# 概率框架下对角算子的熵数Entropy Number of Diagonal Operators under Probabilistic Framework

DOI: 10.12677/AAM.2021.104105, PDF, HTML, XML, 下载: 16  浏览: 39

Abstract: In this paper, we mainly talked about the entropy numbers of the finite dimensional diagonal operators which satisfied , and infinite dimensional diagonal operators which satisfied and estimated its asymptotic order.

1. 引言及主要结果

${\epsilon }_{n,\delta }\left(W,\mu ,X\right)=\underset{G}{\mathrm{inf}}{\epsilon }_{n}\left(W\G,X\right)$

${\epsilon }_{n,\delta }\left(T:X\to Y,\mu \right)=\underset{G\in B}{\mathrm{inf}}{\epsilon }_{n}\left(T\left(X\G\right),Y\right)$

${\gamma }_{m}\left(G\right)={\left(2\pi \right)}^{-\frac{m}{2}}{\int }_{G}\mathrm{exp}\left(-\frac{1}{2}{‖x‖}_{{l}_{2}^{m}}^{2}\right)\text{d}x\text{ }\text{ }.$

G是 ${ℝ}^{m}$ 中任意一个Borel可测集，易见 ${\gamma }_{m}\left({ℝ}^{m}\right)=1$。那么， ${ℝ}^{m}$${l}_{q}^{m}$ 空间中关于标准高斯测度 ${\gamma }_{m}$ 的熵数就可表示为：

${\epsilon }_{n,\delta }\left({ℝ}^{m},{\gamma }_{m},{l}_{q}^{m}\right)=\underset{G}{\mathrm{inf}}{\epsilon }_{n}\left({ℝ}^{m}\G,{l}_{q}^{m}\right),$

G是跑遍所有满足 ${\gamma }_{m}\left(G\right)\le \delta$ 的Borel子集。

${e}_{n}\left(n\in \stackrel{0}{ℕ}=ℕ-\left\{0\right\}\right)$ 对应的特征值为 ${\lambda }_{n}={n}^{-\rho }$ $\left(\rho >1\right)$，即

${C}_{\mu }{e}_{n}={\lambda }_{n}{e}_{n},n\in \stackrel{0}{ℕ}$

${y}_{1},\cdots ,{y}_{n}$${l}_{2}$ 中任一正交系， ${\xi }_{j}=〈{C}_{\mu }{y}_{j},{y}_{j}〉$$j=1,\cdots ,n$，B为 ${ℝ}^{n}$ 中任一Borel集，则 ${l}_{2}$ 中柱集

$G=\left\{x\in {l}_{2}|\text{\hspace{0.17em}}\left(〈x,{y}_{1}〉,\cdots ,〈x,{y}_{n}〉\right)\in B\right\}$

$\mu \left(G\right)=\underset{j=1}{\overset{n}{\prod }}{\left(2\pi {\xi }_{j}\right)}^{-\frac{1}{2}}{\int }_{B}\mathrm{exp}\left(-\underset{j=1}{\overset{n}{\sum }}\frac{{|{\mu }_{j}|}^{2}}{2{\xi }_{j}}\right)\text{d}{\mu }_{1}\cdots \text{d}{\mu }_{n}.$

${S}_{k}=\left\{n\in \stackrel{0}{ℕ}|{2}^{k-1}\le n<{2}^{k}\right\}.$

${F}_{k}=span\left\{{e}_{n}|n\in {S}_{k}\right\}.$

${I}_{k}:{F}_{k}\to {ℝ}^{{m}_{k}}$

$x=\underset{n\in {S}_{k}}{\sum }{x}_{n}{e}_{n}↦{I}_{k}x=\underset{j=1}{\overset{{m}_{k}}{\sum }}{x}_{{2}^{k-1}+j-1}{{e}^{\prime }}_{j},$

${I}_{k}$${F}_{k}$${ℝ}^{{m}_{k}}$ 上的线性同构，且

${‖x‖}_{{l}_{q}}={‖{I}_{k}x‖}_{{l}_{q}^{{m}_{{}_{k}}}}={‖{\left\{〈x,{e}_{{2}^{k-1}+j-1}〉\right\}}_{j=1}^{{m}_{k}}‖}_{{l}_{q}^{{m}_{{}_{k}}}}\left(1\le p\le \infty \right).$

