(t\right)\in C[0,1], \mathrm{for} \mathrm{any} t\in [0,1].$$

This also means that $\phi$ is a function given by

$$\phi :[0,1]\to \left(C[0,1],\Vert ·\Vert \right).$$

The different kinds of Bernstein approximations are provided below.

Theorem 4.

Let $(U,\Vert ·{\Vert }_U)$ be a normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$, and let $({\overline{\mathcal{F}}}_k\left(U\right),d)$ be a metric space. Consider the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\overline{\mathcal{F}}}_k\left(U\right),d\right).$$

Then, for any fixed $x^{*}\in {\overline{S}}^{*}$, the function $s_{\left(B_n\tilde{f}\left(t\right)\right)}(·,x^{*})$ of variable t on $[0,1]$ is the n-th Bernstein polynomial of function $s_{\tilde{f}\left(t\right)}(·,x^{*})$ of variable t on $[0,1]$. Suppose that the metric space $({\overline{\mathcal{F}}}_k\left(U\right),d)$ is sup-compatible with distance. Then, we also have the following properties:

(i) 

Suppose that the fuzzy function $\tilde{f}$ is continuous at $t_0\in [0,1]$. Then, the function $s_{\tilde{f}\left(t\right)}(·,x^{*})$ of variable t is continuous at $t_0$ for any $x^{*}\in {\overline{S}}^{*}$. Also, the sequence ${\left\{s_{B_n\tilde{f}\left(t_0\right)}(·,x^{*})\right\}}_{n=1}^{\infty }$ of functions of variable $x^{*}$ converges to the function $s_{\tilde{f}\left(t_0\right)(·,x^{*})}$ of variable $x^{*}$ uniformly on ${\overline{S}}^{*}$. In other words, the sequence ${\left\{s_{B_n\tilde{f}\left(t\right)}(·,x^{*})\right\}}_{n=1}^{\infty }$ of functions of variable t is the pointwise Bernstein approximation of function $s_{\tilde{f}\left(t\right)}(·,x^{*})$ of variable t at $t_0$ for any $x^{*}\in {\overline{S}}^{*}$.

(ii) 

Suppose that the fuzzy function $\tilde{f}$ is continuous on $[0,1]$. Then, the function $s_{\tilde{f}\left(t\right)}(·,x^{*})$ of variable t is uniformly continuous on $[0,1]$ for any $x^{*}\in {\overline{S}}^{*}$. Also, the sequence ${\left\{s_{B_n\tilde{f}\left(t\right)}(·,x^{*})\right\}}_{n=1}^{\infty }$ of functions of variable t is the uniform Bernstein approximation of function $s_{\tilde{f}\left(t\right)}(·,x^{*})$ of variable t for any $x^{*}\in {\overline{S}}^{*}$.

Proof .

By referring to the support function of fuzzy set $\left(B_n\tilde{f}\right)\left(t\right)$ given in (12), for any fixed $x_0^{*}\in {\overline{S}}^{*}$, we write $\phi \left(t\right)=s_{\tilde{f}\left(t\right)}(·,x_0^{*})$. Part (ii) of Proposition 4 says that we can define a function

$$\phi :[0,1]\to \left(C[0,1],\Vert ·\Vert \right) \mathrm{by} \phi \left(t\right)=s_{\tilde{f}\left(t\right)}(·,x_0^{*}).$$

The continuity of the norm $\Vert ·\Vert$ and the compact set $[0,1]$ say that the following supremum is attained:

$$\underset{t\in [0,1]}{sup}\Vert \phi \left(t\right)\Vert =\underset{t\in [0,1]}{max}\Vert \phi \left(t\right)\Vert =\Vert \phi \left(t_0\right)\Vert =\Vert s_{\tilde{f}\left(t_0\right)}(·,x_0^{*})\Vert$$

for some $t_0\in [0,1]$. Since the function $s_{\tilde{f}\left(t_0\right)}(·,x_0^{*})$ is continuous on the compact set $[0,1]$ by part (ii) of Proposition 4, the following supremum is also attained:

$$\begin{array}{cc}\hfill \Vert s_{\tilde{f}\left(t_0\right)}(·,x_0^{*})\Vert & =\underset{\alpha \in [0,1]}{sup}\left|s_{\tilde{f}\left(t_0\right)}(\alpha ,x_0^{*})\right|\hfill \\ \hfill & =\underset{\alpha \in [0,1]}{max}\left|s_{\tilde{f}\left(t_0\right)}(\alpha ,x_0^{*})\right|=\left|s_{\tilde{f}\left(t_0\right)}({\alpha }_0,x_0^{*})\right|<+\infty \hfill \end{array}$$

for some ${\alpha }_0\in [0,1]$, which shows that the function $\phi$ is bounded in the sense of

$$\Vert \phi \left(t\right)\Vert \le \underset{t\in [0,1]}{sup}\Vert \phi \left(t\right)\Vert =\Vert s_{\tilde{f}\left(t_0\right)}(·,x_0^{*})\Vert <+\infty , \mathrm{for} \mathrm{all} t\in [0,1].$$

