The Fuzzy Width Theory in the Finite-Dimensional Space and Sobolev Space

Abstract

This paper aims to fuzzify the width problem of classical approximation theory. New concepts of fuzzy Kolmogorov n-width and fuzzy linear n-width are introduced on the basis of $\alpha$-fuzzy distance which is induced by the fuzzy norm. Furthermore, the relationship between the classical widths in linear normed space and the fuzzy widths in fuzzy linear normed space is discussed. Finally, the exact asymptotic orders of the fuzzy Kolmogorov n-width and fuzzy linear n-width corresponding to a given fuzzy norm in finite-dimensional space and Sobolev space are estimated.

1. Introduction

The width problem is an important research topic in approximation theory, usually involving seeking the best approximation set and method in a certain sense, i.e., for a given set A and the set family $\mathbb{B}$ in X, the quantity of $E_{\mathbb{B}}\left(A\right)=in{f}_{F\in \mathbb{B}}E{(A,F)}_X$ is required to estimate, where $E{(A,F)}_X$ is the approximation degree of A from F in worst case setting. Kolmogorov [1] first introduced the width theory in linear normed space (the classical width theory), which is closely related to the computational complexity and learning theory. Then, based on the norm in linear space, numerous scholars have conducted a series of works on the classical width theory, including the width theory of point sets in abstract space, the estimation of the width in function classes with basic significance in analysis at a certain scale, etc. [2]. However, with the development of computer technology, the generation of massive data and the emergence of artificial intelligence, more and more practical problems need to be solved with fuzzy ideas.

Zadeh [3] proposed the concept of fuzzy set, and then scholars combined the fuzzy sets with classical mathematics to form different branches of fuzzy mathematics. Katsaras [4] combined the concept of the fuzzy set with a norm and proposed the notation of fuzzy topological vector space by introducing a fuzzy norm in linear space, which laid the foundation of fuzzy functional analysis. Felbin [5] introduced the concept of a fuzzy normed linear space and proved that fuzzy norms were the same upto fuzzy equivalence in a finite dimensional fuzzy normed linear space. Xiao and Zhu [6], Bag and Samanta [7,8] introduced the definition of the fuzzy norm of a linear operator from one fuzzy normed linear space into another and studied various properties of this operator. Golet [9] defined generalized fuzzy norms on a set of objects with linear spatial structure, and analyzed the relationship between these fuzzy norms and fuzzy metrics, as well as the relationship between these fuzzy norms and the topological structure on the basic linear space. Mohiuddine [10] combined the idea of difference operators to introduced weighted statistical convergence and strong weighted summability of order $\beta$ for sequences of fuzzy numbers. In the process of combining the concept of fuzziness with various branches of mathematics, scholars always consider the membership function of fuzzy numbers. Unfortunately, the membership functions of fuzzy numbers are often complex, which makes numerous scholars try to find the corresponding substitutes of fuzzy numbers, and then the problem of fuzzy number approximation arises.

Previous literatures combined the concept of approximation with that of fuzziness to form fuzzy approximation theory becoming another powerful tool in the investigation of real world. Veeramani [11] first introduced an idea of t-best approximations in fuzzy metric spaces. Vaezpour and Karimi [12] proposed the notations of t-approximinal sets and F-approximations to investigate the t-best approximations in the fuzzy normed space. Lee [13] considered approximation properties in the fuzzy normed space. Gal [14] generalized some main results of the approximation theory in linear normed space, such as Weierstrass and Stone–Weierstrass-type results, quantitative estimates in approximation by polynomials, interpolation results, best approximation results, etc., to fuzzy normed linear space. approximation properties in Felbin fuzzy normed spaces are considered. Kim [15] proposed new concepts of approximation properties in Felbin fuzzy normed spaces and made a comparative study among approximation properties in Bag and Samanta fuzzy normed spaces and Felbin fuzzy normed spaces. In addition, they developed the representation of finite rank bounded operators.

Unfortunately, few scholars combine the width theory with that of fuzziness to form fuzzy width theory. Therefore, we attempt to investigate the concept of width from the perspective of fuzzy theory and hope that this survey will stimulate future research in the field of fuzzy width theory. On the basis of $\alpha$-fuzzy distance, this paper studies the width problem of functional approximation theory in the fuzzy normed linear space by defining the fuzzy Kolmogorov n-width and fuzzy linear n-width on the basis of $\alpha$-fuzzy distance and the relationship between the classical widths in linear normed space and the fuzzy widths in fuzzy linear normed space.

This paper is organized as follows. In Section 2, a brief review of definitions and some results that will be used in this paper is presented. Moreover, the notation of $\alpha$-fuzzy distance introduced by a fuzzy norm is proposed, which is crucial for defining the fuzzy Kolmogorov n-width and fuzzy linear n-width in this paper. In Section 3, the relationship between the classical n-widths in linear normed space and the fuzzy n-widths in fuzzy linear normed space equipped with a given fuzzy norm is discussed, and the exact asymptotic orders of the fuzzy Kolmogorov n-width and the fuzzy linear n-width corresponding to the given fuzzy norm in finite-dimensional space and Sobolev space are estimated, which are the main contents and innovations of this paper. In Section 4, the conclusions are presented.

2. Preliminaries

2.1. Fuzzy Normed Linear Space

Previous works in the literature have conducted in-depth research on fuzzy normed spaces, and different scholars have proposed different fuzzy normed spaces based on different fuzzy norms [5,6,7,8]. Various properties in corresponding fuzzy normed spaces are studied, such as convergence [16], continuity, boundedness and compactness of fuzzy linear operators [17,18,19,20]. This paper aims to fuzzify the classical width theory based on Bag and Samanta fuzzy normed spaces, and study the properties of fuzzy linear width and estimate exact asymptotic orders of the fuzzy widths based on the various properties of fuzzy linear operators.

Definition 1 

([7]). Let X be a linear space, θ be the zero element of X. A fuzzy subset N of $X\times \mathbb{R}$ is called a fuzzy norm on X, if for any $x,y\in X,c\in \mathbb{R}$, following conditions are satisfied:

(N1) $\forall t\in \mathbb{R}$ with $t\le 0$, then $N(x,t)=0$;

(N2) $\forall t\in \mathbb{R}$ with $t>0$, then $N(x,t)=1$ iff $x=\theta$;

(N3) $\forall t\in \mathbb{R}$ with $t>0$, if $c\ne 0$, then $N(cx,t)=N(x,\frac{t}{\left|c\right|})$;

(N4) $\forall s,t\in \mathbb{R},x,y\in X$, then

$$N(x+y,s+t)\ge min\left\{N\right(x,s),N(y,t\left)\right\};$$

(N5) $N(x,·)$ is a non-decreasing function on $\mathbb{R}$, and ${lim}_{t\to +\infty }N(x,t)=1$.

And the pair of $(X,N)$ can be regards as a fuzzy normed linear space.

Theorem 1 

([7]). Let $(X,N)$ be a fuzzy normed linear space with a fuzzy norm N satisfying the condition of

(N6) $\forall t>0,N(x,t)>0\Rightarrow x=\theta .$

Define ${\parallel x\parallel }_{\alpha }=\wedge \{t>0\mid N(x,t)\ge \alpha \},\alpha \in (0,1)$. Then $\left\{{\parallel ·\parallel }_{\alpha }:\alpha \in (0,1)\right\}$ are called α-norms on X corresponding to the fuzzy norm N on X, and ${\parallel x\parallel }_{\alpha }\le {\parallel x\parallel }_{\beta },0<\alpha \le \beta <1$.

A novel notation of $\alpha$-fuzzy distance is proposed in the following definition, which is crucial to extend the width theory in normed linear space to the fuzzy width theory in fuzzy linear normed space.

Definition 2. 

Let $(X,N)$ be a fuzzy normed linear space. For $\alpha \in (0,1)$, $x,y\in X$, define the α-fuzzy distance from x to y as

$$d^{\alpha }(x,y)=\wedge \{t>0\mid N(x-y,t)\ge \alpha \}.$$

Remark 1. 

(1) It is obvious that if the fuzzy norm N satisfies the condition of (N6), then the α-fuzzy distance $d^{\alpha }(x,\theta )$ is a norm ${\parallel x\parallel }_{\alpha }$ in the normed linear space X, i.e., $d^{\alpha }(x,\theta )={\parallel x\parallel }_{\alpha }$. Moreover, $d^{\alpha }(x,\theta )\le d^{\beta }(x,\theta )$, for any $0<\alpha \le \beta <1$.

(2) According to Definition 2, it is easy to obtain the following conclusions:

(i) If $\forall x,y\in X$, then $d^{\alpha }(x,y)\ge 0$. In addition, $x=y\Rightarrow d^{\alpha }(x,y)=0$.