${\xi }_{n}=〈{C}_{\mu }{e}_{n},{e}_{n}〉={n}^{-\rho },$

$\frac{1}{{2}^{k\rho }}<{\xi }_{n}\le \frac{{2}^{\rho }}{{2}^{k\rho }}.$

${{\xi }^{\prime }}_{k}=\frac{1}{{2}^{k\rho }}$${{\xi }^{″}}_{k}=\frac{{2}^{\rho }}{{2}^{k\rho }}$${\stackrel{¯}{\xi }}_{k}=\left({\xi }_{{2}^{k-1}},\cdots ,{\xi }_{{2}^{k}-1}\right)$

$x=\left({x}_{1},\cdots ,{x}_{m}\right),y=\left({y}_{1},\cdots ,{y}_{m}\right),\beta \in ℝ,M\subset {ℝ}^{m}$$1\le q\le \infty$，记

$xy=\left({x}_{1}{y}_{1},\cdots ,{x}_{m}{y}_{m}\right),{x}^{\beta }=\left({x}_{1}^{\beta },\cdots ,{x}_{m}^{\beta }\right),e\left(x,M,{l}_{q}^{m}\right)=\underset{y\in M}{\mathrm{inf}}{‖x-y‖}_{{l}_{q}^{m}}.$

2. 主要结果的证明

${\epsilon }_{n,\delta }\left({I}_{m}\right):={\epsilon }_{n,\delta }\left({I}_{m}:{ℝ}^{m}\to {l}_{q}^{m},{\gamma }_{m}\right)={\epsilon }_{n,\delta }\left({ℝ}^{m},{l}_{q}^{m},{\gamma }_{m}\right)\succ \prec {2}^{-\frac{n}{m}}{m}^{\frac{1}{q}-\frac{1}{2}}\sqrt{m+\mathrm{ln}\frac{1}{\delta }}\text{\hspace{0.17em}}.$

$\underset{1\le k\le m}{\mathrm{sup}}{2}^{-\frac{n}{k}}{\left({\sigma }_{1}\cdots {\sigma }_{k}\right)}^{\frac{1}{k}}{k}^{\frac{1}{q}-\frac{1}{p}}\ll {\epsilon }_{n}\left({D}_{m}\right)\ll \underset{1\le k\le m}{\mathrm{sup}}{2}^{-\frac{n}{k}}{\left({\sigma }_{1}\cdots {\sigma }_{k}\right)}^{\frac{1}{k}}{m}^{\frac{1}{q}-\frac{1}{p}}.$

$\underset{1\le k\le m}{\mathrm{sup}}{2}^{-\frac{n}{k}}{\left({\sigma }_{1}\cdots {\sigma }_{k}\right)}^{\frac{1}{k}}\cdot {k}^{\frac{1}{q}-\frac{1}{2}}\sqrt{k+\mathrm{ln}\frac{1}{\delta }}\ll {\epsilon }_{n,\delta }\left({D}_{m}:{ℝ}^{m}\to {l}_{q}^{m},{\gamma }_{m}\right)\ll \underset{1\le k\le m}{\mathrm{sup}}{2}^{-\frac{n}{k}}{\left({\sigma }_{1}\cdots {\sigma }_{k}\right)}^{\frac{1}{k}}\cdot {m}^{\frac{1}{q}-\frac{1}{2}}\sqrt{m+\mathrm{ln}\frac{1}{\delta }}\text{\hspace{0.17em}}.$

${\gamma }_{m}\left(x\in {ℝ}^{m}{|\text{ }‖x‖}_{{l}_{2}^{m}}>{C}_{0}\sqrt{m+\mathrm{ln}\frac{1}{\delta }}\right)\le \delta .$

$\begin{array}{c}{\epsilon }_{n,\delta }\left({D}_{m}:{ℝ}^{m}\to {l}_{q}^{m},{\gamma }_{m}\right)\le {\epsilon }_{n}\left({D}_{m}\left({C}_{0}\sqrt{m+\mathrm{ln}\frac{1}{\delta }}\right){B}_{2}^{m},{l}_{q}^{m}\right)\\ ={C}_{0}\sqrt{m+\mathrm{ln}\frac{1}{\delta }}{\epsilon }_{n}\left({D}_{m}\left({B}_{2}^{m}\right),{l}_{q}^{m}\right)\\ \ll \underset{1\le k\le m}{\mathrm{sup}}{2}^{-\frac{n}{k}}{\left({\sigma }_{1}\cdots {\sigma }_{k}\right)}^{\frac{1}{k}}\cdot {m}^{\frac{1}{q}-\frac{1}{2}}\sqrt{m+\mathrm{ln}\frac{1}{\delta }}\text{\hspace{0.17em}}.\end{array}$