Since $\phi (j/n)=s_{\tilde{f}(j/n)}(·,x_0^{*})$, using part (i) of Proposition 3, we have

$$\begin{array}{cc}\hfill s_{\left(B_n\tilde{f}\right)\left(t\right)}(·,x_0^{*})& =\sum _{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)t^j{(1-t)}^{n-j}{s}_{\tilde{f}(j/n)}(·,x_0^{*})\hfill \\ \hfill & =\sum _{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)t^j{(1-t)}^{n-j}\phi (j/n)\equiv \left(B_n\phi \right)\left(t\right),\hfill \end{array}$$

which says that the function $s_{\left(B_n\tilde{f}\right)\left(t\right)}(·,x_0^{*})$ of variable t is the n-th Bernstein polynomial of the function $\phi \left(t\right)=s_{\tilde{f}\left(t\right)}(·,x_0^{*})$.

Given any $\tilde{A},\tilde{B}\in {\overline{\mathcal{F}}}_k\left(U\right)$, the assumption of sup-compatibility with distance says

$$d(\tilde{A},\tilde{B})\ge \Vert s_{\tilde{A}}-s_{\tilde{B}}\Vert .$$

(29)

Suppose that $\tilde{f}$ is continuous at $t_0\in [0,1]$. Using (29), given any $ϵ>0$, there exists $\delta >0$ such that $|t-t_0|<\delta$ implies

$$\begin{array}{cc}\hfill \Vert \phi \left(t\right)-\phi \left(t_0\right)\Vert & =\Vert s_{\tilde{f}\left(t\right)}(·,x_0^{*})-s_{\tilde{f}\left(t_0\right)}(·,x_0^{*})\Vert \hfill \\ \hfill & =\underset{\alpha \in [0,1]}{sup}\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}({\alpha }_0,x^{*})-s_{\tilde{f}\left(t_0\right)}({\alpha }_0,x^{*})\right|\hfill \\ \hfill & \le \underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{sup}\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,x^{*})-s_{\tilde{f}\left(t_0\right)}(\alpha ,x^{*})\right|\hfill \\ \hfill & =\Vert s_{\tilde{f}\left(t\right)}-s_{\tilde{f}\left(t_0\right)}\Vert \le d(\tilde{f}\left(t\right),\tilde{f}\left(t_0\right))<{\displaystyle \frac{ϵ}{2}},\hfill \end{array}$$

which shows that the function $\phi$ is continuous at $t_0$. We also see that the argument of proving the continuity of $\phi$ at $t_0$ is independent of $x_0^{*}$.

Given any $ϵ>0$, using the boundedness of function $\phi$ and the proof of part (i) of Theorem 1 by referring to (15) and (20), in which $\zeta$ is replaced by $\phi$ and the argument is independent of $x^{*}$, we conclude that there exists an integer N such that

$$n\ge N \mathrm{implies} \left|\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(·,x^{*})-s_{\tilde{f}\left(t_0\right)}(·,x^{*})\right|\right|<{\displaystyle \frac{ϵ}{2}}, \mathrm{for} \mathrm{all} x^{*}\in {\overline{S}}^{*},$$

which shows that the sequence ${\left\{s_{B_n\tilde{f}\left(t_0\right)}(·,x^{*})\right\}}_{n=1}^{\infty }$ of functions of variable $x^{*}$ converges to the function $s_{\tilde{f}\left(t_0\right)(·,x^{*})}$ of variable $x^{*}$ uniformly on ${\overline{S}}^{*}$. In other words, the sequence ${\left\{s_{B_n\tilde{f}\left(t\right)}(·,x^{*})\right\}}_{n=1}^{\infty }$ of functions of variable t is the pointwise Bernstein approximation of the function $s_{\tilde{f}\left(t\right)}(·,x^{*})$ of variable t at $t_0$ for any $x^{*}\in {\overline{S}}^{*}$, which proves part (i).

To prove part (ii), since the fuzzy function $\tilde{f}$ is continuous on $[0,1]$, part (i) says that the function $\phi$ is continuous on the compact set $[0,1]$. It follows that the function $\phi \left(t\right)=s_{\tilde{f}\left(t\right)}(·,x^{*})$ is also uniformly continuous on $[0,1]$ for any $x^{*}\in {\overline{S}}^{*}$. From the proof of part (ii) of Theorem 1, we can also obtain

$$\left|\left|s_{\left(B_n\tilde{f}\right)\left(t\right)}(·,x^{*})-s_{\tilde{f}\left(t\right)}(·,x^{*})\right|\right|\to 0 \mathrm{a}\mathrm{s} n\to \infty$$

uniformly on $[0,1]$ with respect to the variable t for any $x^{*}\in {\overline{S}}^{*}$. This completes the proof. □

