0 引言

记L_{[0, 1]}^p(1≤p＜∞)为[0, 1]上p次幂可积函数的全体，赋以L_{[0, 1]}^p空间中的函数以范数：

$ {\left\| f \right\|_{L_{\left[ {0,1} \right]}^P}}: = {\left( {\int_0^1 {{{\left| {f\left( x \right)} \right|}^p}} } \right)^{\frac{1}{p}}}. $

记C^{[0, 1]}为[0, 1]上连续函数的全体，本文有时也将L_{[0, 1]}^∞理解为C^{[0, 1]}，对于任意f∈L_{[0, 1]}^p，定义其在L^p范数下的连续模如下：

$ \begin{array}{l} \omega {\left( {f,\delta } \right)_{L_{\left[ {0,1} \right]}^P}}: = \\ \;\;\;\;\;\;\;\;\mathop {\sup }\limits_{0 < h \le \delta } {\left\{ {\int_0^{1 - h} {{{\left| {f\left( {x + h} \right) - f\left( x \right)} \right|}^p}{\rm{d}}x} } \right\}^{\frac{1}{P}}},\;\;\;1 \le p < \infty . \end{array} $

简便起见，记

$ \omega {\left( {f,\delta } \right)_{L_{\left[ {0,1} \right]}^P}} = \omega {\left( {f,\delta } \right)_p},{\left\| f \right\|_{L_{\left[ {0,1} \right]}^P}} = {\left\| f \right\|_p}. $

记

$ { \wedge _n}: = \left\{ {{\lambda _1},{\lambda _2}, \cdots ,{\lambda _n}} \right\}, $

$ \begin{array}{l} {R_n}\left( \wedge \right): = \\ \;\;\;\;\;\;\left\{ {\frac{{P\left( x \right)}}{{Q\left( x \right)}}:P\left( x \right),Q\left( x \right) \in {\rm{Span}}\left\{ {{x^{{\lambda _k}}}} \right\},{\lambda _k} \in { \wedge _n}} \right\}, \end{array} $

这里Span{x^λ_k}为{x^λ_k}中元素线性组合的全体，如果Q(0)=0, 则要求$\mathop {\lim }\limits_{x \to 0} \frac{{P\left( x \right)}}{{Q\left( x \right)}}$存在且有限，R_n(∧)中的元素即为周知的对应于Müntz系统{x^λ_n}的n阶Müntz有理函数的全体，定义n阶最佳Müntz有理逼近如下：

$ {R_n}\left( {f, \wedge } \right): = \mathop {\inf }\limits_{r \in {R_n}\left( \wedge \right)} {\left\| {f - r} \right\|_p}. $

由于Müntz有理函数是非线性的(即使对加法也是不封闭的)，估计Müntz有理逼近速度是一个非常困难的问题.1976年，匈牙利数学家SOMORJAI^[1]证明了一个令人惊奇的漂亮结果：对任意0≤λ₁≤λ₂≤…, R_n(∧)在C^[0.1]中总是稠密的.1978年，BAK等^[2]证得：如果{λ_n}为趋于零的正数数列，则R_n(∧)在C_{[0, 1]}中稠密，其后，ZHOU^[3]证明了λ_n≥0的假设亦可去掉，在估计Müntz有理逼近速度方面有许多经典的结论^[4-7].最近相关研究也取得了许多有意义的进展^[8-13].其中，王军霞等^[8]建立了以下定理：

定理1  给定M≥0.如果0≤λ₁＜λ₂…＜λ_n＜…, 且λ_n+1-λ_n≥M_n, n=1, 2, …, 那么对任意f∈C_{[0, 1]}, n=1, 2…, 存在r(x)∈R_n(∧)使得

$\left| f\left( x \right)-r\left( x \right) \right|\le {{C}_{M,{{\rho }^{\omega }}{{\psi }^{\theta }}}}\left( f,\frac{\Delta _{n}^{1-\theta }\left( x \right)}{n} \right),x\in \left[ 0,1 \right],$

其中0≤θ≤1, ${\Delta _n}\left( x \right) = \frac{1}{{{n^p}}} + \psi \left( x \right)$, N∈Z⁺, $\psi \left( x \right) = {x^{\frac{N}{{N + 1}}}}$, 而ω_ψ^θ(f, t)为f的Ditzian-Totik型连续模：