(ii) If $\forall x,y\in X$, $\forall r\in \mathbb{R}$, then

$$d^{\alpha }(rx,ry)=\left|r\right|d^{\alpha }(x,y).$$

(1)

(iii) If $\forall x,y,z\in X$, then $d^{\alpha }(x,y)\le d^{\alpha }(x,z)+d^{\alpha }(z,y).$

(3) The α-fuzzy distance of $d^{\alpha }(x,y)$ represents the shortest distance from x to y when the truth value is no less than α.

(4) If $d^{\alpha }(x,y)=0$, we could not draw the conclusion of $x=y$. Seeing the following example:

Example 1. 

Let $(X,\parallel ·\parallel )$ be a normed linear space. For every $x\in X$, $t\in \mathbb{R}$, define

$$N(x,t)=\left\{\begin{array}{cc}1,& t>\parallel x\parallel ;\\ \frac{\mathrm{t}}{\mathrm{t}+\parallel x\parallel }+\frac{1}{3},& 0<t\le \parallel x\parallel ;\\ 0,& t\le 0.\end{array}\right.$$

Then N is a fuzzy norm on X and $d^{\alpha }(x,y)=\wedge \{t>0\mid N(x-y,t)\ge \alpha \}$ is α-fuzzy distance from x to y in the fuzzy normed linear space $(X,N)$. If $x=\theta$, then $d^{\alpha }(x,\theta )=0$. Unfortunately, the α-fuzzy distance of $d^{\alpha }(x,\theta )=0$ does not imply $x=\theta$.

Proof of Example 1. 

We first present that N is a fuzzy norm.

(N1) It is obvious that $t\le 0$ implies $N(x,t)=0$ for any $x\in X$.

(N2) For every $t>0$, if $N(x,t)=1$, we can obtain that $\parallel x\parallel =0$ by the definition of fuzzy norm of N, which implies that $x=\theta$. Alternatively, if $x=\theta$, then for every $t>0$, $N(x,t)=1$ holds according to the definition of fuzzy norm N.

(N3) Since $c\ne 0$, $\parallel cx\parallel \le t\iff \parallel x\parallel \le \frac{t}{\left|c\right|}$, and $\parallel cx\parallel \ge t\iff \parallel x\parallel \ge \frac{t}{\left|c\right|}$, the condition of $N(cx,t)=N(x,\frac{t}{\left|c\right|})$ holds.

(N4) If $s+t<0:s<0,t<0$, then $N(x+y,s+t)\ge min\left\{N\right(x,s),N(y,t\left)\right\}$ holds.

If $s=t=0$, then $N(x+y,s+t)\ge min\left\{N\right(x,s),N(y,t\left)\right\}$ holds.

If $s+t>0$:

(i) when $s>0,t<0$ or $s<0,t>0$, we can easily find that $N(x+y,s+t)\ge min\left\{N\right(x,s),N(y,t\left)\right\}$ holds.

(ii) when $s>0,t>0$, let $p=N(x,s),q=N(y,t)$.

If $p=0$ and $q=0$, then the condition of (N4) clearly holds.

If $p<q$, then $N(x+y,s+t)=\frac{s+t}{s+t+\parallel x+y\parallel }+\frac{1}{3}\ge \frac{s}{s+\parallel x\parallel }+\frac{1}{3}=p=min\{N(x,s),N(y,t)\},$ which shows that the condition of (N4) holds.

If $p\ge q$, we can also find that the condition of (N4) holds.

(N5) It is obvious that $N(x,·)$ is a non-decreasing function of $\mathbb{R}$ and ${lim}_{t\to +\infty }N(x,s)=1$.

Therefore, we prove that N is a fuzzy norm.

According to the definition of the fuzzy norm, we can easily find that if $x=\theta$, then $d^{\alpha }(x,\theta )=0$.

Let $\parallel x\parallel =1$. Then $d^{\frac{1}{3}}(x,\theta )=\wedge \left\{t>0\mid N(x,t)\ge \frac{1}{3}\right\}=0$, but $x\ne \theta$. Therefore, we could obtain the fact that the $\alpha$-fuzzy distance of $d^{\alpha }(x,\theta )=0$ does not imply $x=\theta$. □

Definition 3 

([7,16]). Let $(X,N)$ be a fuzzy normed linear space, $\left\{x_n\right\}$ be a sequence in X. Then $\left\{x_n\right\}$ is said to be fuzzy convergent in X if there is a $x\in X$, such that ${lim}_{n\to \infty }N(x_n-x,t)=1,\forall t>0.$ In that case, x is called the fuzzy limit of $\left\{x_n\right\}$ and denoted by $x_n\stackrel{N}{⟶}x.$

Definition 4. 

Let $(X,N)$ be a fuzzy normed linear space, $\left\{x_n\right\}$ be a sequence in X and $\alpha \in (0,1)$. Then $\left\{x_n\right\}$ is said to be convergence in the α-fuzzy distance in X if there is a $x\in X$, such that ${lim}_{n\to \infty }{d}^{\alpha }(x_n,x)=0.$ In that case, x is called the α-fuzzy limit of $\left\{x_n\right\}$ by α-fuzzy distance and denoted by $x_n\stackrel{d^{\alpha }}{⟶}x$.

Example 2. 

Let $X={\mathbb{R}}^m$ with the norm defined as ${\parallel x\parallel }_2=({\sum }_{i=1}^m|x_i^2{\left|\right)}^{1/2}$ for any $x=(x_1,\cdots ,x_m)$ in ${\mathbb{R}}^m$. For every $x\in X$, $t\in R$, define

$$N(x,t)=\left\{\begin{array}{cc}\frac{t}{t+\parallel x\parallel },& t>0;\\ 0,& t\le 0.\end{array}\right.$$

It is obvious that $({\mathbb{R}}^m,N)$ is a fuzzy normed linear space. Suppose that $\left\{x_n\right\}$ is a convergent sequence in $({\mathbb{R}}^m,\parallel x{\parallel }_2)$ and ${lim}_{n\to \infty }{\parallel x_n-x_0\parallel }_2=0$. Then $x_0$ is the α-fuzzy limit of $\left\{x_n\right\}$ by α-fuzzy distance. In fact, for $t>0$ and $\alpha \in (0,1)$, we have $N(x_n-x_0,t)\ge \alpha >0$ which means $t\ge \frac{\alpha }{1-\alpha }{\parallel x_n-x_0\parallel }_2$. Since ${lim}_{n\to \infty }{\parallel x_n-x_0\parallel }_2=0$, it is obtained that ${lim}_{n\to \infty }{d}^{\alpha }(x_n,x_0)=0$. Thus, $x_0$ is the α-fuzzy limit of $\left\{x_n\right\}$ by α-fuzzy distance.

Proposition 1. 

Let $(X,N)$ be a fuzzy normed linear space and $\left\{x_n\right\}$ be a sequence in X. If $x_0\in X$, then $x_n\stackrel{N}{⟶}{x}_0$ iff $x_n\stackrel{d^{\alpha }}{⟶}{x}_0$, $\forall \alpha \in (0,1)$.

Proof. 

If $x_n\stackrel{N}{⟶}{x}_0$, then ${lim}_{n\to \infty }N(x_n-x_0,t)=1$, for every $t>0$. So, $\forall t>0$ and $\forall \alpha >0$, $\exists n_0=n_0(\alpha ,t)$, such that for $\forall n\ge n_0$,

$$N(x_n-x_0,t)\ge \alpha .$$

(2)

By Definition 2, we also have

$$d^{\alpha }(x_n,x_0)=\wedge \{s>0|N(x_n-x_0,s)\ge \alpha \},\forall n\in Z.$$

(3)

According to (2) and (3), for any $n\ge n_0$, we obtain $d^{\alpha }(x_n,x_0)\le t$. Since t is arbitrary, we could prove that ${lim}_{n\to \infty }{d}^{\alpha }(x_n,x_0)=0$, i.e., $x_n\stackrel{d^{\alpha }}{⟶}{x}_0$.

On the other hand, if $\forall t\ge 0,\forall \alpha \in (0,1),{lim}_{n\to \infty }{d}^{\alpha }(x_n,x_0)=0$, then $\exists n_0=n_0(\alpha ,t)$, for $\forall n\ge n_0$, we have $d^{\alpha }(x_n,x_0)<t$. Thus, $\exists s_0>0$ and $s_0<t$, such that $N(x_n-x_0,s_0)\ge \alpha$ for any $n\ge n_0$ by the definition of $\alpha$-fuzzy distance. Since $N(x,·)$ is a non-decreasing function by the condition of (N5), we can obtain that $N(x_n-x_0,t)\ge N(x_n-x_0,s_0)\ge \alpha$ for any $n\ge n_0$. Therefore, it can be proved that ${lim}_{n\to \infty }N(x_n-x_0,t)=1$, i.e., $x_n\stackrel{N}{\to }{x}_0$. □

Definition 5. 