4. Bernstein Approximation for Fuzzy Functions

Now, we are in a position to study the Bernstein approximation for fuzzy functions by using the Bernstein approximation for support functions obtained above. We consider the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k\left(U\right),d\right).$$

By referring to (10), we have a sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ of fuzzy functions that consist of Bernstein polynomials of fuzzy function $\tilde{f}$. Two concepts of Bernstein approximation are defined below:

• We say that the sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ of fuzzy functions is the pointwise Bernstein approximation of fuzzy function $\tilde{f}$ at $t_0\in [0,1]$ when we have

  $$\underset{n\to \infty }{lim}d\left(\left(B_n\tilde{f}\right)\left(t_0\right),\tilde{f}\left(t_0\right)\right)=0.$$
• We say that the sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ of fuzzy functions is the uniform Bernstein approximation of fuzzy function $\tilde{f}$ when the sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ converges to $\tilde{f}$ uniformly on $[0,1]$. More precisely, given any $ϵ>0$, there exists an integer N such that

  $$n\ge N \mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{s} d\left(\left(B_n\tilde{f}\right)\left(t\right),\tilde{f}\left(t\right)\right)<ϵ, \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} t\in [0,1],$$

  where N is independent of t.

We first study the Bernstein approximation for the fuzzy function

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k^C\left(U\right),d\right)=\left({\overline{\mathcal{F}}}_k\left(U\right),d\right),$$

which is presented below.

Theorem 5.

Let $(U,\Vert ·{\Vert }_U)$ be a separable normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$, and let $({\mathcal{F}}_k^C\left(U\right),d)$ be a metric space. Consider the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k^C\left(U\right),d\right).$$

Suppose that the metric space $({\mathcal{F}}_k^C\left(U\right),d)$ is compatible with distance. Then, we have the following approximations:

(i) 

Suppose that the fuzzy function $\tilde{f}$ is continuous at $t_0\in [0,1]$. Then, the sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ of fuzzy functions is the pointwise Bernstein approximation of fuzzy function $\tilde{f}$ at $t_0$.

(ii) 

Suppose that the fuzzy function $\tilde{f}$ is continuous on $[0,1]$. Then, the sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ of fuzzy functions is the uniform Bernstein approximation of fuzzy function $\tilde{f}$.

Proof .

To prove part (i), continued from the proof of part (i) of Theorem 1, we have

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}d\left(\tilde{f}\left(t_0\right),\left(B_n\tilde{f}\right)\left(t_0\right)\right)& =\underset{n\to \infty }{lim}\Vert s_{\tilde{f}\left(t_0\right)}-s_{\left(B_n\tilde{f}\right)\left(t_0\right)}\Vert (\mathrm{using} (16\left)\right)\hfill \\ \hfill & =\underset{n\to \infty }{lim}\Vert \zeta \left(t_0\right)-\left(B_n\zeta \right)\left(t_0\right)\Vert =0(\mathrm{using} (15) \mathrm{and} (21\left)\right).\hfill \end{array}$$

To prove part (ii), continued from the proof of part (ii) of Theorem 1, we have

$$\begin{array}{cc}\hfill d\left(\tilde{f}\left(t\right),\left(B_n\tilde{f}\right)\left(t\right)\right)& =\Vert s_{\tilde{f}\left(t\right)}-s_{\left(B_n\tilde{f}\right)\left(t\right)}\Vert (\mathrm{using} (16\left)\right)\hfill \\ \hfill & =\Vert \zeta \left(t\right)-\left(B_n\zeta \right)\left(t\right)\Vert \to 0(\mathrm{using} (15\left)\right)\hfill \end{array}$$

uniformly on $[0,1]$ as $n\to \infty$. This completes the proof. □

Without assuming compatibility with distance, we consider the subfamily ${\mathcal{F}}_{kc}^C\left(U\right)$ of ${\mathcal{F}}_{kc}\left(U\right)$ such that, for each $\tilde{A}\in {\mathcal{F}}_{kc}^C\left(U\right)$, the function ${\eta }_{\tilde{A}}$ defined in (3) is continuous with respect to the Pompeiu–Hausdorff metric $d_H$. Then, we can also show that ${\mathcal{F}}_{kc}^C\left(U\right)$ is closed under addition and scalar multiplication. In this case, we can consider the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_{kc}^C\left(U\right),d_{\mathcal{F}}\right).$$

Then, we also have the following interesting approximation without considering compatibility with distance.

Theorem 6.

Let $(U,\Vert ·{\Vert }_U)$ be a separable normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$, and let $({\mathcal{F}}_{kc}^C\left(U\right),d_{\mathcal{F}})$ be a metric space. Consider the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_{kc}^C\left(U\right),d_{\mathcal{F}}\right).$$

Then, we have the following approximations:

(i) 

Suppose that the fuzzy function $\tilde{f}$ is continuous at $t_0\in [0,1]$. Then, the sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ of fuzzy functions is the pointwise Bernstein approximation of fuzzy function $\tilde{f}$ at $t_0$.