$1\le k\le m$

${I}_{k}:{l}_{2}^{k}\to {l}_{2}^{m}$

$\left({x}_{1},\cdots ,{x}_{k}\right)↦\left({x}_{1},\cdots ,{x}_{k},0,\cdots ,0\right),$

${P}_{k}:{l}_{q}^{m}\to {l}_{q}^{k}$

$\left({x}_{1},\cdots ,{x}_{k},\cdots ,{x}_{m}\right)↦\left({x}_{1},\cdots ,{x}_{k}\right),$

${D}_{k}:{l}_{2}^{k}\to {l}_{q}^{k}$，则 ${D}_{k}={P}_{k}{D}_{m}{I}_{k}$

${\epsilon }_{n,\delta }\left({D}_{m}:{ℝ}^{m}\to {l}_{q}^{m},{\gamma }_{m}\right)={\epsilon }_{n}\left({D}_{m}\left({Q}_{m}\right),{l}_{q}^{m}\right).$

${D}_{m}\left({Q}_{m}\right)\subset \underset{j=1}{\overset{l}{\cup }}\left\{{y}_{j}+{\epsilon }_{n,\delta }{B}_{q}^{m}\right\}\text{ }\text{ }.$

${Q}_{k}={P}_{k}{Q}_{m}$，则 ${\gamma }_{k}\left({Q}_{k}\right)\ge 1-\delta$。若不然， ${\gamma }_{k}\left({Q}_{k}\right)<1-\delta$，令

$F=\left\{\left({x}_{1},\cdots ,{x}_{k},{x}_{k+1},\cdots ,{x}_{m}\right)|\left({x}_{1},\cdots ,{x}_{k}\right)\in {Q}_{k},-\infty <{x}_{j}<\infty ,j=k+1,\cdots ,m\right\},$

${\gamma }_{m}\left(F\right)={\gamma }_{k}\left({Q}_{k}\right)<1-\delta$，而 ${Q}_{m}\subset F$，矛盾。

${\epsilon }_{n,\delta }\left({D}_{k}:{ℝ}^{k}\to {l}_{q}^{k},{\gamma }_{k}\right)\le {\epsilon }_{n,\delta }\left({D}_{k}\left({Q}_{k}\right),{l}_{q}^{k},{\gamma }_{k}\right).$

${D}_{k}\left({Q}_{k}\right)\subset \underset{j=1}{\overset{l}{\cup }}\left\{{P}_{k}{y}_{j}+{\epsilon }_{n,\delta }{B}_{q}^{k}\right\}\text{ }\text{ }.$

${\epsilon }_{n,\delta }\left({D}_{k}:{ℝ}^{k}\to {l}_{q}^{k},{\gamma }_{k}\right)\ll {\epsilon }_{n,\delta }\left({D}_{m}:{ℝ}^{m}\to {l}_{q}^{m},{\gamma }_{m}\right).$

$D=G\cap {G}_{{t}_{1}},{D}_{1}=G-D,{D}_{2}={G}_{{t}_{1}}-D$。则

$\left\{\begin{array}{l}{‖x‖}_{{l}_{2}^{k}}\le {t}_{1}\text{ }x\in {D}_{1}\\ {‖x‖}_{{l}_{2}^{k}}\ge {t}_{1}\text{ }x\in {D}_{2}\end{array}$

$Vo{l}_{k}\left({ℝ}^{k}-G\right)\ge Vo{l}_{k}\left({ℝ}^{k}-{G}_{{t}_{1}}\right),$