$ \begin{array}{l} \omega _\psi ^\theta \left( {f,t} \right) = \\ \;\;\;\;\;\;\mathop {\sup }\limits_{0 < h \le t} \left\| {f\left( {x + \frac{{h{\psi ^\theta }\left( x \right)}}{2}} \right) - f\left( {x - \frac{{h{\psi ^\theta }\left( x \right)}}{2}} \right)} \right\|. \end{array} $

对于L^p空间的Müntz有理逼近，YU等^[11]得到了以下定理：

定理2  给定M≥0.如果0≤λ₁＜λ₂＜…＜λ_n＜…, 且λ_n+1-λ_n≥M_n, n=1, 2, …, 那么对任意f∈L_{[0, 1]}^p, 1≤p＜∞, n=1, 2…, 存在r(x)∈R_n(∧)使得

$ {\left\| {f - r} \right\|_p} \le {C_{M,N,{p^\omega }\psi }}{\left( {f,\frac{1}{n}} \right)_p}, $

这里C_{M, N, p}为仅依赖于M, N和p的正常数，ω_ψ(f, t)_p定义为：

$ {\omega _\psi }{\left( {f,t} \right)_p}: = \mathop {\sup }\limits_{0 < h \le t} {\left\| {{\Delta _{h\psi }}f} \right\|_p}, $

而

$ {\Delta _{h\psi \left( x \right)}}f\left( x \right): = \left\{ \begin{array}{l} f\left( {x + \frac{{h\psi \left( x \right)}}{2}} \right) - f\left( {x - \frac{{h\psi \left( x \right)}}{2}} \right),\\ x \pm \frac{{h\psi \left( x \right)}}{2} \in \left[ {0,1} \right],\\ 0,\;\;\;\;\;其他, \end{array} \right. $

当ψ(x)≡1时, 定理2即为文献[10]中的结论.

本文的主要目的是将定理2的结论推广到加权L^p空间.令ω(x)：=x^α(1-x)^α, α≥0, 0≤x≤1.置L_ω^p：=f：{ωf∈L_{[0, 1]}^p}, 1≤p≤∞.赋L_ω^p空间中的函数以范数‖f‖p, ω：=‖ωf‖_p.

首先，有以下结论：

定理3  假设{λ_n}满足定理2的条件.如果f∈L_ω^p, 1≤p＜∞, α＞0, 则存在r(x)∈R_n(∧)以及仅依赖于M，N, α和p的正常数C_{M, N, α, p}使得

$ {\left\| {f - r} \right\|_{p,\omega }} \le {C_{M,N,\alpha ,{p^\omega }\psi }}{\left( {f,\frac{1}{n}} \right)_{p,\omega }}, $

这里

$ \begin{array}{l} {\omega _\psi }{\left( {f,t} \right)_{p,\omega }}: = \\ \mathop {\sup }\limits_{0 < h \le t} {\left\| {\omega {\Delta _{h\psi }}f} \right\|_{L_{\left[ {C{t^{N + 1}},1} \right]}^p}} + \mathop {\sup }\limits_{0 < h \le t} {\left\| {\omega {{\vec \Delta }_h}f} \right\|_{L_{\left[ {0,C{t^{N + 1}}} \right]}^p}}, \end{array} $

而${{\vec \Delta }_h}f\left( x \right) = f\left( {x + h} \right) - f\left( x \right)$.

当ω(x)≡1, 1≤p＜∞时, 定理3即为定理2.

置L_ω^{p, r}：={f：ωf^(r)∈L_ω^p}.第2个结论为：

定理4  假设{λ_n}满足定理2的条件.如果f∈L_ω^{p, r}, r≥1, α＞0, 则存在r(x)∈R_n(∧)以及仅依赖于M, N, α, p使得

$ \begin{array}{l} {\left\| {f - r} \right\|_{p,\omega }} \le \\ \;\;\;{C_{M,N,\alpha ,p}}\sum\limits_{\mu = 1}^r {\frac{{{{\left\| {{f^{\left( \mu \right)}}} \right\|}_{p,\omega }}}}{{{n^\mu }\mu !}} + \frac{{{C_{M,N\alpha ,p}}}}{{{n^r}}}{\omega _\psi }{{\left( {{f^{\left( r \right)}},\frac{1}{n}} \right)}_{p,\omega }}} . \end{array} $

方便起见，本文中以C表示正常数，其值可能依赖于M, N, p和α等参数，但不依赖于f和x.它们的值在不同的地方可以不同.