(ii) 

Suppose that the fuzzy function $\tilde{f}$ is continuous on $[0,1]$. Then, the sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ of fuzzy functions is the uniform Bernstein approximation of fuzzy function $\tilde{f}$.

Proof. 

Part (iii) of Proposition 3 says that the metric space $({\mathcal{F}}_{kc}^C\left(U\right),d_{\mathcal{F}})$ is compatible with distance. Therefore, the proof of Theorem 5 is still valid. This completes the proof. □

Theorem 7.

Let $(U,\Vert ·{\Vert }_U)$ be a separable normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$, and let $({\mathcal{F}}_k\left(U\right),d)$ be a metric space. Consider the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k\left(U\right),d\right).$$

Suppose that the metric space $({\mathcal{F}}_k\left(U\right),d)$ is compatible with distance. Then, we have the following properties:

(i) 

Suppose that the fuzzy function $\tilde{f}$ is continuous at $t_0\in [0,1]$. Then, the sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ of fuzzy functions is the pointwise Bernstein approximation of fuzzy function $\tilde{f}$ at $t_0$.

(ii) 

Suppose that the fuzzy function $\tilde{f}$ is continuous on $[0,1]$. Then, the sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ of fuzzy functions is the uniform Bernstein approximation of fuzzy function $\tilde{f}$.

Proof. 

To prove part (i), continued from the proof of part (i) of Theorem 3, using (27), we have

$$n\ge N \mathrm{implies} \underset{\alpha \in [0,1]}{sup}\left|\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,·)-s_{\tilde{f}\left(t_0\right)}(\alpha ,·)\right|\right|\le {\displaystyle \frac{ϵ}{2}}<ϵ.$$

Therefore, we obtain

$$\begin{array}{cc}\hfill 0& =\underset{n\to \infty }{lim}\underset{\alpha \in [0,1]}{sup}\left|\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,·)-s_{\tilde{f}\left(t_0\right)}(\alpha ,·)\right|\right|\hfill \\ \hfill & =\underset{n\to \infty }{lim}\underset{\alpha \in [0,1]}{sup}\underset{x^{*}\in {\overline{S}}^{*}}{sup}\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,x^{*})-s_{\tilde{f}\left(t_0\right)}(\alpha ,x^{*})\right|\hfill \\ \hfill & =\underset{n\to \infty }{lim}\underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{sup}\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,x^{*})-s_{\tilde{f}\left(t_0\right)}(\alpha ,x^{*})\right|\hfill \\ \hfill & =\underset{n\to \infty }{lim}\left|\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}-s_{\tilde{f}\left(t_0\right)}\right|\right|.\hfill \end{array}$$

(30)

Using the assumption of compatibility with distance, we obtain

$$\underset{n\to \infty }{lim}d(\left(B_n\tilde{f}\right)\left(t_0\right),\tilde{f}\left(t_0\right))=\underset{n\to \infty }{lim}\left|\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}-s_{\tilde{f}\left(t_0\right)}\right|\right|=0.$$

To prove part (ii), continued from the proof of part (ii) of Theorem 3, using (28), we have

$$\underset{\alpha \in [0,1]}{sup}\left|\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,·)-s_{\tilde{f}\left(t_0\right)}(\alpha ,·)\right|\right|\to 0$$

uniformly on $[0,1]$ with respect to the variable t as $n\to \infty$. By referring to (30), we also have

$$\left|\left|s_{\left(B_n\tilde{f}\right)\left(t\right)}-s_{\tilde{f}\left(t\right)}\right|\right|=\underset{\alpha \in [0,1]}{sup}\left|\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,·)-s_{\tilde{f}\left(t_0\right)}(\alpha ,·)\right|\right|\to 0$$

uniformly on $[0,1]$ with respect to the variable t as $n\to \infty$. Using the assumption of compatibility with distance, we obtain

$$d\left(\left(B_n\tilde{f}\right)\left(t\right),\tilde{f}\left(t\right)\right)=\left|\left|s_{\left(B_n\tilde{f}\right)\left(t\right)}-s_{\tilde{f}\left(t\right)}\right|\right|\to 0$$

uniformly on $[0,1]$ as $n\to \infty$. This completes the proof. □

The above Theorem 7 considers the separable normed space. When the normed space is not separable, we need to assume that the fuzzy function $\tilde{f}$ is bounded, which is presented below.

Theorem 8.

Let $(U,\Vert ·{\Vert }_U)$ be a normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$, and let $({\mathcal{F}}_k\left(U\right),d)$ be a metric space. Consider the following fuzzy function:

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k\left(U\right),d\right).$$

Suppose that the fuzzy function $\tilde{f}$ is bounded and that the metric space $({\mathcal{F}}_k\left(U\right),d)$ is compatible with distance. Then, we have the following properties:

(i) 

Suppose that the fuzzy function $\tilde{f}$ is continuous at $t_0\in [0,1]$. Then, the sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ of fuzzy functions is the pointwise Bernstein approximation of fuzzy function $\tilde{f}$ at $t_0$.