Let $(X,N)$ be a fuzzy normed linear space and A be a subset of X. Then

(1) the derived set $A^{{}^{\prime }}$ of A is the set which contains all the fuzzy limit points of A;

(2) a set A is said to be fuzzy closed if and only if $A^{{}^{\prime }}\subseteq A$;

(3) the fuzzy closure $\overline{A}$ of A is the union of A and $A^{{}^{\prime }}$.

Definition 6 

([8,16,17,18,19,20]). Let $(U_1,N_1)$ and $(U_2,N_2)$ be two fuzzy normed linear spaces over the same number field ($\mathbb{R}$ or $\mathbb{C}$).

(1) A mapping T from $(U_1,N_1)$ to $(U_2,N_2)$ is called strongly fuzzy continuous at $x_0\in U_1$, if for any $\epsilon >0,\exists \delta >0$, such that $\forall x\in U_1$,

$$N_2(T\left(x\right)-T\left(x_0\right),\epsilon )\ge N_1(x-x_0,\delta ).$$

Moreover, T is called strongly fuzzy continuous on $U_1$ if T is strongly fuzzy continuous at each point of $U_1$.

(2) A mapping T from $(U_1,N_1)$ to $(U_2,N_2)$ is called strongly fuzzy bounded operator if $\exists M>0$ such that for $\forall x\in U_1$ and $\forall s\in R$,

$$N_2(T\left(x\right),s)\ge N_1(x,\frac{s}{M}).$$

Proposition 2 

([8]). Let $(U_1,N_1)$ and $(U_2,N_2)$ be two fuzzy normed linear spaces over the same number field ($\mathbb{R}$ or $\mathbb{C}$). A mapping T from $(U_1,N_1)$ to $(U_2,N_2)$ is strongly fuzzy continuous iff T is strongly fuzzy bounded operator.

2.2. The Classical Widths

In the second part of this section, definitions of Kolmogorov n-width and linear n-width in the normed linear space and the exact asymptotic orders of Kolmogorov n-width and linear n-width in finite dimensional space as well as Sobolev space are recalled.

Definition 7 

([2]). Let $(X,\parallel ·\parallel )$ be a normed linear space, and A be a subset in X. Then

(1) the Kolmogorov n-width of A in X is given by

$$d_n\left(A\right):=d_n(A,X):=\underset{X_n}{inf}\underset{x\in A}{sup}\underset{y\in X_n}{inf}\parallel x-y\parallel ,$$

where the leftmost infimum is taken over all subspaces $X_n$ of X with the dimension at most n.

(2) the linear n-width of A in X is given by

$${\delta }_n\left(A\right):={\delta }_n(A,X):=\underset{T_n}{inf}\underset{x\in A}{sup}\parallel x-T_n\left(x\right)\parallel ,$$

where the infimum is taken over all bounded linear operators $T_n$ from X to X with the rank at most n.

More details about Kolmogorov n-widths and linear n-widths can be easily referred to Pinkus [2].

The finite dimensional space ${\ell }_p^m(1\le p\le \infty )$ is a m-dimensional linear normed space of vectors $x=(x_1,x_2,\dots ,x_m)\in {\mathbb{R}}^m$, with the norm ${\parallel ·\parallel }_{{\ell }_p^m}$ being given by

$${\parallel x\parallel }_{{\ell }_p^m}=\left\{\begin{array}{cc}{\left(\sum _{i=1}^m{\left|x_i\right|}^p\right)}^{1/p},& 1\le p<\infty ;\\ \underset{1\le i\le m}{max}\left|x_i\right|,& p=\infty .\end{array}\right.$$

$B_p^m\left(\rho \right):=\{x\in {\ell }_p^m\mid {\parallel x\parallel }_{{\ell }_p^m}\le \rho \}$ is a ball of ${\ell }_p^m$ with radius $\rho$, and $B_p^m$ represents the unit ball in ${\ell }_p^m$.

Then the asymptotic order of Kolmogorov n-width and linear n-width in finite dimensional space are represented in the following theorem.

Theorem 2 

([2]). If $1\le p,q\le \infty$ and $m,n\in N,n\le m$, then

(1) the exact asymptotic order of Kolmogorov n-width in finite dimensional space can be estimated as

$$d_n\left(B_p^m,{\ell }_q^m\right)=\left\{\begin{array}{cc}{(m-n)}^{\frac{1}{q}-\frac{1}{p}},& 1\le q\le p\le \infty ,\\ min{\left\{1,m^{\frac{1}{q}}{n}^{-\frac{1}{2}}\right\}}^{(\frac{1}{p}-\frac{1}{q})(\frac{1}{2}-\frac{1}{q})},& 2\le p<q\le \infty ,\\ max\left\{m^{\frac{1}{q}-\frac{1}{p}},min\left\{1,m^{\frac{1}{q}}{n}^{-\frac{1}{2}}\right\}·\sqrt{1-n/m}\right\},& 1\le p<2\le q\le \infty ,\\ max\left\{m^{\frac{1}{q}-\frac{1}{p}},{\left(\sqrt{1-n/m}\right)}^{(\frac{1}{p}-\frac{1}{q})(\frac{1}{p}-\frac{1}{2})}\right\},& 1\le p<q\le 2.\end{array}\right.$$

(2) the exact asymptotic order of linear n-width in finite dimensional space can be estimated as

$${\delta }_n\left(B_p^m,{\ell }_q^m\right)=\left\{\begin{array}{cc}{(m-n)}^{1/q-1/p},\hfill & 1\le q\le p\le \infty ,\\ \mathsf{\Phi }(m,n,p,q),\hfill & 1\le p<q<\infty ,\\ \mathsf{\Phi }\left(m,n,q^{\prime },p^{\prime }\right),\hfill & max\left\{p,p^{\prime }\right\}<q\le \infty ,\end{array}\right.$$

where

$$\mathsf{\Phi }(m,n,p,q)=\left\{\begin{array}{cc}min{\left\{1,m^{1/q}{n}^{-1/2}\right\}}^{(1/p-1/q)(1/2-1/q)},& 2\le p<q\le \infty ,\\ max\left\{m^{1/q-1/p},min\left\{1,m^{1/q}{n}^{-1/2}\right\}·\sqrt{1-n/m}\right\},& 1\le p<2\le q\le \infty ,\\ max\left\{m^{1/q-1/p},{\left(\sqrt{1-n/m}\right)}^{(1/p-1/q)(1/p-1/2)},\right.& 1\le p<q\le 2,\end{array}\right.$$

$$\mathsf{\Psi }(m,n,p,q)=\left\{\begin{array}{cc}\mathsf{\Phi }(m,n,p,q),\hfill & 1\le p<q<\infty ,\\ \mathsf{\Phi }\left(m,n,q^{\prime },p^{\prime }\right),\hfill & max\left\{p,p^{\prime }\right\}<q\le \infty ,\end{array}\right.$$

and $1/p+1/p^{\prime }=1,1/q+1/q^{\prime }=1,1/p^{\prime }=1-1/p<1/2$.

For $1\le p\le \infty$, denote by $L_p\left(T\right)$, the classical space of $p-th$ powers integrable $2\pi$-periodic functions defined on the $T=[0,2\pi ]$ equipped with the usual norm ${\parallel ·\parallel }_{L_p}$. Then Sobolev space $W_p^r\left(T\right)$ of real-valued on $T=[0,2\pi ]$ can be defined by

$$W_p^r=W_p^r\left(T\right)=\{f:f^{(r-1)}abs.cont.,f^{\left(r\right)}\in L_p\left(T\right)\},$$

with the norm ${\parallel f\parallel }_{W_p^r}:={\parallel f\parallel }_{L_p}+{\parallel f^{\left(r\right)}\parallel }_{L_p}$. Setting $B_p^r\left(\rho \right)=\{f:f\in W_p^r{,\parallel f\parallel }_{W_p^r}\le \rho \}$ be a ball of $W_p^r$ with radius $\rho$. In particular, denote by $B_p^r:=B_p^r\left(1\right)$, the unit ball in the Sobolev space $W_p^r$. Then the asymptotic order of Kolmogorov n-width and linear n-width in Sobolev space could be represented in the following theorem.