$\begin{array}{c}{2}^{\frac{n}{k}}{\epsilon }_{n}\left({D}_{k}\left({ℝ}^{k}-{G}_{{t}_{1}}\right),{l}_{q}^{k}\right)\ge {\left(\frac{Vo{l}_{k}\left({D}_{k}\left({ℝ}^{k}-{G}_{{t}_{1}}\right)\right)}{Vo{l}_{k}\left({B}_{q}^{k}\right)}\right)}^{\frac{1}{k}}\\ \ge {\left({\sigma }_{1}\cdots {\sigma }_{k}\right)}^{\frac{1}{k}}{\left(\frac{Vo{l}_{k}\left(\left({ℝ}^{k}-{G}_{{t}_{1}}\right)\right)}{Vo{l}_{k}\left({B}_{q}^{k}\right)}\right)}^{\frac{1}{k}}\\ \ge {\left({\sigma }_{1}\cdots {\sigma }_{k}\right)}^{\frac{1}{k}}{\left(\frac{{t}_{1}^{k}Vo{l}_{k}\left({B}_{2}^{k}\right)}{Vo{l}_{k}\left({B}_{q}^{k}\right)}\right)}^{\frac{1}{k}}\\ \ge {\left({\sigma }_{1}\cdots {\sigma }_{k}\right)}^{\frac{1}{k}}{k}^{\frac{1}{q}-\frac{1}{2}}\sqrt{k+\mathrm{ln}\frac{1}{\delta }}\text{ }\text{ }.\end{array}$

${\epsilon }_{n}\left({D}_{k}\left({ℝ}^{k}-G\right),{l}_{q}^{k}\right)\ge {2}^{-\frac{n}{k}}{\left({\sigma }_{1}\cdots {\sigma }_{k}\right)}^{\frac{1}{k}}{k}^{\frac{1}{q}-\frac{1}{2}}\sqrt{k+\mathrm{ln}\frac{1}{\delta }},$

${\epsilon }_{n,\delta }\left({D}_{k}:{ℝ}^{k}\to {l}_{q}^{k},{\gamma }_{k}\right)\gg {2}^{-\frac{n}{k}}{\left({\sigma }_{1}\cdots {\sigma }_{k}\right)}^{\frac{1}{k}}{k}^{\frac{1}{q}-\frac{1}{2}}\sqrt{k+\mathrm{ln}\frac{1}{\delta }},$

${\epsilon }_{n,\delta }\left({D}_{m}:{ℝ}^{m}\to {l}_{q}^{m},{\gamma }_{m}\right)\gg \underset{1\le k\le m}{\mathrm{sup}}{2}^{-\frac{n}{k}}{\left({\sigma }_{1}\cdots {\sigma }_{k}\right)}^{\frac{1}{k}}{k}^{\frac{1}{q}-\frac{1}{2}}\sqrt{k+\mathrm{ln}\frac{1}{\delta }}.$

${\sigma }_{1}\ge {\sigma }_{2}\ge \cdots \ge {\sigma }_{k}\ge \cdots >0,n\in ℕ,\delta \in \left(0,\frac{1}{2}\right],n\in \stackrel{0}{ℕ}$

$\left\{{n}_{k}\right\}$ 为非负整数列， $\left\{{\delta }_{k}\right\}$ 为非负数列，且满足

$\underset{k=1}{\overset{\infty }{\sum }}{n}_{k}\le n,\text{ }\underset{k=1}{\overset{\infty }{\sum }}{\delta }_{k}\le \delta .$

${\epsilon }_{n,\delta }\left(D\right):={\epsilon }_{n}\left(D:{l}_{2}\to {l}_{q},\mu \right)\ll \underset{k=1}{\overset{\infty }{\sum }}{2}^{-\frac{k}{2}\rho }{\epsilon }_{{n}_{k},{\delta }_{k}}\left({D}_{k}:{ℝ}^{{m}_{k}}\to {l}_{q}^{{m}_{k}},{\gamma }_{{m}_{k}}\right).$

${D}_{k}:{l}_{2}^{{m}_{k}}\to {l}_{q}^{{m}_{k}}$

$x=\left({x}_{1},\cdots ,{x}_{{m}_{k}}\right)↦{D}_{k}x=\left({\sigma }_{{2}^{k-1}}{x}_{1},\cdots ,{\sigma }_{{2}^{k}-1}{x}_{{2}^{k}-1}\right).$

${\gamma }_{{m}_{k}}\left(\left\{y\in {ℝ}^{{m}_{k}}|e\left({D}_{k}y,{M}_{k},{l}_{q}^{{m}_{k}}\right)>{\epsilon }_{{n}_{k},{\delta }_{k}}\right\}\right)\le {\delta }_{k}.$