(ii) 

Suppose that the fuzzy function $\tilde{f}$ is continuous on $[0,1]$. Then, the sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ of fuzzy functions is the uniform Bernstein approximation of fuzzy function $\tilde{f}$.

Proof. 

To prove part (i), continued from the proof of part (i) of Theorem 2, using (24), we also have

$$n\ge N \mathrm{implies} \underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{sup}\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,x^{*})-s_{\tilde{f}\left(t_0\right)}(\alpha ,x^{*})\right|\le {\displaystyle \frac{ϵ}{2}}<ϵ.$$

Therefore, we obtain

$$\begin{array}{cc}\hfill 0& =\underset{n\to \infty }{lim}\underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{sup}\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,x^{*})-s_{\tilde{f}\left(t_0\right)}(\alpha ,x^{*})\right|\hfill \\ \hfill & =\underset{n\to \infty }{lim}\left|\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}-s_{\tilde{f}\left(t_0\right)}\right|\right|.\hfill \end{array}$$

Using the assumption of compatibility with distance, we obtain

$$\underset{n\to \infty }{lim}d(\left(B_n\tilde{f}\right)\left(t_0\right),\tilde{f}\left(t_0\right))=\underset{n\to \infty }{lim}\left|\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}-s_{\tilde{f}\left(t_0\right)}\right|\right|=0.$$

To prove part (ii), continued from the proof of part (ii) of Theorem 2, using (25), we have

$$\left|\left|s_{\left(B_n\tilde{f}\right)\left(t\right)}-s_{\tilde{f}\left(t\right)}\right|\right|=\underset{(\alpha ,x^{*})\in [0,1]\times {\overline{S}}^{*}}{sup}\left|s_{\left(B_n\tilde{f}\right)\left(t_0\right)}(\alpha ,x^{*})-s_{\tilde{f}\left(t_0\right)}(\alpha ,x^{*})\right|\to 0$$

uniformly on $[0,1]$ with respect to the variable t as $n\to \infty$. Using the assumption of compatibility with distance, we obtain

$$d\left(\left(B_n\tilde{f}\right)\left(t\right),\tilde{f}\left(t\right)\right)=\left|\left|s_{\left(B_n\tilde{f}\right)\left(t\right)}-s_{\tilde{f}\left(t\right)}\right|\right|\to 0$$

uniformly on $[0,1]$ as $n\to \infty$. This completes the proof. □

5. Practical Examples

We take $U=\mathbb{R}$. Then, given any $\tilde{A}\in {\mathcal{F}}_{kc}^C\left(\mathbb{R}\right)$, the $\alpha$-level sets ${\tilde{A}}_{\alpha }$ are bounded closed intervals for $\alpha \in [0,1]$. In this case, we write

$${\tilde{A}}_{\alpha }=\left[{\tilde{A}}_{\alpha }^L,{\tilde{A}}_{\alpha }^U\right].$$

Given any $\tilde{A},\tilde{B}\in {\mathcal{F}}_{kc}^C\left(\mathbb{R}\right)$, it is clear to see that

$$d_H\left({\tilde{A}}_{\alpha },{\tilde{B}}_{\alpha }\right)=max\left\{\left|{\tilde{A}}_{\alpha }^L-{\tilde{A}}_{\alpha }^L\right|,\left|{\tilde{A}}_{\alpha }^U-{\tilde{A}}_{\alpha }^U\right|\right\}.$$

Therefore, we obtain

$$d_{\mathcal{F}}\left(\tilde{A},\tilde{B}\right)=\underset{\alpha \in [0,1]}{sup}{d}_H\left({\tilde{A}}_{\alpha },{\tilde{B}}_{\alpha }\right)=\underset{\alpha \in [0,1]}{sup}max\left\{\left|{\tilde{A}}_{\alpha }^L-{\tilde{A}}_{\alpha }^L\right|,\left|{\tilde{A}}_{\alpha }^U-{\tilde{A}}_{\alpha }^U\right|\right\}.$$

(31)

