﻿ 模糊数对角算子宽度

# 模糊数对角算子宽度Width of Fuzzy Number Diagonal Operator

Abstract: Based on the classical width theory, the set of fuzzy numbers is regarded as the approximated set. The asymptotic order of the width of diagonal matrix is discussed to 1≤s< . This paper continues the above work. Using the function’s Zadeh’s expansion principle to discuss the width of diagonal matrix asymptotic order when s= . In particular, when fuzzy number set restrictions in real number set, the error estimation and the classical theory of the width of the corresponding results are consistent.

1. 引言

2. 预备知识

2.1. 模糊数

1) u是正规模糊集；

2) u是上半连续的；

3) u的承集 $\text{supp}\text{\hspace{0.17em}}u=cl\left\{x\in {R}^{N}:u\left(x\right)>0\right\}$

4) u是凸模糊集，即

$u\left(\lambda x+\left(1-\lambda \right)y\right)\ge \mathrm{min}\left\{u\left(x\right),u\left(y\right)\right\}$$0\le \lambda \le 1$ 对任意的 $x,y\in {R}^{N}$，则称u为一个模糊数，令 ${R}^{N}$ 是N维欧式空间， ${E}^{N}$ 是全体模糊数的集合。 ${R}^{N}$ 可以嵌入到 ${E}^{N}$ 中，对 $\forall u\in {R}^{N}$ 定义

$\stackrel{^}{u}\left(x\right)=\left\{\begin{array}{l}1,\text{ }\text{ }\text{ }\text{ }u=x;\\ 0,\text{ }\text{ }u\ne x.\end{array}$

${\left[u\right]}^{\alpha }:=\left\{\begin{array}{l}\left\{x\in {R}^{N}:u\left(x\right)\ge \alpha \right\},\text{ }\text{\hspace{0.17em}}0<\alpha \le 1;\\ \text{supp}\text{\hspace{0.17em}}u,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\alpha =0.\end{array}$

${E}^{N}$ 中定义代数运算：

${\left[u+v\right]}^{\alpha }={\left[u\right]}^{\alpha }+{\left[v\right]}^{\alpha },{\left[ku\right]}^{\alpha }=k{\left[u\right]}^{\alpha },k\in R,\alpha \in \left[0,1\right]$

$f:{R}^{N}\to {R}^{N}$ 是一个函数，我们通过 $\stackrel{˜}{f}:{E}^{N}\to {E}^{N}$ 函数定义f的Zadeh扩张

$\stackrel{˜}{f}\left(u\right)\left(x\right)=\left\{\begin{array}{l}\underset{z\in {f}^{-1}\left(x\right)}{\mathrm{sup}}u\left(z\right),\text{ }\text{\hspace{0.17em}}{f}^{-1}\left(x\right)\ne \phi ;\\ 0,\text{ }\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=\phi .\end{array}$

$x=\left({x}_{1},\cdots ,{x}_{N}\right)\in {R}^{N}$ 的n维赋范线性空间 $\left({R}^{N},‖•‖\right)$ 记为 ${l}_{p}^{N}$，定义范数如下：

${‖x‖}_{p}=\left\{\begin{array}{l}{\left(\underset{i=1}{\overset{N}{\sum }}{|{x}_{i}|}^{p}\right)}^{1/p},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}1\le p<\infty ;\\ \underset{1\le i\le N}{\mathrm{max}}|{x}_{i}|,\text{ }\text{ }\text{ }\text{ }p=\infty .\end{array}$

2.2. 对角矩阵n-宽度

$D=diag\left\{{D}_{1},\cdots ,{D}_{N}\right\}$ 是一个 $N×N$ 阶的对角矩阵，假设 ${D}_{1}\ge {D}_{2}\ge \cdots \ge {D}_{N}>0$，令 ${D}_{p}=\left\{Dx:{‖x‖}_{p}\le 1\right\}$${D}_{p}$ 的n宽度在文献 [2] [3] [13] 中可以找到。

${d}_{n}\left({D}_{p};{l}_{p}^{N}\right)={d}^{n}\left({D}_{p};{l}_{p}^{N}\right)={b}_{n}\left({D}_{p};{l}_{p}^{N}\right)={\delta }_{n}\left({D}_{p};{l}_{p}^{N}\right)={D}_{n+1}$

${d}_{n}\left({D}_{p};{l}_{q}^{N}\right)={d}^{n}\left({D}_{p};{l}_{q}^{N}\right)={\delta }_{n}\left({D}_{p};{l}_{q}^{N}\right)={\left(\underset{k=n+1}{\overset{N}{\sum }}{D}_{k}^{r}\right)}^{\frac{1}{r}}$