1 引理

置

$ {p_j}\left( x \right): = {p_{n,j}}\left( x \right) = {x^{\lambda j}}\prod\limits_{l = 1}^j {x_l^{ - \Delta \lambda t}} ,j = 1,2 \cdots ,n, $

其中Δλ₁=λ₁, Δλ_κ=λ_κ-λ_κ-1, κ=2, 3，…

令x(t)=t^N+1, t∈[0, 1].取[0, 1]上的结点$\left\{ {{x_k} = x\left( {\frac{\kappa }{n}} \right)} \right\}_{k = 1}^n$.定义算子L_n(f, x)如下：

$ {L_n}\left( {f,x} \right) = \sum\limits_{\kappa = 2}^{n - 1} {\frac{1}{{{x_k} - {x_{k - 1}}}}\int_{{x_{k - 1}}}^{{x_k}} {f\left( t \right){\rm{d}}t{r_k}\left( x \right)} } , $

(1)

其中，

$ {r_k}\left( x \right): = \frac{{{p_k}\left( x \right)}}{{\sum\limits_{l = 2}^{n - 1} {{p_l}\left( x \right)} }},k = 2,3, \cdots ,n - 1. $

显然，L_n(f, x)∈R_n(∧)为正线性算子.这一算子在本文结论的证明中起关键性作用.

引理1  对x∈[x_j-1, x_j], 1≤j≤n, 有如下不等式成立：

$ {r_k}\left( x \right) \le C{{\rm{e}}^{ - {C_M}\left( {N + 1} \right)\left| {\kappa - j} \right|}},\kappa = 1,2, \cdots ,n - 1. $

(2)

证明  参考文献[12]引理1的证明，便可得到此结论.

引理2  对任意x∈[x_j-1, x_j], 2≤j≤n, 有

$\left| x-{{x}_{k}} \right|\le C\frac{{{\left( \left| k-j \right|+1 \right)}^{N+1}}}{n}\psi \left( \mu \right),\mu \in \left[ x,{{x}_{k}} \right]$

或

$ \left[ {{x_k},x} \right],\;\;\;k = 1,2, \cdots ,n. $

(3)

对x∈(0, x₁], 1≤j≤n有

$ \left| {x - {x_j}} \right| \le C\frac{{{{\left( {j + 1} \right)}^{N + 1}}}}{{{n^{N + 1}}}}, $

(4)

记x_j为距离x最近的结点，则有

$ \left| {x - {x_j}} \right| \le C\frac{{\psi \left( x \right)}}{n}. $

(5)

式(3)和(4)包含在文献[11]的引理1中，而式(5)为文献[12]中的结论.

引理3  对f(x)∈L_ω^p, 定义加权K泛函：

$ \begin{array}{l} {K_\psi }{\left( {f,t} \right)_{p,\omega }}: = \\ \mathop {\inf }\limits_{g \in AC\left[ {0,1} \right]} \left\{ {{{\left\| {f - g} \right\|}_{p,\omega }} + t{{\left\| {\psi g'} \right\|}_{p,\omega }} + {t^{N + 1}}{{\left\| {g'} \right\|}_{p,\omega }}} \right\}, \end{array} $

则有

$ {K_\psi }{\left( {f,t} \right)_{p,\omega }} \sim {\omega _\psi }{\left( {f,t} \right)_{p,\omega }}, $

这里A~B表示存在正常数C使得C^-1≤A/B≤C.

当ω(x)≡1时，引理3为文献[5]中的theorem 3.1.2, 而其他情形可以套用文献[6]中的方法得到，在此略去详细过程.