We consider the fuzzy function

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_{kc}^C\left(\mathbb{R}\right),d_{\mathcal{F}}\right)$$

such that it is continuous on $[0,1]$ with respect to the Pompeiu–Hausdorff metric $d_{\mathcal{F}}$. By referring to (10), the n-th Bernstein polynomial of the fuzzy function $\tilde{f}$ is a fuzzy function

$$\left(B_n\tilde{f}\right):[0,1]\to \left({\mathcal{F}}_{kc}^C\left(U\right),d_{\mathcal{F}}\right)$$

given by

$$\left(B_n\tilde{f}\right)\left(t\right)=\underset{j=0}{\overset{n}{⨁}}{b}_{nj}\left(t\right)\tilde{f}\left({\displaystyle \frac{j}{n}}\right) \mathrm{with} b_{nj}\left(t\right)=\left(\begin{array}{c}n\\ j\end{array}\right)·t^j·{(1-t)}^{n-j},$$

where

$$\tilde{f}\left({\displaystyle \frac{j}{n}}\right)={\left({\displaystyle \frac{j}{n}}\right)}^p{\tilde{A}}^{\left(p\right)}\oplus {\left({\displaystyle \frac{j}{n}}\right)}^{p-1}{\tilde{A}}^{(p-1)}\oplus \left({\displaystyle \frac{j}{n}}\right){\tilde{A}}^{\left(1\right)}$$

for $t\in [0,1]$ and $j=1,\dots ,p$. Using Theorem 6, the sequence ${\left\{B_n\tilde{f}\right\}}_{n=1}^{\infty }$ of fuzzy functions is the uniform Bernstein approximation of fuzzy function $\tilde{f}$. More precisely, given any $ϵ>0$, there exists an integer N such that

$$n\ge N \mathrm{implies} d_{\mathcal{F}}\left(\left(B_n\tilde{f}\right)\left(t\right),\tilde{f}\left(t\right)\right)<ϵ, \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} t\in [0,1],$$

(32)

where N is independent of t.

Example 2.

Given any $\tilde{A}\in {\mathcal{F}}_{kc}^C\left(\mathbb{R}\right)$, we consider the following fuzzy-valued function:

$$\tilde{f}\left(t\right)=texp\left(\tilde{A}\right)\oplus sin\left(\tilde{A}\right), \mathrm{f}\mathrm{o}\mathrm{r} t\in [0,1],$$

(33)

where $texp\left(\tilde{A}\right)$ means scalar multiplication. It is not difficult to show that $\tilde{f}$ is continuous on $[0,1]$ with respect to the Pompeiu–Hausdorff metric $d_{\mathcal{F}}$. The α-level sets of $exp\left(\tilde{A}\right)$ and $sin\left(\tilde{A}\right)$ are bounded closed intervals given by

$${\left(exp\left(\tilde{A}\right)\right)}_{\alpha }=\left\{exp\left(x\right):x\in {\tilde{A}}_{\alpha }\right\}=\left[exp\left({\tilde{A}}_{\alpha }^L\right),exp\left({\tilde{A}}_{\alpha }^L\right)\right]$$

and

$${\left(sin\left(\tilde{A}\right)\right)}_{\alpha }=\left\{sin\left(x\right):x\in {\tilde{A}}_{\alpha }\right\}=\left[\underset{x\in {\tilde{A}}_{\alpha }}{min}sin\left(x\right),\underset{x\in {\tilde{A}}_{\alpha }}{max}sin\left(x\right)\right].$$

for $\alpha \in [0,1]$. The α-level sets of $\tilde{f}\left(t\right)$ defined in (33) are also the bounded closed intervals with end-points given by

$${\left(\tilde{f}\left(t\right)\right)}_{\alpha }^L\equiv {\tilde{f}}_{\alpha }^L\left(t\right)=texp\left({\tilde{A}}_{\alpha }^L\right)+\underset{x\in {\tilde{A}}_{\alpha }}{min}sin\left(x\right)$$

and

$${\left(\tilde{f}\left(t\right)\right)}_{\alpha }^L\equiv {\tilde{f}}_{\alpha }^L\left(t\right)=texp\left({\tilde{A}}_{\alpha }^U\right)+\underset{x\in {\tilde{A}}_{\alpha }}{max}sin\left(x\right).$$

Similarly, for $\left(B_n\tilde{f}\right)\left(t\right)$, we also have

$${\left(B_n\tilde{f}\left(t\right)\right)}_{\alpha }^L\equiv B_n{\tilde{f}}_{\alpha }^L\left(t\right)=\sum _{j=0}^n{b}_{nj}\left(t\right){\tilde{f}}_{\alpha }^L\left({\displaystyle \frac{j}{n}}\right)=\sum _{j=0}^n{b}_{nj}\left(t\right)\left(\left({\displaystyle \frac{j}{n}}\right)exp\left({\tilde{A}}_{\alpha }^L\right)+\underset{x\in {\tilde{A}}_{\alpha }}{min}sin\left(x\right)\right)$$

and

$${\left(B_n\tilde{f}\left(t\right)\right)}_{\alpha }^U\equiv B_n{\tilde{f}}_{\alpha }^U\left(t\right)=\sum _{j=0}^n{b}_{nj}\left(t\right){\tilde{f}}_{\alpha }^U\left({\displaystyle \frac{j}{n}}\right)=\sum _{j=0}^n{b}_{nj}\left(t\right)\left(\left({\displaystyle \frac{j}{n}}\right)exp\left({\tilde{A}}_{\alpha }^U\right)+\underset{x\in {\tilde{A}}_{\alpha }}{max}sin\left(x\right)\right).$$