Theorem 3 

([2]). If $r\ge 2,1\le p,q\le \infty$ and $1/p+1/p^{\prime }=1$, then

(1) the exact asymptotic order of Kolmogorov n-width in Sobolev space can be estimated as

$$d_n\left(B_p^r,L^q\right)\approx \left\{\begin{array}{cc}{n}^{-r},& 1\le q\le p\le \infty ,\\ n^{-r},& 2\le p\le q\le \infty ,\\ n^{-r+1/p-1/2},& 1\le p\le 2\le q\le \infty ,\\ n^{-r+1/p-1/q}& 1\le p\le q\le 2;\end{array}\right.$$

(2) the exact asymptotic order of linear n-width in Sobolev space can be estimated as

$${\delta }_n\left(B_p^r,L^q\right)\approx \left\{\begin{array}{cc}{n}^{-r},& 1\le q\le p\le \infty ,\\ n^{-r+1/p-1/q},& 1\le p\le q\le 2,\\ n^{-r+1/p-1/q},& 2\le p\le q\le \infty ,\\ n^{-r+1/p-1/2},& 1\le p\le 2\le q\le \infty ,p^{\prime }\ge q,\\ n^{-r+1/2-1/q},& 1\le p\le 2\le q\le \infty ,p^{\prime }\le q.\end{array}\right.$$

3. The Fuzzy n-Widths in Fuzzy Normed Linear Space

In this section, the fuzzy n-widths in fuzzy normed linear space are investigated, which are the main contents and innovations of this paper. There are two achievements about fuzzy n-widths as below: (1) In Section 3.1, the definition of fuzzy Kolmogorov n-width and fuzzy linear n-width in fuzzy normed linear space are proposed. Moreover, some basic properties of the two fuzzy n-widths being used to estimate the asymptotic order of fuzzy n-width are presented. (2) In Section 3.2, the relationship of the fuzzy n-widths and the classic n-widths and further estimate the exact asymptotic order of fuzzy Kolmogorov n-width and fuzzy linear n-width in fuzzy normed linear space equipped with a given fuzzy norm are considered.

3.1. The Fuzzy n-Widths

Definition 8. 

Let $(X,N)$ be a fuzzy normed linear space, A be a subset in X, and $\alpha \in (0,1)$. Then

(1) the fuzzy Kolmogorov n-width of A in X is given by

$$d_n^{\alpha }\left(A\right):=d_n(A,X,N):=\underset{X_n}{inf}\underset{x\in A}{sup}\underset{y\in x_n}{inf}{d}^{\alpha }(x,y),$$

where the leftmost infimum is taken over all subspaces $X_n$ of X with the dimension at most n.

(2) the fuzzy linear n-width of A in X is given by

$${\delta }_n^{\alpha }\left(A\right):={\delta }_n^{\alpha }(A,X,N):=\underset{T_n}{inf}\underset{x\in A}{sup}{d}^{\alpha }(x,T_{n}x),$$

where the infimum is taken over all strongly fuzzy continuous linear operators $T_n$ from X to X with the rank at most n.

Remark 2. 

(1) If the fuzzy norm N of X satisfies the condition of (N6), then the fuzzy normed linear space X becomes the normed linear space with the norm ${\parallel ·\parallel }_{\alpha }$ induced by the fuzzy norm N. Thus, the fuzzy Kolmogorov n-width and the fuzzy linear n-width of A are equal to the classical Kolmogorov n-width and linear n-width of A, respectively, i.e., $d_n^{\alpha }\left(A\right)=d_n\left(A\right)$ and ${\delta }_n^{\alpha }\left(A\right)={\delta }_n\left(A\right)$.

(2) The fuzzy Kolmogorov n-width $d_n^{\alpha }\left(A\right)$ can be used to represent the best approximation of A by n-dimensional subspace of X when the truth value is no less than α.

(3) The fuzzy linear n-width ${\delta }_n^{\alpha }\left(A\right)$ can be used to represent the best approximation of A by strongly fuzzy continuous linear operators from X to X with the rank at most n when the truth value is no less than α.

With the above definitions of fuzzy Kolmogorov n-width and fuzzy linear n-width, some basic properties of fuzzy n-widths are further investigated in the following propositions.

Proposition 3. 

Let $(X,N)$ be a fuzzy normed linear space, and $A_1,A_2$ be nonempty subsets in X. Then some basic properties of fuzzy Kolmogorov n-width can be obtained as follows.

(1) If $A_1\subseteq A_2$, then $d_n^{\alpha }\left(A_1\right)\le d_n^{\alpha }\left(A_2\right)$.

(2) If β is a scalar, then $d_n^{\alpha }\left(\beta A\right)=\left|\beta \right|d_n^{\alpha }\left(A\right)$.

(3) $d_n^{\alpha }\left(\overline{A}\right)=d_n^{\alpha }\left(A\right)$, where $\overline{A}$ is the closure of A.

(4) $d_n^{\alpha }\left(coA\right)=d_n^{\alpha }\left(A\right)$, where $coA$ is the convex hull of A.

(5) $d_n^{\alpha }\left(A\right)$ is monotonically decreasing with respect to n, i.e., $d_{n+1}^{\alpha }\left(A\right)\le d_n^{\alpha }\left(A\right).$

(6) Let $A,B$ be nonempty subsets of X, and $B\subseteq A$.

Denote by $E^{\alpha }(A,B)={sup}_{x\in A}{inf}_{y\in B}{d}^{\alpha }(x,y)$. Then

$$d_n^{\alpha }\left(A\right)-E^{\alpha }(A,B)\le d_n^{\alpha }\left(B\right)\le d_n^{\alpha }\left(A\right).$$

(7) Suppose that $(X,N)$ and $(Y,N)$ are two fuzzy linear normed spaces equipped with the same fuzzy norm. If $X\subseteq Y$, and $\varnothing \ne A\subseteq X$, then $d_n^{\alpha }(A,X)\ge d_n^{\alpha }(A,Y).$

Proof. 

(1) Taking a linear subspace $X_n$ of X with the dimension at most n, if $A_1\subseteq A_2$, then we have

$$\{\underset{y\in X_n}{inf}{d}^{\alpha }(x,y)\mid x\in A_1\}\subseteq \{\underset{y\in x_n}{inf}{d}^{\alpha }(x,y)\mid x\in A_2\}.$$

Thus, ${sup}_{y\in A_1}{inf}_{y\in X_n}{d}^{\alpha }(x,y)\le su{p}_{y\in A_2}{inf}_{y\in X_n}{d}^{\alpha }(x,y)$ holds.

According to the definition of fuzzy Kolmogorov n-width and the arbitrariness of $X_n$, we can obtain that

$$d_n^{\alpha }\left(A\right)=\underset{X_n}{inf}\underset{x\in A_1}{sup}\underset{y\in x_n}{inf}{d}^{\alpha }(x,y)\le \underset{X_n}{inf}\underset{x\in A_2}{sup}\underset{y\in x_n}{inf}{d}^{\alpha }(x,y)=d_n^{\alpha }\left(A_2\right).$$

(2) If $\beta =0$, then the conclusion holds obviously.

If $\beta \ne 0$, then

$$d_n^{\alpha }\left(\beta A\right)=\underset{X_n}{inf}\underset{x\in A}{sup}\underset{y\in X_n}{inf}{d}^{\alpha }(\beta x,y)=\underset{X_n}{inf}\underset{x\in A}{sup}\underset{y\in X_n}{inf}\left|\beta \right|d^{\alpha }\left(x,\frac{y}{\beta }\right)=\left|\beta \right|d_n^{\alpha }\left(A\right),$$

where $X_n$ is a linear subspace of X with the dimension at most n.

(3) Since $\overline{A}=A\cup A^{{}^{\prime }}\supseteq A$, we have $d_n^{\alpha }\le d_n^{\alpha }\left(\overline{A}\right)$ by Proposition 3 (1). So, we just need to verify $d_n^{\alpha }\ge d_n^{\alpha }\left(\overline{A}\right)$.

In fact, we can assume that $\overline{A}\setminus A\ne \varnothing$ (if $\overline{A}\setminus A=\varnothing$, then $A=\overline{A}$, which shows that the conclusion holds obviously). For every $x_0\in \overline{A}\setminus A$, we have $x_0\in A^{{}^{\prime }}$. So, there is a sequence $\left\{x_n\right\}$ of A, such that $x_n\stackrel{N}{\to }{x}_0$. Then we can deduce $x_n\stackrel{d^{\alpha }}{\to }{x}_0$ by Proposition 1, i.e., for $\forall \epsilon >0,\exists n_0\in \mathbb{N}$, such that $d_{\alpha }(x_{n_0},x_0)<\epsilon$, for every $n\ge n_0$. Taking a linear subspace $X_n$ of X with the dimension at most n, we have

$$\underset{y\in X_n}{inf}{d}^{\alpha }\left(x_0,y\right)\le \underset{y\in X_n}{inf}\left\{d^{\alpha }\left(x_{n_0},x_0\right)+d^{\alpha }\left(x_{n_0},y\right)\right\}\le \underset{y\in X_n}{inf}{d}^{\alpha }\left(x_{n_0},y\right)+\epsilon .$$

Hence, ${inf}_{y\in X_n}{d}^{\alpha }\left(x_0,y\right)\le {sup}_{x\in A}{inf}_{y\in X_n}{d}^{\alpha }(x,y)+\epsilon$ holds.