引理4  对任意μ＞0, x∈[0, 1]，以下不等式成立：

$ \sum\limits_{\kappa = 2}^{n - 1} {{{\left| {x - {x_{\tilde k}}} \right|}^\mu }{r_k}\left( x \right)} \le \frac{C}{{{n^\mu }}},\;\;\;\tilde k = k,k - 1. $

(6)

证明  若x∈[0, x₁]，则由式(2)和(4)得

$ \begin{array}{l} \sum\limits_{\kappa = 1}^n {{{\left| {x - {x_k}} \right|}^\mu }{r_k}\left( x \right)} \le \\ \;\;\;\;\;\;\frac{C}{{{n^{\left( {N + 1} \right)\mu }}}}\sum\limits_{\kappa = 1}^n {{{\left( {k + 1} \right)}^{\left( {N + 1} \right){\mu _e} - {C_M}\left( {N + 1} \right)\left( {k - 1} \right)}}} \le \frac{C}{{{n^{\left( {N + 1} \right)\mu }}}}. \end{array} $

若x∈[x₁, 1], 则由式(2)和(3)得

$ \begin{array}{l} \sum\limits_{\kappa = 1}^n {{{\left| {x - {x_k}} \right|}^\mu }{r_k}\left( x \right)} \le \\ C{\left( {\frac{{\psi \left( x \right)}}{n}} \right)^\mu }\sum\limits_{k = 1}^n {{{\left( {\left| {j - \kappa } \right| + 1} \right)}^{\left( {N + 1} \right){\mu _e} - {C_M}\left( {N + 1} \right)\left| {j - k} \right|}}} \le \\ C{\left( {\frac{{\psi \left( x \right)}}{n}} \right)^\mu } \le \frac{C}{{{n^\mu }}}. \end{array} $

因此，式(6)在$\tilde k = k$时成立得证.

当$\tilde k = k - 1$时，利用

$ \left| {{x_k} - {x_{k - 1}}} \right| \le \frac{{{k^{N + 1}} - {{\left( {k - 1} \right)}^{N + 1}}}}{{{n^{N + 1}}}} \le \frac{{{2^{N + 1}}{{\left( {k - 1} \right)}^N}}}{{{n^{N + 1}}}}, $

(7)

以及$\tilde k = k$时的结论，即得

$ \begin{array}{l} \sum\limits_{k = 2}^{n - 1} {{{\left| {x - {x_{k - 1}}} \right|}^\mu }{r_k}\left( x \right)} \le \\ \sum\limits_{k = 2}^{n - 1} {{{\left| {x - {x_k}} \right|}^\mu }{r_k}\left( x \right)} + \sum\limits_{k = 2}^{n - 1} {{{\left| {x - {x_{k - 1}}} \right|}^\mu }{r_k}\left( x \right)} \le \frac{C}{{{n^\mu }}}. \end{array} $

引理5  对任意x∈[x_j-1, x_j]，有以下不等式成立：

$ \frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}} \le C{\left( {\left| {j - k} \right| + 1} \right)^{\alpha \left( {N + 1} \right)}},k = 2,3, \cdots ,n - 1. $

(8)

证明  分以下几种情形来证明结论.

情形1  $0＜x＜{{x}_{k}}\le \frac{1}{2}$或$\frac{1}{2}\le {{x}_{k}}\le x＜1$.因为ω(x)在$\left[ 0,\frac{1}{2} \right]$单调递增，在$\left[ \frac{1}{2},1 \right]$单调递减，所以总有

$ \frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}} \le 1 \le {\left( {\left| {j - k} \right| + 1} \right)^{\alpha \left( {N + 1} \right)}}. $

因此，式(8)成立.

情形2  $0＜{{x}_{k}}\le x\le \frac{1}{2}$.此时，有(注意到k≤j-1)

$ \begin{array}{*{20}{c}} {\frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}} = \frac{{{x^\alpha }{{\left( {1 - x} \right)}^\alpha }}}{{x_k^\alpha {{\left( {1 - {x_k}} \right)}^\alpha }}} \le \frac{{x_j^\alpha {{\left( {1 - {x_{j - 1}}} \right)}^\alpha }}}{{x_k^\alpha {{\left( {1 - {x_k}} \right)}^\alpha }}} \le \frac{{x_j^\alpha }}{{x_k^\alpha }} = }\\ {{{\left( {\frac{{j - k + k}}{k}} \right)}^{\alpha \left( {N + 1} \right)}} \le {{\left( {\left| {j - k} \right| + 1} \right)}^{\alpha \left( {N + 1} \right)}}.} \end{array} $