The Bernstein approximation involves the computations given in (32) and (31). More precisely, we have

$$d_{\mathcal{F}}\left(\left(B_n\tilde{f}\right)\left(t\right),\tilde{f}\left(t\right)\right)=\underset{\alpha \in [0,1]}{sup}max\left\{\left|B_n{\tilde{f}}_{\alpha }^L\left(t\right)-{\tilde{f}}_{\alpha }^L\left(t\right)\right|,\left|B_n{\tilde{f}}_{\alpha }^U\left(t\right)-{\tilde{f}}_{\alpha }^U\left(t\right)\right|\right\},$$

where

$$B_n{\tilde{f}}_{\alpha }^L\left(t\right)-{\tilde{f}}_{\alpha }^L\left(t\right)=\sum _{j=0}^n{b}_{nj}\left(t\right)\left(\left({\displaystyle \frac{j}{n}}\right)exp\left({\tilde{A}}_{\alpha }^L\right)+\underset{x\in {\tilde{A}}_{\alpha }}{min}sin\left(x\right)\right)-texp\left({\tilde{A}}_{\alpha }^L\right)-\underset{x\in {\tilde{A}}_{\alpha }}{min}sin\left(x\right)$$

and

$$B_n{\tilde{f}}_{\alpha }^U\left(t\right)-{\tilde{f}}_{\alpha }^U\left(t\right)=\sum _{j=0}^n{b}_{nj}\left(t\right)\left(\left({\displaystyle \frac{j}{n}}\right)exp\left({\tilde{A}}_{\alpha }^U\right)+\underset{x\in {\tilde{A}}_{\alpha }}{max}sin\left(x\right)\right)-texp\left({\tilde{A}}_{\alpha }^U\right)-\underset{x\in {\tilde{A}}_{\alpha }}{max}sin\left(x\right).$$

Example 3.

Given any ${\tilde{A}}^{\left(i\right)}\in {\mathcal{F}}_{kc}^C\left(\mathbb{R}\right)$ for $i=1,\dots ,p$, we consider the following fuzzy-valued function:

$$\tilde{f}\left(t\right)=t^p{\tilde{A}}^{\left(p\right)}\oplus t^{p-1}{\tilde{A}}^{(p-1)}\oplus t{\tilde{A}}^{\left(1\right)}, \mathrm{f}\mathrm{o}\mathrm{r} t\in [0,1],$$

(34)

where $t^j{\tilde{A}}^{\left(j\right)}$ means scalar multiplication. It is not difficult show that $\tilde{f}$ is continuous on $[0,1]$ with respect to the Pompeiu–Hausdorff metric $d_{\mathcal{F}}$.

Given any ${\tilde{A}}^{\left(i\right)}\in {\mathcal{F}}_{kc}^C\left(\mathbb{R}\right)$ for $i=1,\dots ,p$, the α-level sets of $\tilde{f}\left(t\right)$ defined in (34) are also the bounded closed intervals with end-points given by

$${\left(\tilde{f}\left(t\right)\right)}_{\alpha }^L\equiv {\tilde{f}}_{\alpha }^L\left(t\right)={\tilde{A}}_{\alpha p}^L{t}^p+\dots +{\tilde{A}}_{\alpha j}^L{t}^j+\dots +{\tilde{A}}_{\alpha 1}^{L}t$$

and

$${\left(\tilde{f}\left(t\right)\right)}_{\alpha }^U\equiv {\tilde{f}}_{\alpha }^U\left(t\right)={\tilde{A}}_{\alpha p}^U{t}^p+\dots +{\tilde{A}}_{\alpha j}^U{t}^j+\dots +{\tilde{A}}_{\alpha 1}^{U}t,$$

where

$${\left({\tilde{A}}^{\left(j\right)}\right)}_{\alpha }=\left[{\left({\tilde{A}}^{\left(j\right)}\right)}_{\alpha }^L,{\left({\tilde{A}}^{\left(j\right)}\right)}_{\alpha }^U\right]\equiv \left[{\tilde{A}}_{\alpha j}^L,{\tilde{A}}_{\alpha j}^U\right].$$