Considering that $\overline{A}=(\overline{A}\setminus A)\cup A$, we can obtain

$$\underset{x\in \overline{A}}{sup}\underset{y\in X_n}{inf}{d}^{\alpha }(x,y)\le \underset{x\in A}{sup}\underset{y\in X_n}{inf}{d}^{\alpha }(x,y)+\epsilon .$$

Therefore, $d_n^{\alpha }\left(\overline{A}\right)\le d_n^{\alpha }\left(A\right)+\epsilon$ for every $\epsilon$, which implies that $d_n^{\alpha }\left(\overline{A}\right)\le d_n^{\alpha }\left(A\right)$ holds.

(4) Since $A\subseteq coA$, we can obtain $d_n^{\alpha }\left(A\right)\le d_n^{\alpha }\left(coA\right)$ by Proposition 3 (1). Therefore, we just need to verify $d_n^{\alpha }\left(A\right)\ge d_n^{\alpha }\left(coA\right)$.

According to the definition of $\alpha$-fuzzy distance, we can find the fact that

$$\begin{array}{cc}\hfill d^{\alpha }(x+y,z)& =d^{\alpha }\left(x-{\lambda }_{1}z,y-{\lambda }_{2}z\right)\hfill \\ & \le d^{\alpha }\left(x-{\lambda }_{1}z,\theta \right)+d^{\alpha }\left(y-{\lambda }_{2}z,\theta \right)\hfill \\ & =d^{\alpha }\left(x,{\lambda }_{1}z\right)+d^{\alpha }\left(y,{\lambda }_{2}z\right),\hfill \end{array}$$

where $0<{\lambda }_1,{\lambda }_2<1$ and ${\lambda }_1+{\lambda }_2=1$.

Since $coA=\left\{{\sum }_{i=1}^m{\lambda }_i{x}_i\mid {\lambda }_i\ge 0,{\sum }_{i=1}^m{\lambda }_i=1,x_i\in A,m\in \mathrm{N}\right\},$ we can obtain the conclusion that

$$\begin{array}{cc}\hfill d_n^{\alpha }\left(coA\right)& =\underset{X_n}{inf}\underset{x=\left({\sum }_{i=1}^m{\lambda }_i{x}_i\right)\in \left(coA\right)}{sup}\underset{y\in X_n}{inf}{d}^{\alpha }(\sum _{i=1}^m({\lambda }_i{x}_i,y))\hfill \\ & \le \underset{X_n}{inf}\underset{x=\left({\sum }_{i=1}^m{\lambda }_i{x}_i\right)\in \left(coA\right)}{sup}\underset{y\in X_n}{inf}\sum _{i=1}^m{d}^{\alpha }({\lambda }_i{x}_i,{\lambda }_i{y}_i)\hfill \\ & \le \underset{X_n}{inf}\sum _{i=1}^m{\lambda }_i\underset{x_i\in A}{sup}\underset{y\in X_n}{inf}{d}^{\alpha }(x_i,y)=\underset{X_n}{inf}\underset{x\in A}{sup}\underset{y\in X_n}{inf}{d}^{\alpha }(x,y)d_n^{\alpha }\left(A\right).\hfill \end{array}$$

Therefore, it is proved that $d_n^{\alpha }\left(coA\right)=d_n^{\alpha }\left(A\right)$.

(5) This conclusion can be verified easily by the definition of fuzzy Kolmogorov n-width and the fact that every linear subspace $X_n$ of X with the dimension at most n must be a linear subspace of $X_{n+1}$ with the dimension at most $n+1$ in X.

(6) Since $B\subseteq A$, we have $d_n^{\alpha }\left(B\right)\le d_n^{\alpha }\left(A\right)$ by Proposition 3 (1). Thus, we just need to verify $d_n^{\alpha }\left(A\right)-E^{\alpha }(A,B)\le d_n^{\alpha }\left(B\right)$. Suppose that $X_n$ is a linear subspace of X with the the dimension at most n. For every $x\in B$ and $b\in B$, we have

$$\begin{array}{cc}\hfill \underset{y\in X_n}{inf}{d}^{\alpha }(x,y)& \le \underset{y\in X_n}{inf}\left\{d^{\alpha }(x,b)+d^{\alpha }(y,b)\right\}\hfill \\ & =d^{\alpha }(x,b)+\underset{y\in X_n}{inf}{d}^{\alpha }(y,b)\hfill \\ & \le d^{\alpha }(x,b)+\underset{b\in B}{sup}\underset{y\in X_n}{inf}{d}^{\alpha }(y,b)\hfill \end{array}$$

Thus ${inf}_{y\in X_n}{d}^{\alpha }(x,y)\le {inf}_{b\in B}{d}^{\alpha }(x,b)+{sup}_{b\in B}{inf}_{y\in X_n}{d}^{\alpha }(y,b)$ holds by the arbitrary of b. Therefore,

$$\begin{array}{cc}\hfill \underset{x\in A}{sup}\underset{y\in X_n}{inf}{d}^{\alpha }(x,y)& \le \underset{x\in A}{sup}\underset{b\in B}{inf}{d}^{\alpha }(x,b)+\underset{b\in B}{sup}\underset{y\in X_n}{inf}{d}^{\alpha }(y,b)\hfill \\ & =E^{\alpha }(A,B)+\underset{b\in B}{sup}\underset{y\in X_n}{inf}{d}^{\alpha }(y,b).\hfill \end{array}$$

This illustrates that $d_n^{\alpha }\left(A\right)\le E^{\alpha }(A,B)+d_n^{\alpha }\left(B\right)$, i.e., $d_n^{\alpha }\left(A\right)-E^{\alpha }(A,B)\le d_n^{\alpha }\left(B\right)$.

(7) This conclusion can be verified easily by the definition of fuzzy Kolmogorov n-width and the fact that every linear subspace $X_n$ of X must be a linear subspace of Y. □

Proposition 4. 

Let $(X,N)$ be a fuzzy normed linear space, and $A_1,A_2,A_3$ be nonempty subsets in X. Then some basic properties of fuzzy linear n-width can be obtained as follows.

(1) If $A_1\subseteq A_2$, then ${\delta }_n^{\alpha }\left(A_1\right)\le {\delta }_n^{\alpha }\left(A_2\right)$.

(2) If β is a scalar, then ${\delta }_n^{\alpha }\left(\beta A\right)=\left|\beta \right|{\delta }_n^{\alpha }\left(A\right)$.

(3) ${\delta }_n^{\alpha }\left(\overline{A}\right)={\delta }_n^{\alpha }\left(A\right)$, where $\overline{A}$ is the closure of A.

(4) ${\delta }_n^{\alpha }\left(b_n\left(A\right)\right)={\delta }_n^{\alpha }\left(A\right)$, where $b_n\left(A\right)$ is the balanced set of A, i.e., $b_n\left(A\right)=\{\alpha x:x\in A,|\alpha |\le 1\}$.

(5) ${\delta }_n^{\alpha }\left(coA\right)={\delta }_n^{\alpha }\left(A\right)$, where coA is the convex hull of A.

(6) ${\delta }_n^{\alpha }\left(A\right)$ is monotonically decreasing with respect to n, i.e., ${\delta }_{n+1}^{\alpha }\left(A\right)\le {\delta }_n^{\alpha }\left(A\right)$.

Proof. 

(1) Let $T_n$ be a strongly fuzzy continuous linear operator from X to X with the rank at most n. Then ${sup}_{x\in A_1}{d}^{\alpha }(x,T_{n}x)\le {sup}_{x\in A_2}{d}^{\alpha }(x,T_{n}x)$ for $A_1\subseteq A_2$.

Thus, we can obtain

${\delta }_n^{\alpha }\left(B\right)={inf}_{T_n}{sup}_{x\in B}{d}^{\alpha }(x,T_{n}x)\le {inf}_{T_n}{sup}_{x\in A}{d}^{\alpha }(x,T_{n}x)={\delta }_n^{\alpha }\left(A\right)$, i.e., ${\delta }_n^{\alpha }\left(B\right)\le {\delta }_n^{\alpha }\left(A\right)$.