情形3  $\frac{1}{2} ＜ x \le {x_k} ＜ 1$.此时，有(注意到k≥j)

$ \begin{array}{l} \frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}} = \frac{{{x^\alpha }{{\left( {1 - x} \right)}^\alpha }}}{{x_k^\alpha {{\left( {1 - {x_k}} \right)}^\alpha }}} \le \frac{{x_j^\alpha {{\left( {1 - {x_{j - 1}}} \right)}^\alpha }}}{{x_k^\alpha {{\left( {1 - {x_k}} \right)}^\alpha }}} \le \\ \;\;\;\;\;\;\;\;\frac{{{{\left( {1 - {x_{j - 1}}} \right)}^\alpha }}}{{{{\left( {1 - {x_k}} \right)}^\alpha }}} = \frac{{{{\left( {1 - {{\left( {\frac{{j - 1}}{n}} \right)}^{\left( {N + 1} \right)}}} \right)}^\alpha }}}{{{{\left( {1 - {{\left( {\frac{k}{n}} \right)}^{\left( {N + 1} \right)}}} \right)}^\alpha }}} = \\ \;\;\;\;\;\;\;\;{\left( {1 + \frac{{{k^{\left( {N + 1} \right)}} - {{\left( {j - 1} \right)}^{\left( {N + 1} \right)}}}}{{{n^{\left( {N + 1} \right)}} - {k^{\left( {N + 1} \right)}}}}} \right)^\alpha } = \\ \;\;\;\;\;\;\;\;{\left( {1 + \frac{{\xi _1^N\left( {k - j + 1} \right)}}{{\xi _2^N\left( {n - k} \right)}}} \right)^\alpha } \le \\ \;\;\;\;\;\;\;\;{\left( {1 + \left( {k - j + 1} \right)\frac{{\xi _1^N}}{{\xi _2^N}}} \right)^\alpha } \le {\left( {1 + \left| {k - j} \right|} \right)^\alpha }, \end{array} $

这里ξ₁∈(j-1, k), ξ₂∈(k, n).

情形4  $0 \le x \le \frac{1}{2} \le {x_k} ＜ 1$.此时有1-x_j-1≤1, 因此，

$ \begin{array}{l} \frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}} \le \frac{{x_j^\alpha {{\left( {1 - {x_{j - 1}}} \right)}^\alpha }}}{{x_k^\alpha {{\left( {1 - {x_k}} \right)}^\alpha }}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;{2^\alpha }\frac{{x_j^\alpha }}{{{{\left( {1 - {x_k}} \right)}^\alpha }}} = {2^\alpha }{\left( {\frac{{{j^{N + 1}}}}{{{n^{N + 1}} - {k^{N + 1}}}}} \right)^\alpha }. \end{array} $

置$A = \frac{1}{2}\left( {1 + {{\left( {\frac{1}{2}} \right)}^{1/\left( {N + 1} \right)}}} \right)$, 则${\left( {\frac{1}{2}} \right)^{1/\left( {N + 1} \right)}} ＜ A ＜ 1$.因为${x_k} \ge \frac{1}{2}$, 所以$k \ge {\left( {\frac{1}{2}} \right)^{1/\left( {N + 1} \right)}}n$.

如果${\left( {\frac{1}{2}} \right)^{1/\left( {N + 1} \right)}}n \le k \le {A_n}$, 那么

$ \begin{array}{l} \frac{{{j^{N + 1}}}}{{{n^{N + 1}} - {k^{N + 1}}}} \le \frac{{{j^{N + 1}}}}{{\left( {1 - {A^{N + 1}}} \right){n^{N + 1}}}} \le \\ \;\;\;\;\;\;\;\frac{{{A^{N + 1}}}}{{1 - {A^{N + 1}}}}{\left( {\frac{j}{k}} \right)^{N + 1}} \le C{\left( {\left| {j - k} \right| + 1} \right)^{N + 1}}. \end{array} $

当k＞A_n时，分2种情况来估计.