3. 模糊数宽度

${\stackrel{˜}{X}}_{n}=\left\{u:u\in {E}^{N};{\left[u\right]}^{0}\subseteq {X}_{n}\right\};$

${\stackrel{˜}{L}}^{n}=\left\{u:u\in {E}^{N};{\left[u\right]}^{0}\subseteq {L}^{n}\right\};$

$S\left({\stackrel{˜}{X}}_{n+1}\right)=\left\{u:d\left(u,\stackrel{^}{0}\right)\le 1,u\in {\stackrel{˜}{X}}_{n+1}\right\}.$

${\stackrel{˜}{P}}_{n}$ 是秩为n的连续线性算子 ${P}_{n}:{R}^{N}\to {R}^{N}$ 的Zadeh扩张。

1) A在 ${E}^{N}$ 中的Kolmogorov n-宽度定义为

${d}_{n}\left(A;{E}^{N}\right)=\underset{{\stackrel{˜}{X}}_{n}}{\mathrm{inf}}\underset{u\in A}{\mathrm{sup}}\underset{v\in {\stackrel{˜}{X}}_{n}}{\mathrm{inf}}d\left(u,v\right).$

2) A在 ${E}^{N}$ 中的Bernstein n-宽度定义为

${b}_{n}\left(A;{E}^{N}\right)=\underset{{\stackrel{˜}{X}}_{n+1}}{\mathrm{sup}}\mathrm{sup}\left\{\lambda :\lambda S\left({\stackrel{˜}{X}}_{n+1}\right)\subseteq A\right\}=\underset{{\stackrel{˜}{X}}_{n+1}}{\mathrm{sup}}\underset{x\in \partial \left(A\cap {\stackrel{˜}{X}}_{n+1}\right)}{\mathrm{inf}}d\left(u,\stackrel{^}{0}\right)$

3) A在 ${E}^{N}$ 中的Gelfand n-宽度定义为

${d}^{n}\left(A;{E}^{N}\right)=\underset{{\stackrel{˜}{L}}^{n}}{\mathrm{inf}}\underset{x\in A\cap {\stackrel{˜}{L}}^{n}}{\mathrm{sup}}d\left(u,\stackrel{^}{0}\right)$

4) A在 ${E}^{N}$ 中的线性n-宽度定义为

${\delta }_{n}\left(A;{E}^{N}\right)=\underset{v\in {\stackrel{˜}{P}}_{n}\left(A\right)}{\mathrm{inf}}\underset{u\in A}{\mathrm{sup}}d\left(u,v\right).$

1) ${\delta }_{n}\left(A;{E}^{N}\right)\ge {d}_{n}\left(A;{E}^{N}\right)$

2) ${\delta }_{n}\left(A;{E}^{N}\right)\ge {d}^{n}\left(A;{E}^{N}\right)$

4. $\stackrel{˜}{D}$ n-宽度

$\kappa \left({R}^{N}\right)$${l}_{p}^{N}$ 中的非空紧集全体构成的空间，如果 $A,B\in \kappa \left({R}^{N}\right)\text{\hspace{0.17em}}$$1\le p<\infty$ 则A与B的Hausdorf距离定义为

${d}_{H}^{p}\left(A,B\right)=\mathrm{max}\left\{\underset{a\in A}{\mathrm{sup}}\underset{b\in B}{\mathrm{inf}}{‖a-b‖}_{p},\underset{b\in B}{\mathrm{sup}}\underset{a\in A}{\mathrm{inf}}{‖a-b‖}_{p}\right\}$

$u,v\in {E}^{N},1\le p<\infty ,1\le s<\infty$$\alpha \in \left[0,1\right]$，我们定义 $1\le p<\infty ,1\le s<\infty$

${d}_{s}^{p}\left(u,v\right)={\left({\int }_{0}^{1}{d}_{H}^{p}{\left({\left[u\right]}^{\alpha },{\left[v\right]}^{\alpha }\right)}^{s}\text{d}\alpha \right)}^{1/s}.$

${L}_{s,p}^{N}=\left({E}^{N},{d}_{s,p}\right)$${d}_{s,p}$${E}^{N}$ 上的 ${L}_{s,p}^{N}$ 度量。