(2) If $\beta =0$, then the conclusion holds obviously. If $\beta \ne 0$, then

$$\begin{array}{cc}\hfill {\delta }_n^{\alpha }\left(\beta A\right)& =\underset{T_n}{inf}\underset{x\in A}{sup}{d}^{\alpha }\left(\beta x,T_{n}x\right)=\underset{T_n}{inf}\underset{x\in A}{sup}\left|\beta \right|d^{\alpha }\left(x,\frac{T_{n}x}{\beta }\right)\hfill \\ & =\left|\beta \right|\underset{T_n}{inf}\underset{x\in A}{sup}{d}^{\alpha }\left(x,T_{n}x\right)=\left|\beta \right|{\delta }_n^{\alpha }\left(A\right),\hfill \end{array}$$

where $T_n$ is a strongly fuzzy continuous linear operator from X to X with the rank at most n.

(3) Since $\overline{A}=A\cup A^{{}^{\prime }}\supseteq A$, we can obtain ${\delta }_n^{\alpha }\le {\delta }_n^{\alpha }\left(\overline{A}\right)$ by the first result of Proposition 4 (1). So, we just need to verify ${\delta }_n^{\alpha }\left(A\right)\ge {\delta }_n^{\alpha }\left(\overline{A}\right)$. In fact, we can assume that $\overline{A}\setminus A\ne \varnothing$ (if $\overline{A}\setminus A=\varnothing$, then $A=\overline{A}$, which shows that the conclusion holds obviously).

For every $x_0\in \overline{A}\setminus A$, then $x_0\in A^{{}^{\prime }}$. So there is a sequence $\left\{x_n\right\}$ of A, such that $x_n\stackrel{N}{\to }{x}_0$, which deduces that $x_n\stackrel{d^{\alpha }}{\to }{x}_0$ by the Proposition 1. That is, $\forall \epsilon >0$, $\exists n_0\in \mathbb{N}$, such that $d^{\alpha }(x_{n_0},x_0)<\epsilon$, for every $n\ge n_0$. Assume T is a strongly fuzzy continuous linear operator from X to X with the rank at most n, we have $T{x}_n\stackrel{N}{\to }T{x}_0$ which implies $T{x}_n\stackrel{d^{\alpha }}{\to }T{x}_0$. Hence, there exists a $n_0\in \mathbb{N}$ and $n_0\ge n_1$, such that $d^{\alpha }(T{x}_{n_0},T{x}_0)<\epsilon$.

Thus, we have

$$\begin{array}{cc}\hfill d^{\alpha }\left(x_0,T{x}_0\right)& \le d^{\alpha }\left(x_0,x_{n_0}\right)+d^{\alpha }\left(x_{n_0},T{x}_{n_0}\right)+d^{\alpha }\left(T{x}_{n_0},T{x}_0\right)\hfill \\ & \le d^{\alpha }\left(x_{n_0},T{x}_{n_0}\right)+\epsilon +\epsilon =d^{\alpha }\left(x_{n_0},T{x}_{n_0}\right)+2\epsilon .\hfill \end{array}$$

By the arbitrary of $\epsilon$, we can obtain $d^{\alpha }(x_0,T{x}_0)\le d^{\alpha }(x_{n_0},T{x}_{n_0})$. Then $d^{\alpha }(x_0,T{x}_0)\le {sup}_{x\in A}{d}^{\alpha }(x,Tx)$, which implies that ${sup}_{x\in A^{{}^{\prime }}}{d}^{\alpha }(x,Tx)\le {sup}_{x\in A}{d}^{\alpha }(x,Tx)$. Therefore, we have

$$\underset{x\in \overline{A}}{sup}{d}^{\alpha }(x,Tx)=\underset{x\in A\cup A^{{}^{\prime }}}{sup}{d}^{\alpha }(x,Tx)\le \underset{x\in A}{sup}{d}^{\alpha }(x,Tx).$$

Thus ${\delta }_n^{\alpha }\left(\overline{A}\right)\le {\delta }_n^{\alpha }\left(A\right)$ holds by the arbitrary taking of the strongly fuzzy continue operator. So, ${\delta }_n^{\alpha }\left(\overline{A}\right)={\delta }_n^{\alpha }\left(A\right)$ holds.

(4) Since $A\subset b_n\left(A\right)$, we have ${\delta }_n^{\alpha }\left(b_n\left(A\right)\right)\ge {\delta }_n^{\alpha }\left(A\right)$. Thus, we just need to verify ${\delta }_n^{\alpha }\left(b_n\left(A\right)\right)\le {\delta }_n^{\alpha }\left(A\right)$.

If $\forall x\in b_n\left(A\right)$, then $\exists x_1\in A$, such that $x=\alpha x_1$, and $d^{\alpha }(x,T_{n}x)=d^{\alpha }(\alpha x_1,T_n\alpha x_1)=\alpha d^{\alpha }(x_1,T_n{x}_1)$. So we can obtain

$$\underset{x\in b_n\left(A\right)}{sup}{d}^{\alpha }(x,T_{n}x)\le \underset{x\in A}{sup}{d}^{\alpha }(x,T_{n}x)\Rightarrow {\delta }_n^{\alpha }\left(b_n\left(A\right)\right)\le {\delta }_n^{\alpha }\left(A\right).$$

Therefore, ${\delta }_n^{\alpha }\left(b_n\left(A\right)\right)\ge {\delta }_n^{\alpha }\left(A\right)$ holds.

(5) Since $A\subseteq coA$, we have ${\delta }_n^{\alpha }\left(A\right)\le {\delta }_n^{\alpha }\left(coA\right)$. Thus, we just need to show ${\delta }_n^{\alpha }\left(A\right)\ge {\delta }_n^{\alpha }\left(coA\right)$.

In fact, noticing that $coA=\{{\sum }_{i=1}^m{\lambda }_i{x}_i\mid {\lambda }_i\ge 0,{\sum }_{i=1}^m{\lambda }_i=1,x_i\in A,i\in N\}$ and $d^{\alpha }(x+y,z)\le d^{\alpha }(x-{\lambda }_{1}z,\theta )+d^{\alpha }(y-{\lambda }_{2}z,\theta )=d^{\alpha }(x,{\lambda }_{1}z)+d^{\alpha }(y,{\lambda }_{2}z),$ where $0<{\lambda }_1,{\lambda }_2<1$ and ${\lambda }_1+{\lambda }_2=1$, we can obtain that

$$\begin{array}{cc}\hfill {\delta }_n^{\alpha }\left(coA\right)& =\underset{T_n}{inf}\underset{x=\left({\sum }_{i=1}^m{\lambda }_i{x}_i\right)\in \left(coA\right)}{sup}{d}^{\alpha }(\sum _{i=1}^m{\lambda }_i{x}_i,T_n(\sum _{i=1}^m{\lambda }_i{x}_i))\hfill \\ & \le \underset{T_n}{inf}\underset{x=\left({\sum }_{i=1}^m{\lambda }_i{x}_i\right)\in \left(coA\right)}{sup}\sum _{i=1}^m{d}^{\alpha }\left({\lambda }_i{x}_i,T_n{\lambda }_i{x}_i\right)\hfill \\ & \le \underset{T_n}{inf}\sum _{i=1}^m{\lambda }_i\underset{x_i\in A}{sup}{d}^{\alpha }\left(x_i,T{x}_i\right)\le \underset{T_n}{inf}\underset{x\in A}{sup}{d}^{\alpha }\left(x,T_{n}x\right)={\delta }_n^{\alpha }\left(A\right).\hfill \end{array}$$

Thus, it can be proved that ${\delta }_n^{\alpha }\left(coA\right)={\delta }_n^{\alpha }\left(A\right)$.

(6) The conclusion can be verified easily by the definition of fuzzy linear n-width and the fact that every strongly fuzzy continuous operator $T_n$ from X to X with the rank at most n must be contained in a strongly fuzzy continuous operator $T_{n+1}$ from X to X with the rank at most $n+1$. □

3.2. The Exact Asymptotic Order of Fuzzy n-Widths

3.2.1. The Relationship of the Fuzzy n-Widths and Classical n-Widths

In this section, the relationship between the fuzzy n-widths and the classical n-widths is explored, which is crucial for estimating the exact asymptotic orders of the fuzzy n-widths in the finite dimensional space and the Sobolev space. The fuzzy n-widths here are induced by a given fuzzy norm $N(x,t)$ (see, [7]) which is widely used in fuzzy mathematics.

Definition 9 

([7]). Let $(X,\parallel ·\parallel )$ be a linear normed space. For every $x\in X$, $t\in R$, define

$$N(x,t)=\left\{\begin{array}{cc}\frac{t}{t+\parallel x\parallel },& t>0;\\ 0,& t\le 0.\end{array}\right.$$

(4)

Then N is a fuzzy norm on X.

Through the definition of fuzzy norm N in (4), we draw the conclusion of the relationship between the fuzzy n-widths and the classical n-widths.

Theorem 4. 