${D}^{k}:{l}_{2}\to {l}_{q}$

$x=\left({x}_{n}\right)↦{D}^{k}x=\left(0,\cdots ,0,\cdots ,{\sigma }_{{2}^{k-1}}{x}_{{2}^{k-1}},\cdots ,{\sigma }_{{2}^{k}-1}{x}_{{2}^{k}-1},0,\cdots \right),$

$x\in {l}_{2}$，则 ${D}^{k}x\in {F}_{k}$。易见

$e\left({D}^{k}x,{I}_{k}^{-1}{M}_{k},{l}_{q}\right)=e\left({\left\{〈{D}^{k}x,{e}_{{2}^{k-1}+j-1}〉\right\}}_{j=1}^{{m}_{k}},{M}_{k},{l}_{q}^{{m}_{k}}\right).$

${G}_{k}=\left\{x\in {l}_{2}|e\left({D}^{k}x,{I}_{k}^{-1}{M}_{k},{l}_{q}\right)>{{\xi }^{″}}_{k}{}^{\text{\hspace{0.17em}}\frac{1}{2}}{\epsilon }_{{n}_{k},{\delta }_{k}}\right\}.$

$\begin{array}{c}\mu \left({G}_{k}\right)=\mu \left(\left\{x\in {l}_{2}|e\left({\left\{〈{D}^{k}x,{e}_{{2}^{k-1}+j-1}〉\right\}}_{j=1}^{{m}_{k}},{M}_{k},{l}_{q}^{{m}_{k}}\right)\ge {{\xi }^{″}}_{k}^{\frac{1}{2}}{\epsilon }_{{n}_{k},{\delta }_{k}}\right\}\right)\\ ={\gamma }_{{m}_{k}}\left(\left\{y\in {ℝ}^{{m}_{k}}|e\left(e\left({D}_{k}y\right){\stackrel{¯}{\xi }}_{k}^{\frac{1}{2}},{M}_{k}^{}{\stackrel{¯}{\xi }}^{\frac{1}{2}},{l}_{q}^{{m}_{k}}\right)>{{\xi }^{″}}_{k}^{\frac{1}{2}}{\epsilon }_{{n}_{k},{\delta }_{k}}\right\}\right)\\ \le {\gamma }_{{m}_{k}}\left(\left\{y\in {ℝ}^{{m}_{k}}|{{\xi }^{″}}_{k}^{\frac{1}{2}}e\left({D}_{k}y,{M}_{k},{l}_{q}^{{m}_{k}}\right)>{{\xi }^{″}}_{k}^{\frac{1}{2}}{\epsilon }_{{n}_{k},{\delta }_{k}}\right\}\right)\\ ={\gamma }_{{m}_{k}}\left(\left\{y\in {ℝ}^{{m}_{k}}|e\left({D}_{k}y,{M}_{k},{l}_{q}^{{m}_{k}}\right)>{\epsilon }_{{n}_{k},{\delta }_{k}}\right\}\right)\\ \le {\delta }_{k}.\end{array}$

$G=\underset{k=1}{\overset{\infty }{\cup }}{G}_{k},M=\sum {I}_{k}^{-1}{M}_{k}$

$\mu \left(G\right)\le \sum \mu \left({G}_{n}\right)\le \sum {\delta }_{k}\le \delta ,$

$|M|\le {2}^{\underset{n=1}{\overset{\infty }{\sum }}{n}_{k}}\le {2}^{n}.$

$\begin{array}{c}{\epsilon }_{n,\delta }\left(D:{l}_{2}\to {l}_{q},\mu \right)\ll \underset{x\in {l}_{2}\G}{\mathrm{sup}}e\left(Dx,M,{l}_{q}\right)\\ \ll \underset{x\in {l}_{2}\G}{\mathrm{sup}}\sum e\left({D}_{k}x,{I}_{k}^{-1}{M}_{k},{l}_{q}\right)\\ \ll \underset{k}{\sum }{2}^{-\frac{k}{2}\rho }{\epsilon }_{{n}_{k},{\delta }_{k}}\left({D}_{k}:{ℝ}^{{m}_{k}}\to {l}_{q}^{{m}_{k}},{\gamma }_{{m}_{k}}\right).\end{array}$

${\epsilon }_{n,\delta }\left(D:{l}_{2}\to {l}_{q},\mu \right)\gg {2}^{-\frac{k}{2}\rho }{\epsilon }_{n,\delta }\left({D}_{k}:{ℝ}^{{m}_{k}}\to {l}_{q}^{{m}_{k}},{\gamma }_{{m}_{k}}\right),$