${d}_{\infty ,p}\left(u,v\right)=\underset{\alpha \in \left[0,1\right]}{\mathrm{sup}}{d}_{H}^{p}\left({\left[u\right]}^{\alpha },{\left[v\right]}^{\alpha }\right)$，为模糊数空间 ${E}^{N}$ 上的上确界度量(或一致Hausdorf度量) [14]。

${L}_{\infty ,p}^{N}=\left({E}^{N},{d}_{\infty ,p}\right)$${d}_{\infty ,p}$${E}^{N}$ 上的 ${L}_{\infty ,p}^{N}$ 度量。

$\begin{array}{l}{d}_{\infty ,p}\left(\stackrel{˜}{D}{\left[u\right]}^{\alpha },k{\left[\stackrel{^}{0}\right]}^{\alpha }\right)\\ =\underset{\alpha \in \left[0,1\right]}{\mathrm{sup}}{d}_{H}^{p}\left(D{\left[u\right]}^{\alpha },k{\left[\stackrel{^}{0}\right]}^{\alpha }\right)\\ =\underset{\alpha \in \left[0,1\right]}{\mathrm{sup}}\mathrm{max}\left\{\underset{a\in {\left[u\right]}^{\alpha }}{\mathrm{sup}}\underset{b\in k{\left[\stackrel{^}{0}\right]}^{\alpha }}{\mathrm{inf}}{‖Da-b‖}_{p},\underset{b\in k{\left[\stackrel{^}{0}\right]}^{\alpha }}{\mathrm{sup}}\underset{a\in {\left[u\right]}^{\alpha }}{\mathrm{inf}}{‖Da-b‖}_{p}\right\}\\ =\underset{\alpha \in \left[0,1\right]}{\mathrm{sup}}\underset{a\in {\left[u\right]}^{\alpha }}{\mathrm{sup}}{‖Da‖}_{p}\end{array}$

$0\le {r}_{1}\le {r}_{2}\le 1$${\left[u\right]}^{{r}_{2}}\subset {\left[u\right]}^{{r}_{1}}$，于是 ${d}_{\infty ,p}\left(\stackrel{˜}{D}u,k\stackrel{^}{0}\right)=\underset{a\in {\left[u\right]}^{0}}{\mathrm{sup}}{‖Da‖}_{p}$

${\stackrel{˜}{D}}_{P}=\left\{\stackrel{˜}{D}u:u\in {E}^{N},{d}_{\infty ,p}\left(u,\stackrel{^}{0}\right)\le 1\right\}$${\stackrel{˜}{D}}_{p}^{-1}=\left\{{\stackrel{˜}{D}}^{-1}u:u\in {E}^{N},{d}_{\infty ,p}\left(u,\stackrel{^}{0}\right)\le 1\right\}$

${\delta }_{n}\left({\stackrel{˜}{D}}_{p};{L}_{\infty ,p}^{N}\right)={d}_{n}\left({\stackrel{˜}{D}}_{p};{L}_{\infty ,p}^{N}\right)={d}^{n}\left({\stackrel{˜}{D}}_{p};{L}_{\infty ,p}^{N}\right)={b}_{n}\left({\stackrel{˜}{D}}_{p};{L}_{\infty ,p}^{N}\right)={D}_{n+1}$

${\delta }_{n}\left({\stackrel{˜}{D}}_{P};{L}_{\infty ,q}^{N}\right)={d}_{n}\left({\stackrel{˜}{D}}_{P};{L}_{\infty ,q}^{N}\right)={d}^{n}\left({\stackrel{˜}{D}}_{P};{L}_{\infty ,q}^{N}\right)={\left(\underset{k=n+1}{\overset{N}{\sum }}{D}_{k}^{r}\right)}^{\frac{1}{r}}$

1) ${\delta }_{n}\left({\stackrel{˜}{D}}_{P};{L}_{\infty ,P}^{N}\right)\ge {d}_{n}\left({\stackrel{˜}{D}}_{P};{L}_{\infty ,P}^{N}\right)\ge {b}_{n}\left({\stackrel{˜}{D}}_{P};{L}_{\infty ,P}^{N}\right)$

2) ${\delta }_{n}\left({\stackrel{˜}{D}}_{P};{L}_{\infty ,p}^{N}\right)\ge {d}^{n}\left({\stackrel{˜}{D}}_{P};{L}_{\infty ,p}^{N}\right)\ge {b}_{n}\left({\stackrel{˜}{D}}_{P};{L}_{\infty ,p}^{N}\right)$