Let $(X,\parallel ·\parallel )$ be a linear normed space. The fuzzy norm N is defined by (4). If A is a nonempty subset of X, $\alpha \in (0,1)$, then the relationship with the fuzzy Kolmogorov n-width and the classical Kolmogorov n-width is as follows

$$d_n^{\alpha }(A,X,N)=\frac{\alpha }{1-\alpha }{d}_n(A,X).$$

Proof. 

According to the definition of the $\alpha$-fuzzy distance $d^{\alpha }(x,y)$, we have

$$\begin{array}{cc}\hfill d^{\alpha }(x,y)& =\wedge \{t>0\mid N(x-y,t)\ge \alpha \}\hfill \\ & =\wedge \left\{t>0\mid \frac{t}{t+\parallel x-y\parallel }\ge \alpha \right\}=\frac{\alpha }{1-\alpha }\parallel x-y\parallel ,\hfill \end{array}$$

for every $\alpha \in (0,1)$ and every $x,y\in X$.

Thus, we can obtain

$$\begin{array}{cc}\hfill d_n^{\alpha }(A,X,N)& =\underset{X_n}{inf}\underset{x\in A}{sup}\underset{y\in X_n}{inf}{d}^{\alpha }(x,y)=\underset{X_n}{inf}\underset{x\in A}{sup}\underset{y\in X_n}{inf}\frac{\alpha }{1-\alpha }\parallel x-y\parallel \hfill \\ & =\frac{\alpha }{1-\alpha }\underset{X_n}{inf}\underset{x\in A}{sup}\underset{y\in X_n}{inf}\parallel x-y\parallel =\frac{\alpha }{1-\alpha }{d}_n(A,X),\hfill \end{array}$$

where $X_n$ is a linear subspace of X with the dimension no more than n. □

In order to consider the relationship between the fuzzy linear n-width and the classical linear n-width, we first introduce the following proposition.

Proposition 5. 

Let $(X,\parallel ·\parallel )$ be a normed linear space, T be a linear operator from X to X and N be a fuzzy norm defined by (4). Then T is a strongly fuzzy continuous linear operator on $(X,N)$ if and only if T is a bounded linear operator on $(X,\parallel ·\parallel )$.

Proof. 

According to the result of Proposition 2, we just need to testify that T is a strongly fuzzy bounded linear operator on $(X,N)$ if and only if T is a bounded linear operator on $(X,\parallel ·\parallel )$. In fact, if T is a strongly fuzzy bounded linear operator on $(X,N)$, then there is a positive number $M>0$, such that $N(Tx,t)\ge N(x,\frac{t}{M})$, for every $x\in X$. According to the definition of the fuzzy norm N by (4), we have $\frac{t}{t+\parallel Tx\parallel }\ge \frac{\frac{t}{M}}{\frac{t}{M}+\parallel x\parallel }$, for every $x\in X$, and every $t>0$.

Thus, for the above mentioned $M>0$, we can obtain $\parallel Tx\parallel \le M\parallel x\parallel$, $\forall x\in X$, $\forall t>0$, which implies that T is a bounded linear operator on $(X,\parallel ·\parallel )$. Alternatively, if T is a bounded linear operator on $(X,\parallel ·\parallel )$, then $\exists M>0$, such that $\parallel Tx\parallel \le M\parallel x\parallel$, for every $x\in X$. So, for every $x\in X$, and every $t>0$, we obtain that $\frac{t}{t+\parallel Tx\parallel }\ge \frac{t}{t+M\parallel x\parallel }=\frac{\frac{t}{M}}{\frac{t}{M}+\parallel x\parallel }$, which means T is a strongly fuzzy bounded linear operator on $(X,N)$. □

Theorem 5. 

Let $(X,\parallel ·\parallel )$ be a normed linear space. N is the fuzzy norm defined by (4). If A is a nonempty subset of X, then

$${\delta }_n^{\alpha }(A,X,N)=\frac{\alpha }{1-\alpha }{\delta }_n(A,X),$$

where ${\delta }_n(A,X)$ is the classical linear n-width in $(X,\parallel ·\parallel )$.

Proof. 

According to the definition of the $\alpha$-fuzzy distance, we have

$$\begin{array}{cc}\hfill d^{\alpha }(x,Tx)& =\wedge \{t>0\mid N(x-Tx,t)\ge \alpha \}\hfill \\ & =\wedge \{t>0\mid \frac{t}{t+\parallel x-Tx\parallel }\ge \alpha \}=\frac{\alpha }{1-\alpha }\parallel x-Tx\parallel ,\hfill \end{array}$$

for every $\alpha \in (0,1)$ and every $x\in X$. From the definitions of fuzzy linear n-width and classical linear n-width as well as Proposition 4, we can obtain that

$$\begin{array}{cc}\hfill {\delta }_n^{\alpha }(A,X,N)& =\underset{T}{inf}\underset{x\in .A}{sup}{d}^{\alpha }(x,Tx)\hfill \\ & =\underset{T}{inf}\underset{x\in A}{sup}\frac{\alpha }{1-\alpha }\parallel x-Tx\parallel \hfill \\ & =\frac{\alpha }{1-\alpha }{\delta }_n(A,X),\hfill \end{array}$$

where T is a strong fuzzy continuous linear operator on $(X,N)$. □

3.2.2. The Exact Asymptotic Order of Fuzzy n-Widths

In this section, we further set the normed linear space X as a finite dimensional space or Sobolev space, and estimate the exact asymptotic order of the fuzzy n-widths in the finite dimensional space and the Sobolev space.

We consider the fuzzy normed linear space $({\ell }_p^m,N_p)$, for $1\le p\le \infty$, where the fuzzy norm $N_p$ is defined as

$$N_p:=N_p(x,t)=\left\{\begin{array}{ccc}\frac{t}{t+{\parallel x\parallel }_{{\ell }_p^m}},& t>0,& x\in {\ell }_p^m,\\ 0,& t\le 0,& x\in {\ell }_p^m.\end{array}\right.$$

With the fuzzy norm $N_p$ in fuzzy normed linear space $({\ell }_p^m,N_p)$ and the definition of the $\alpha$-fuzzy distance, we could obtain that the $\alpha$-fuzzy distance $d_p^{\alpha }(x,y)$ from x to y equals to $\frac{\alpha }{1-\alpha }{\parallel x-y\parallel }_{{\ell }_p^m}$. Setting $B^{\alpha }\left(p\right):=\{x\in {\ell }_p^m\mid d_p^{\alpha }(x,\theta )\le 1\}$ be the unit ball in ${\ell }_p^m$ with the fuzzy norm $N_p$, we can find the fact that the unit ball of $B^{\alpha }\left(p\right)$ in fuzzy normed linear space $({\ell }_p^m,N_p)$ becomes the ball of radius $\frac{1-\alpha }{\alpha }$ in normed linear space $({\ell }_p^m,\parallel ·{\parallel }_{{\ell }_p^m})$, i.e., $B^{\alpha }\left(p\right)=B_p^m\left(\frac{1-\alpha }{\alpha }\right)$, which plays the key role in estimating the exact asymptotic order of the fuzzy n-widths in the finite dimensional space.

Now, the exact asymptotic orders of the fuzzy Kolmogrov n-width and fuzzy linear n-width in the finite dimensional space are estimated in the following theorem.

Theorem 6. 

Let $1\le p,q\le \infty$. Then

(1) the exact asymptotic order of fuzzy Kolmogorov n-width in the finite dimensional space can be estimated as

$$d_n^{\alpha }\left(B^{\alpha }\left(p\right),{\ell }_q^m,N_q\right)=\left\{\begin{array}{cc}{(m-n)}^{\frac{1}{q}-\frac{1}{p}},& 1\le q\le p\le \infty ,\\ min{\left\{1,m^{\frac{1}{q}}{n}^{-\frac{1}{2}}\right\}}^{(\frac{1}{p}-\frac{1}{q})(\frac{1}{2}-\frac{1}{q})},& 2\le p<q\le \infty ,\\ max\left\{m^{\frac{1}{q}-\frac{1}{p}},min\{1,m^{\frac{1}{q}}{n}^{-\frac{1}{2}}\}·\sqrt{1-\frac{n}{m}}\right\},& 1\le p<2\le q\le \infty ,\\ max\left\{m^{\frac{1}{q}-\frac{1}{p}},{\left(\sqrt{1-\frac{n}{m}}\right)}^{\left(\frac{1}{p}-\frac{1}{q}\right)\left(\frac{1}{p}-\frac{1}{2}\right)}\right\},& 1\le p<q\le 2,\end{array}\right.$$

(2) the exact asymptotic order of fuzzy linear n-width in the finite dimensional space can be estimated as

${\delta }_n^{\alpha }\left(B^{\alpha }\left(p\right),{\ell }_q^m,N_q\right)=\left\{\begin{array}{cc}{(m-n)}^{1/q-1/p},\hfill & 1\le q\le p\le \infty ,\\ \mathsf{\Phi }(m,n,p,q),\hfill & 1\le p<q<\infty ,\\ \mathsf{\Phi }\left(m,n,q^{\prime },p^{\prime }\right),\hfill & max\left\{p,p^{\prime }\right\}<q\le \infty ,\end{array}\right.$where $\mathsf{\Phi }(m,n,p,q)$ and $\mathsf{\Phi }(m,n,q^{{}^{\prime }},p^{{}^{\prime }})$ are defined in Theorem 2.