${D}_{k}:{l}_{2}^{{m}_{k}}\to {l}_{q}^{{m}_{k}}$

$x=\left({x}_{1},\cdots ,{x}_{{m}_{k}}\right)↦{D}_{k}x=\left({\sigma }_{{2}^{k-1}}{x}_{1},\cdots ,{\sigma }_{{2}^{k}-1}{x}_{{2}^{k}-1}\right).$

$\forall j\in S$${\xi }_{j}=〈{C}_{\mu }{e}_{j},{e}_{j}〉={j}^{-\rho }$，则 $\frac{1}{{2}^{\rho k}{2}^{\rho }}<{\xi }_{j}\le \frac{{2}^{\rho }}{{2}^{\rho k}}$。令

${{\xi }^{\prime }}_{k}=\frac{1}{{2}^{\rho k}},{{\xi }^{″}}_{k}=\frac{{2}^{\rho }}{{2}^{\rho k}},{\stackrel{¯}{\xi }}_{k}=\left({\xi }_{{2}^{k-1}},\cdots ,{\xi }_{{2}^{k}-1}\right).$

${M}_{1}$${l}_{q}\cap {F}_{s}$ 中的子集，且 $|M|\le {2}^{n}$，则

$\mu \left\{x\in {l}_{2}\cap {F}_{s}|e\left({D}_{k}x,{M}_{1},{l}_{q}\cap {F}_{s}\right)>{\epsilon }_{n,\delta }\right\}\le \delta .$

${I}_{s}:{F}_{s}\to {ℝ}^{{m}_{k}}$

$x=\underset{j\in s}{\sum }{x}_{j}{e}_{j}↦{I}_{s}x=\underset{j=1}{\overset{{m}_{k}}{\sum }}{x}_{{2}^{k-1}+j-1}{{e}^{\prime }}_{j}.$

${I}_{s}$${F}_{s}$${ℝ}^{{m}_{k}}$ 线性算子，且

${‖x‖}_{{l}_{q}}=‖{I}_{s}x‖={‖{\left\{〈x,{e}_{{2}^{k-1}+j-1}〉\right\}}_{j=1}^{{m}_{k}}‖}_{{l}_{q}^{m}}.$

$G=\left\{y\in {ℝ}^{{m}_{k}}|e\left\{\left({D}_{k}y\right),\left({I}_{s}{M}_{1}\right),{l}_{q}^{{m}_{k}}\right\}>{{\xi }^{\prime }}_{k}^{{}^{-\frac{1}{2}}}{\epsilon }_{n,\delta }\right\}.$

$\begin{array}{c}{\gamma }_{{m}_{k}}\left(G\right)={\gamma }_{{m}_{k}}\left(\left\{y\in {ℝ}^{{m}_{k}}|e\left(\left({D}_{k}y\right){{\xi }^{\prime }}_{k}^{{}^{\frac{1}{2}}},\left({I}_{s}{M}_{1}\right){{\xi }^{\prime }}_{k}^{{}^{\frac{1}{2}}},{l}_{q}^{{m}_{k}}\right)>{\epsilon }_{n,\delta }\right\}\right)\\ \le {\gamma }_{{m}_{k}}\left(\left\{y\in {ℝ}^{{m}_{k}}|e\left(\left({D}_{k}y\right){\stackrel{¯}{\xi }}_{k}^{{}^{\frac{1}{2}}},\left({I}_{s}{M}_{1}\right){\stackrel{¯}{\xi }}_{k}^{{}^{\frac{1}{2}}},{l}_{q}^{{m}_{k}}\right)>{\epsilon }_{n,\delta }\right\}\right)\\ =\mu \left(\left\{x\in {l}_{2}\cap {F}_{s}|e\left({\left(Dx,{e}_{j}\right)}_{j\in s},{I}_{s}{M}_{1},{l}_{q}^{{m}_{k}}\right)>{\epsilon }_{n,\delta }\right\}\right)\\ \le \mu \left(\left\{x\in {l}_{2}\cap {F}_{s}|e\left(Dx,{M}_{1},{l}_{q}\right)>{\epsilon }_{n,\delta }\right\}\right)<\delta .\end{array}$