Similarly, for $\left(B_n\tilde{f}\right)\left(t\right)$, we also have

$$\begin{array}{cc}\hfill {\left(B_n\tilde{f}\left(t\right)\right)}_{\alpha }^L& \equiv B_n{\tilde{f}}_{\alpha }^L\left(t\right)=\sum _{j=0}^n{b}_{nj}\left(t\right){\tilde{f}}_{\alpha }^L\left({\displaystyle \frac{j}{n}}\right)\hfill \\ \hfill & =\sum _{j=0}^n{b}_{nj}\left(t\right)\left({\tilde{A}}_{\alpha p}^L{\left({\displaystyle \frac{j}{n}}\right)}^p+\dots +{\tilde{A}}_{\alpha j}^L{\left({\displaystyle \frac{j}{n}}\right)}^j+\dots +{\tilde{A}}_{\alpha 1}^L\left({\displaystyle \frac{j}{n}}\right)\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\left(B_n\tilde{f}\left(t\right)\right)}_{\alpha }^U& \equiv B_n{\tilde{f}}_{\alpha }^U\left(t\right)=\sum _{j=0}^n{b}_{nj}\left(t\right){\tilde{f}}_{\alpha }^U\left({\displaystyle \frac{j}{n}}\right)\hfill \\ \hfill & =\sum _{j=0}^n{b}_{nj}\left(t\right)\left({\tilde{A}}_{\alpha p}^U{\left({\displaystyle \frac{j}{n}}\right)}^p+\dots +{\tilde{A}}_{\alpha j}^U{\left({\displaystyle \frac{j}{n}}\right)}^j+\dots +{\tilde{A}}_{\alpha 1}^U\left({\displaystyle \frac{j}{n}}\right)\right)\hfill \end{array}$$

The Bernstein approximation involves the computations given in (32) and (31). More precisely, we have

$$d_{\mathcal{F}}\left(\left(B_n\tilde{f}\right)\left(t\right),\tilde{f}\left(t\right)\right)=\underset{\alpha \in [0,1]}{sup}max\left\{\left|B_n{\tilde{f}}_{\alpha }^L\left(t\right)-{\tilde{f}}_{\alpha }^L\left(t\right)\right|,\left|B_n{\tilde{f}}_{\alpha }^U\left(t\right)-{\tilde{f}}_{\alpha }^U\left(t\right)\right|\right\},$$

where $B_n{\tilde{f}}_{\alpha }^L\left(t\right)-{\tilde{f}}_{\alpha }^L\left(t\right)$ and $B_n{\tilde{f}}_{\alpha }^U\left(t\right)-{\tilde{f}}_{\alpha }^U\left(t\right)$ can be computed using the above formulas.

6. Conclusions

The Bernstein approximations for fuzzy functions have been successfully obtained with the help of considering the support functions of fuzzy sets in which the weak* topology for the normed dual space is involved. Let $(U,\Vert ·{\Vert }_U)$ be a normed space with the dual space $(U^{*},\Vert ·{\Vert }_{U^{*}})$. This paper studies the Bernstein approximations for the following fuzzy functions:

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k\left(U\right),d\right)$$

and

$$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k^C\left(U\right),d\right)=\left({\overline{\mathcal{F}}}_k\left(U\right),d\right).$$

The concepts of pointwise Bernstein approximation and uniform Bernstein approximation are introduced to tell the difference between pointwise convergence and uniform convergence, which are based on the continuity of fuzzy functions at a single point or on a compact set. The Bernstein approximations are summarized below:

• In Theorem 5, by considering the fuzzy function

  $$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k^C\left(U\right),d\right),$$

  we need to assume that the normed space $(U,\Vert ·{\Vert }_U)$ is separable. The pointwise Bernstein approximation of fuzzy function $\tilde{f}$ at $t_0$ can be obtained when the fuzzy function $\tilde{f}$ is continuous at $t_0$, and the uniform Bernstein approximation of fuzzy function $\tilde{f}$ can be obtained when the fuzzy function $\tilde{f}$ is continuous on the compact set $[0,1]$.
• In Theorem 7, by considering the fuzzy function

  $$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k\left(U\right),d\right),$$

  we still need to assume that the normed space $(U,\Vert ·{\Vert }_U)$ is separable. Then, we also have the same pointwise and uniform Bernstein approximations for fuzzy functions. However, in this case, the range of fuzzy functions is ${\mathcal{F}}_k\left(U\right)$, a larger space than ${\mathcal{F}}_k^C\left(U\right)$, adopted in Theorem 5.
• In Theorem 8, by considering the fuzzy function

  $$\tilde{f}:[0,1]\to \left({\mathcal{F}}_k^C\left(U\right),d\right),$$

  the normed space $(U,\Vert ·{\Vert }_U)$ is not necessarily assumed to be separable. However, we need to assume that the fuzzy function $\tilde{f}$ is bounded. In this case, we still can obtain the same pointwise and uniform Bernstein approximations for fuzzy functions. Although Theorems 7 and 8 consider the same fuzzy function, the separability of normed space and the boundedness of fuzzy functions are the important issues for presenting the Bernstein approximations.

When fuzzy uncertainty is detected for engineering and economic problems, their models should involve fuzzy functions. Calculating the function values of fuzzy functions sometimes is time-expensive. Therefore, the Bernstein approximation of fuzzy functions studied in this paper may provide an alternative methodology to calculate the approximated function values for the purpose of reducing time consumption. The reason is that the Bernstein polynomials are in the forms of polynomials, which are easy to compute. In future research, we are going to design some efficient algorithms to compute the fuzzy type of Bernstein polynomials with the help of the $\alpha$-level sets of fuzzy sets and the well-known computational methods for the crisp type of Bernstein polynomials.