Proof. 

(1) According to Theorem 4 and the relationship of the unite ball $B^{\alpha }\left(p\right)$ in the space $({\ell }_p^m,N_p)$ and the ball $B_p^m\left(\frac{1-\alpha }{\alpha }\right)$ in the space $({\ell }_p^m,\parallel ·{\parallel }_{{\ell }_p^m})$, we have

$$d_n^{\alpha }\left(B^{\alpha }\left(p\right),{\ell }_q^m,N_q\right)=\frac{\alpha }{1-\alpha }{d}_n\left(B_p^m\left(\frac{1-\alpha }{\alpha }\right),{\ell }_q^m\right)=d_n\left(B_p^m,{\ell }_q^m\right).$$

Thus we can obtain the exact asymptotic order of the fuzzy Kolmogorov n-width by Theorem 2 (1).

(2) From Theorem 5 and the relationship of the unite ball $B^{\alpha }\left(p\right)$ in the space $({\ell }_p^m,N_p)$ and the ball $B_p^m\left(\frac{1-\alpha }{\alpha }\right)$ in the space $({\ell }_p^m,\parallel ·{\parallel }_{{\ell }_p^m})$, we have

$${\delta }_n^{\alpha }\left(B^{\alpha }\left(p\right),{\ell }_q^m,N_q\right)=\frac{\alpha }{1-\alpha }{\delta }_n\left(B_p^m\left(\frac{1-\alpha }{\alpha }\right),{\ell }_q^m\right)={\delta }_n\left(B_p^m,{\ell }_q^m\right).$$

Thus we can obtain the exact asymptotic order of the fuzzy linear n-width by Theorem 2 (2).

For $1\le p\le \infty$, $L_q\left(T\right)$ is the classical space of p-th powers integrable $2\pi$-periodic functions defined on $T=[0,2\pi ]$ equipped with the usual norm ${\parallel ·\parallel }_{L_p}$. Define

$$N_{L_q}:=N_{L_q}(x,t)=\left\{\begin{array}{cc}\frac{t}{t+{\parallel x\parallel }_{L_q}},& t>0,\\ 0,& t\le 0.\end{array}\right.$$

Then $N_{L_q}$ is a fuzzy norm and $(L_q,N_{L_q})$ is a fuzzy normed linear space. Consider the fuzzy normed linear space $(W_p^r,N_{p,r})$, for $1\le p\le \infty$, where the fuzzy norm $N_{p,r}$ is defined as

$$N_{p,r}:=N_{p,r}(x,t)=\left\{\begin{array}{ccc}\frac{t}{t+{\parallel x\parallel }_{W_p^r}},& t>0,& x\in W_p^r,\\ 0,& t\le 0,& x\in W_p^r.\end{array}\right.$$

Then the $\alpha$-fuzzy distance from x to y in the fuzzy space $(W_p^r,N_{p,r})$ is equal to $\frac{\alpha }{1-\alpha }{\parallel x-y\parallel }_{W_p^r}$. Setting $B^{\alpha }(p,r)=\{x\in W_p^r\left(T\right)\mid d_{p,r}^{\alpha }(x,\theta )\le 1\}$ be the unit ball in the Sobolev space $W_p^r$, we can obtain that the unit ball of $B^{\alpha }(p,r)$ in fuzzy normed linear space $(W_p^r,N_{p,r})$ can be converted into the ball of radius $\frac{1-\alpha }{\alpha }$ in normed linear space $(W_p^r,\parallel ·{\parallel }_{W_p^r})$, i.e., $B^{\alpha }(p,r)=B_p^r\left(\frac{1-\alpha }{\alpha }\right)$, which plays the key role in estimating the exact asymptotic order of the fuzzy n-widths in Sobolev space. □

Now, exact asymptotic orders of the fuzzy Kolmogorov n-width and fuzzy linear n-width in Sobolev space are estimated in the following theorem.

Theorem 7. 

Let $1\le p,q\le \infty$, $r\ge 2$. Then

(1) the exact asymptotic order of fuzzy Kolmogorov n-width in Sobolev space can be estimated as

$$d_n^{\alpha }\left(B^{\alpha }(p,r),L_q\left(T\right),N_{L_q}\right)=\left\{\begin{array}{cc}{n}^{-r},& 1\le q\le p\le \infty ,\\ n^{-r},& 2\le p\le q\le \infty ,\\ n^{-r+1/p-1/2},& 1\le p\le 2\le q\le \infty ,\\ n^{-r+1/p-1/q}& 1\le p\le q\le 2,\end{array}\right.$$

(2) the exact asymptotic order of fuzzy linear n-width in Sobolev space can be estimated as

$${\delta }_n^{\alpha }\left(B^{\alpha }(p,r),L_q\left(T\right),N_{L_q}\right)=\left\{\begin{array}{cc}{n}^{-r},& 1\le q\le p\le \infty ,\\ n^{-r+1/p-1/q},& 1\le p\le q\le 2,\\ n^{-r+1/p-1/q},& 2\le p\le q\le \infty ,\\ n^{-r+1/p-1/2},& 1\le p\le 2\le q\le \infty ,p^{\prime }\ge q,\\ n^{-r+1/2-1/q},& 1\le p\le 2\le q\le \infty ,p^{\prime }\le q.\end{array}\right.$$

Proof. 

Based on the above analysis, it is obvious that the unit ball of $B^{\alpha }(p,r)$ in fuzzy normed linear space $(W_p^r,N_{p,r})$ can be converted into the ball of radius $\frac{1-\alpha }{\alpha }$ in normed linear space $(W_p^r,\parallel ·{\parallel }_{W_p^r})$.

(1) According to the relationship of fuzzy n-widths and the classical n-widths mentioned in Theorem 4, we obtain

$$d_n^{\alpha }\left(B^{\alpha }(p,r),L_q\left(T\right),N_{L_q}\right)=\frac{\alpha }{1-\alpha }{d}_n\left(B_p^r\left(\frac{1-\alpha }{\alpha }\right),L_q\left(T\right)\right)=d_n\left(B_p^r,L_q\left(T\right)\right).$$

(5)

Therefore, the exact asymptotic order of fuzzy Kolmogorov n-width in Sobolev space can be estimated by Equation (5) and Theorem 3 (1).

(2) According to the relationship of fuzzy n-widths and the classical n-widths mentioned in Theorem 5, we obtain

$${\delta }_n^{\alpha }\left(B^{\alpha }(p,r),L_q\left(T\right),N_{L_q}\right)=\frac{\alpha }{1-\alpha }{\delta }_n\left(B_p^r\left(\frac{1-\alpha }{\alpha }\right),L_q\left(T\right)\right)={\delta }_n\left(B_p^r,L_q\left(T\right)\right).$$

(6)

Therefore, the exact asymptotic order of the fuzzy linear n-width can be estimated by Equation (6) and Theorem 3 (2). □

4. Conclusions

The concept of a fuzzy norm on a linear space is an important issues in fuzzy mathematics. In the paper, based on a fuzzy norm on a linear space, the width problem of classical approximation theory is generalized, the $\alpha$-fuzzy distance induced by the fuzzy norm is obtained, and fuzzy Kolmogorov n-width and fuzzy linear n-width are defined. More importantly, in finite-dimensional space and Sobolev space, the exact asymptotic orders of the fuzzy Kolmogorov n-width and fuzzy linear n-width are estimated, and the relationship between the classical widths in linear normed space and the fuzzy widths in fuzzy linear normed space is discussed.

Intuitively, the main results of the classical width theory can be utilized in such as boundary recognition problems of image processing. On the one hand, the fuzzy width theory is a generalization of the classical width problem by considering fuzziness on a normed linear space; on the other hand, the fuzzy width theory may be a useful and effective tool to process boundary recognition problems or image deblurring in a blurred image. In fact, natural blurry images exist extensively due to both incomplete and imprecise information or multi-sources information, and image deblurring technology has be a challenging problem in image processing. The fuzzy width theory studies estimation of subset of fuzzy normed linear spaces, which can be used to improve objective functions of image deblurring methods. All of these will be our future works.