$\begin{array}{c}{\epsilon }_{n,\delta }\left({D}_{k}:{ℝ}^{{m}_{k}}\to {l}_{q}^{{m}_{k}},{\gamma }_{{m}_{k}}\right)\le e\left({D}_{k}\left({ℝ}^{{m}_{k}}\G\right),{I}_{s}{M}_{1},{l}_{q}^{{m}_{k}}\right)\\ =\underset{y\in {ℝ}^{{m}_{k}}\G}{\mathrm{sup}}e\left({D}_{k}y,{I}_{s}{M}_{1},{l}_{q}^{{m}_{k}}\right)\\ \ll {2}^{\frac{\rho }{2}k}{\epsilon }_{n,\delta }.\end{array}$

${\epsilon }_{n,\delta }\left(D:{l}_{2}\to {l}_{q},\mu \right)\gg {2}^{-\frac{\rho }{2}k}{\epsilon }_{n,\delta }\left({D}_{k}:{ℝ}^{{m}_{k}}\to {l}_{q}^{{m}_{k}},{\gamma }_{{m}_{k}}\right).$

${\epsilon }_{n,\delta }\left(D:{l}_{2}\to {l}_{q},\mu \right)\succ \prec {n}^{-\left(\alpha +\frac{\rho }{2}-\frac{1}{q}\right)}\sqrt{1+\frac{1}{n}\mathrm{ln}\frac{1}{\delta }}\text{ }\text{ }.$

$\forall k\in ℕ$，令

${n}_{k}=\left\{\begin{array}{l}{2}^{k}{2}^{\left(1-\beta \right)\left({k}^{\prime }-k\right)},k\le {k}^{\prime }\\ {2}^{k}{2}^{\left(1+\beta \right)\left({k}^{\prime }-k\right)},k>{k}^{\prime }\end{array},{\delta }_{k}=\frac{\delta \cdot {n}_{k}}{n},$

$\sum {n}_{k}\le n,\sum {\delta }_{k}\le \delta$，且由引理2.1及定理2.5，得

$\begin{array}{c}{\epsilon }_{n,\delta }\left(D:{l}_{2}\to {l}_{q},\mu \right)\ll \underset{k=1}{\overset{\infty }{\sum }}{2}^{-\frac{\rho k}{2}}{\epsilon }_{{n}_{k},{\delta }_{k}}\left({D}_{k}:{ℝ}^{{m}_{k}}\to {l}_{q}^{{m}_{k}},{\gamma }_{{m}_{k}}\right)\\ =\underset{k\le {k}^{\prime }}{\sum }{2}^{-\frac{\rho k}{2}}{\epsilon }_{{n}_{k},{\delta }_{k}}\left({D}_{k}:{ℝ}^{{m}_{k}}\to {l}_{q}^{{m}_{k}},{\gamma }_{{m}_{k}}\right)+\underset{k>{k}^{\prime }}{\sum }{2}^{-\frac{\rho k}{2}}{\epsilon }_{{n}_{k},{\delta }_{k}}\left({D}_{k}:{ℝ}^{{m}_{k}}\to {l}_{q}^{{m}_{k}},{\gamma }_{{m}_{k}}\right)\\ ={I}_{1}+{I}_{2}.\end{array}$