$\begin{array}{c}{d}_{H}^{p}\left({\left[\stackrel{˜}{D}u\right]}^{\alpha },{\left[{\stackrel{˜}{P}}_{n}u\right]}^{\alpha }\right)={d}_{H}^{p}\left(D{\left[u\right]}^{\alpha },{P}_{n}{\left[u\right]}^{\alpha }\right)\\ =\mathrm{max}\left\{\underset{a\in {\left[u\right]}^{\alpha }}{\mathrm{sup}}\underset{b\in {\left[u\right]}^{\alpha }}{\mathrm{inf}}{‖Da-{P}_{n}b‖}_{p},\underset{b\in {\left[u\right]}^{\alpha }}{\mathrm{sup}}\underset{a\in {\left[u\right]}^{\alpha }}{\mathrm{inf}}{‖Da-{P}_{n}b‖}_{p}\right\}\\ \underset{a\in {\left[u\right]}^{\alpha }}{=\mathrm{max}}{‖\left(D-{P}_{n}\right)a‖}_{p}\end{array}$ (1)

$\begin{array}{c}{\delta }_{n}\left({\stackrel{˜}{D}}_{p};{L}_{\infty ,p}^{N}\right)=\underset{{\stackrel{˜}{P}}_{n}}{\mathrm{inf}}\underset{u\in {E}^{N}}{\mathrm{sup}}{‖\stackrel{˜}{D}u-{\stackrel{˜}{P}}_{n}u‖}_{p}\le \underset{{d}_{\infty }^{p}\left(u,\stackrel{^}{0}\right)\le 1}{\mathrm{max}}{d}_{\infty ,p}\left(\stackrel{˜}{D}u,{\stackrel{˜}{P}}_{n}u\right)\\ =\underset{u\ne \stackrel{^}{0}}{\mathrm{max}}\frac{{d}_{\infty ,p}\left(\stackrel{˜}{D}u,{\stackrel{˜}{P}}_{n}u\right)}{{d}_{\infty ,p}\left(u,\stackrel{^}{0}\right)}=\underset{u\ne \stackrel{^}{0}}{\mathrm{max}}\frac{\underset{\alpha \in \left[0,1\right]}{\mathrm{sup}}{d}_{H}^{p}\left({\left[\stackrel{˜}{D}u\right]}^{\alpha },{\left[{\stackrel{˜}{P}}_{n}u\right]}^{\alpha }\right)}{\underset{\alpha \in \left[0,1\right]}{\mathrm{sup}}{d}_{H}^{p}\left({\left[u\right]}^{\alpha },{\left[\stackrel{^}{0}\right]}^{\alpha }\right)}\\ =\underset{u\ne \stackrel{^}{0}}{\mathrm{max}}\frac{\underset{a\in {\left[u\right]}^{0}}{\mathrm{max}}{‖\stackrel{˜}{D}-{\stackrel{˜}{P}}_{n}\left(a\right)‖}_{p}}{\underset{a\in {\left[u\right]}^{0}}{\mathrm{max}}{‖a‖}_{p}}=\underset{u\ne \stackrel{^}{0}}{\mathrm{max}}\frac{\underset{a\in {\left[u\right]}^{0}}{\mathrm{max}}{\left({\sum }_{i=n+1}^{N}{|{\stackrel{˜}{D}}_{i}{a}_{i}|}^{p}\right)}^{\frac{1}{p}}}{\underset{a\in {\left[u\right]}^{0}}{\mathrm{max}}{‖a‖}_{p}}\\ \le \underset{u\ne \stackrel{^}{0}}{\mathrm{max}}\frac{{D}_{n+1}\left(\underset{a\in {\left[u\right]}^{0}}{\mathrm{max}}{\left({\sum }_{i=n+1}^{N}{|{a}_{i}|}^{p}\right)}^{\frac{1}{p}}\right)}{\underset{a\in {\left[u\right]}^{0}}{\mathrm{max}}{‖a‖}_{p}}\le {D}_{n+1}\end{array}$

${p}_{n}=diag\left({D}_{1},\cdots ,{D}_{n},0,\cdots ,0\right)$，对任意的 $u\in {E}^{N}$，如定理1中(1)的证明

${d}_{H}^{q}\left({\left[\stackrel{˜}{D}u\right]}^{\alpha },{\left[{\stackrel{˜}{P}}_{n}u\right]}^{\alpha }\right)=\underset{a\in {\left[u\right]}^{\alpha }}{\mathrm{sup}}{‖\left(D-{P}_{n}\right)a‖}_{q}=\underset{a\in {\left[u\right]}^{\alpha }}{\mathrm{sup}}{\left(\underset{k=n+1}{\overset{N}{\sum }}{|{D}_{k}{a}_{k}|}^{q}\right)}^{1/q},$