$\begin{array}{c}{I}_{1}\ll {\underset{k\le {k}^{\prime }}{\sum }{2}^{-\frac{\rho k}{2}}\underset{1\le j\le {m}_{k}}{\mathrm{sup}}{2}^{-\frac{{n}_{k}}{j}}{\epsilon }_{{n}_{k},{\delta }_{k}}\left(\frac{1}{{2}^{\left(k-1\right)\alpha }}\cdots \frac{1}{{\left({2}^{k-1}+j-1\right)}^{\alpha }}\right)}^{\frac{1}{j}}\cdot {m}_{k}^{\frac{1}{q}-\frac{1}{2}}\sqrt{{m}_{k}+\mathrm{ln}\frac{1}{{\delta }_{k}}}\\ \ll \underset{k\le {k}^{\prime }}{\sum }{2}^{-\frac{\rho k}{2}}{2}^{-\frac{{2}^{k}{2}^{\left(1-\beta \right)\left({k}^{\prime }-k\right)}}{{2}^{k-1}}}\cdot \frac{1}{{2}^{k\rho }}\cdot {2}^{\frac{k}{q}}+\underset{k\le {k}^{\prime }}{\sum }{2}^{-\frac{\rho k}{2}}{2}^{-\frac{{2}^{k}{2}^{\left(1-\beta \right)\left({k}^{\prime }-k\right)}}{{2}^{k-1}}}\cdot \frac{1}{{2}^{k\rho }}\cdot {2}^{\left(\frac{1}{q}-\frac{1}{2}\right)k}\sqrt{\mathrm{ln}\frac{1}{\delta }\frac{{2}^{k}}{{2}^{k}{2}^{\left(1-\beta \right)\left({k}^{\prime }-k\right)}}}\\ \ll \underset{k\le {k}^{\prime }}{\sum }{2}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}\right)k}{2}^{-2\cdot {2}^{\left(1-\beta \right)\left({k}^{\prime }-k\right)}}+\underset{k\le {k}^{\prime }}{\sum }{2}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}+\frac{1}{2}\right)k}\cdot {2}^{-2\cdot {2}^{\left(1-\beta \right)\left({k}^{\prime }-k\right)}}\sqrt{\beta \left({k}^{\prime }-k\right)}\\ \ll {2}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}\right){k}^{\prime }}\underset{0\le \xi \le {k}^{\prime }-1}{\sum }{2}^{\left(\frac{\rho }{2}-\frac{1}{q}\right)\xi -{2}^{\left(1-\beta \right)\xi }}+{2}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}+\frac{1}{2}\right){k}^{\prime }}\underset{0\le \xi \le {k}^{\prime }}{\sum }{2}^{\left(\frac{\rho }{2}-\frac{1}{q}+\frac{1}{2}\right)\xi -2\cdot {2}^{\left(1-\beta \right)\xi }}\sqrt{\beta \xi }\sqrt{\text{\hspace{0.17em}}\mathrm{ln}\frac{1}{\delta }}\\ \ll {2}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}\right){k}^{\prime }}+{2}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}+\frac{1}{2}\right){k}^{\prime }}\sqrt{\mathrm{ln}\frac{1}{\delta }}\ll {n}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}\right)}\sqrt{1+\frac{1}{n}\mathrm{ln}\frac{1}{\delta }}.\end{array}$

$\begin{array}{c}{I}_{2}\ll {\underset{k>{k}^{\prime }}{\sum }{2}^{-\frac{\rho k}{2}}\underset{1\le j\le {m}_{k}}{\mathrm{sup}}{2}^{-\frac{{n}_{k}}{j}}\left(\frac{1}{{2}^{\left(k-1\right)\alpha }}\cdots \frac{1}{{\left({2}^{k-1}+j-1\right)}^{\alpha }}\right)}^{j}\cdot {2}^{\left(\frac{1}{q}-\frac{1}{2}\right)k}\sqrt{{2}^{k}+\mathrm{ln}\frac{{2}^{-\beta \left({k}^{\prime }-k\right)}}{\delta }}\\ \ll {2}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}\right){k}^{\prime }}\underset{\xi >0}{\sum }{2}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}\right)\xi -2\cdot {2}^{-\left(1+\beta \right)\xi }}+{2}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}\right){k}^{\prime }}\underset{\xi >0}{\sum }{2}^{-\left(\frac{\rho }{2}+\alpha +\frac{1}{2}-\frac{1}{q}\right)\xi -2\cdot {2}^{\left(1-\beta \right)\xi }}\sqrt{\beta \xi }\sqrt{\text{\hspace{0.17em}}\mathrm{ln}\frac{1}{\delta }}\\ \ll {n}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}\right)}\sqrt{1+\frac{1}{n}\mathrm{ln}\frac{1}{\delta }}.\end{array}$

$\begin{array}{c}{\epsilon }_{n,\delta }\gg {2}^{-\frac{\rho k}{2}}{2}^{-\frac{n}{{2}^{k-1}}}\frac{1}{{\left({2}^{k}-1\right)}^{\alpha }}{\left({2}^{k-1}\right)}^{\frac{1}{q}-\frac{1}{2}}\sqrt{{2}^{k-1}+\mathrm{ln}\frac{1}{\delta }}\\ \gg {2}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}+\frac{1}{2}\right)k}\sqrt{{2}^{k}+\mathrm{ln}\frac{1}{\delta }}\\ \gg {n}^{-\left(\frac{\rho }{2}+\alpha -\frac{1}{q}\right)}\sqrt{1+\frac{1}{n}\mathrm{ln}\frac{1}{\delta }}.\end{array}$

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