${\left(\underset{k=n+1}{\overset{N}{\sum }}{|{D}_{k}{a}_{k}|}^{q}\right)}^{1/q}\le {\left(\underset{k=n+1}{\overset{N}{\sum }}{|{D}_{k}|}^{r}\right)}^{1/r}{\left(\underset{k=n+1}{\overset{N}{\sum }}{|{a}_{k}|}^{p}\right)}^{1/p},$

$\begin{array}{c}{d}_{H}^{q}\left({\left[\stackrel{˜}{D}u\right]}^{\alpha },{\left[{\stackrel{˜}{P}}_{n}u\right]}^{\alpha }\right)\le \underset{a\in {\left[u\right]}^{\alpha }}{\mathrm{sup}}{\left(\underset{k=n+1}{\overset{N}{\sum }}{|{D}_{k}|}^{r}\right)}^{1/r}{\left(\underset{k=n+1}{\overset{N}{\sum }}{|{a}_{k}|}^{p}\right)}^{1/p}\\ \le {\left(\underset{k=n+1}{\overset{N}{\sum }}{|{D}_{k}|}^{r}\right)}^{1/r}\underset{a\in {\left[u\right]}^{\alpha }}{\mathrm{sup}}{\left(\underset{k=1}{\overset{N}{\sum }}{|{a}_{k}|}^{p}\right)}^{1/p}\\ \le {\left(\underset{k=n+1}{\overset{N}{\sum }}{|{D}_{k}|}^{r}\right)}^{1/r}{d}_{H}^{p}\left({\left[u\right]}^{\alpha },{\left[\stackrel{^}{0}\right]}^{\alpha }\right)\end{array}$

$\begin{array}{c}{\delta }_{n}\left({\stackrel{˜}{D}}_{p};{L}_{\infty ,q}^{N}\right)=\underset{{P}_{n}}{\mathrm{inf}}\underset{u\in {E}^{N}}{\mathrm{sup}}{‖\stackrel{˜}{D}u-{\stackrel{˜}{P}}_{n}u‖}_{p}\\ \le \underset{{d}_{\infty }^{p}\left(u,\stackrel{^}{0}\right)\le 1}{\mathrm{max}}{d}_{\infty ,p}^{N}\left({\left[Du\right]}^{\alpha },{\left[{P}_{n}u\right]}^{\alpha }\right)\\ \le \underset{{d}_{\infty }^{p}\left(u,\stackrel{^}{0}\right)\le 1}{\mathrm{max}}\underset{\alpha \in \left[0,1\right]}{\mathrm{sup}}{d}_{H}^{q}\left(D\left({\left[u\right]}^{\alpha }\right),{P}_{n}\left({\left[u\right]}^{\alpha }\right)\right)\\ \le \underset{{d}_{p}\left(u,\stackrel{^}{0}\right)}{\mathrm{max}}{\left(\underset{k=n+1}{\overset{N}{\sum }}{|{D}_{k}|}^{r}\right)}^{1/r}{d}_{\infty ,p}^{N}\left(u,\stackrel{^}{0}\right)\\ \le {\left(\underset{k=n+1}{\overset{N}{\sum }}{|{D}_{k}|}^{r}\right)}^{1/r}\end{array}$ (2)

${d}_{H}^{p}\left({\left[u\right]}^{\alpha },{\left[\stackrel{^}{0}\right]}^{\alpha }\right)=\underset{a\in {\left[u\right]}^{\alpha }}{\mathrm{max}}{\left(\underset{k=1}{\overset{N}{\sum }}{|{a}_{k}|}^{p}\right)}^{1/p}$

${d}^{n}\left({\stackrel{˜}{D}}_{P},{L}_{\infty ,q}^{N}\right)\ge {d}^{n}\left({D}_{P},{L}_{\infty ,q}^{N}\right)={d}^{n}\left({D}_{P},{l}_{q}^{N}\right)={\left(\underset{k=n+1}{\overset{N}{\sum }}{D}_{k}^{r}\right)}^{1/r}$ (3)

${\delta }_{n}\left({\stackrel{˜}{D}}_{p},{L}_{\infty ,q}^{N}\right)={d}^{n}\left({\stackrel{˜}{D}}_{p},{L}_{\infty ,q}^{N}\right)={\left(\underset{k=n+1}{\overset{N}{\sum }}{D}_{k}^{r}\right)}^{1/r}$

2020年“西华杯”大学生创新创业项目(2020108)。

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