如果$1 \le j \le \frac{{A{{\left( {1/2} \right)}^{1/\left( {N + 1} \right)}}}}{{\frac{1}{2}\left( {A{2^{1/\left( {N + 1} \right)}} - 1} \right)}} = :B$(B为常数), 那么

$ \frac{{{j^{N + 1}}}}{{{n^{N + 1}} - {k^{N + 1}}}} \le {j^{N + 1}} \le {B^{N + 1}} \le {B^{N + 1}}{\left( {\left| {j - k} \right| + 1} \right)^{N + 1}}. $

如果$B ＜ j ＜ {\left( {\frac{1}{2}} \right)^{1/\left( {N + 1} \right)}}n + 1$，注意到${x_{j - 1}} ＜ \frac{1}{2}$, 故有$j ＜ {\left( {\frac{1}{2}} \right)^{1/\left( {N + 1} \right)}}n + 1$.则有：

$ \begin{array}{l} k - j \ge {A_n} - j > \left[ {A{2^{1/N + 1}} - 1} \right]j - A{2^{1/\left( {N + 1} \right)}} > \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}\left[ {A{2^{1/N + 1}} - 1} \right]j. \end{array} $

因此

$ \begin{array}{l} \frac{{{j^{N + 1}}}}{{{n^{N + 1}} - {k^{N + 1}}}} \le {j^{N + 1}} \le \\ \;\;\;\;\;\;\;\;\;\;\frac{{{2^{N + 1}}}}{{{{\left[ {A{2^{1/N + 1}} - 1} \right]}^{N + 1}}}}{\left( {\left| {j - k} \right| + 1} \right)^{N + 1}}. \end{array} $

由上面讨论知，式(8)在情形4时亦成立.

情形5  $0 ＜ {x_k} ＜ \frac{1}{2} ＜ x ＜ 1$.有

$ \begin{array}{l} \frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}} \le {2^\alpha }\frac{{x_j^\alpha }}{{x_k^\alpha }} \le C{\left( {\frac{j}{k}} \right)^{\alpha \left( {N + 1} \right)}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C{\left( {\left| {j - k} \right| + 1} \right)^{\alpha \left( {N + 1} \right)}}. \end{array} $

综合以上各种情形的讨论，引理5得证.

引理6  如果{λ_n}满足定理2的条件，则L_n(f, x)在L_ω^p(1≤p≤∞)中有界，即存在正常数C使得

$ {\left\| {{L_n}\left( f \right)} \right\|_{p,\omega }} \le C{\left\| f \right\|_{p,\omega }}. $

(9)

证明  如果x∈[x_j-1, x_j], 1≤j≤n, 那么由中值定理可知存在${\xi _i} \in \left( {\frac{{j - 1}}{n},\frac{j}{n}} \right),{\xi _k} \in \left( {\frac{{k - 1}}{n},\frac{k}{n}} \right)$使得

$ \begin{array}{l} \frac{{{x_j} - {x_{j - 1}}}}{{{x_k} - {x_{k - 1}}}} \le {\left( {\frac{{{\xi _j}}}{{{\xi _k}}}} \right)^{\alpha \left( {N + 1} \right) - 1}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left( {\left| {j - k} \right| + 1} \right)^{\alpha \left( {N + 1} \right) - 1}}. \end{array} $

(10)

利用式(10)和(8)，有

$\begin{array}{*{35}{l}} {{\left\| {{L}_{n}}\left( f \right) \right\|}_{1,\omega }}=\int_{0}^{1}{\omega \left( x \right)\left| \sum\limits_{k=2}^{n-1}{\frac{1}{{{x}_{k}}-{{x}_{k-1}}}}\times \right.} \\ \int_{{{x}_{k-1}}}^{{{x}_{k}}}{f\left( t \right)\text{d}t\left| {{r}_{k}}\left( x \right)\text{d}x \right.}\le \\ \sum\limits_{k=2}^{n-1}{\frac{1}{{{x}_{k}}-{{x}_{k-1}}}}\int_{0}^{1}{\int_{{{x}_{k-1}}}^{{{x}_{k}}}{\omega \left( x \right){{r}_{k}}\left( x \right)\frac{\left| f\left( t \right)\omega \left( t \right) \right|}{\omega \left( t \right)}\text{d}t\text{d}x}}\le \\ C\sum\limits_{j=1}^{n}{\sum\limits_{k=2}^{n-1}{\frac{1}{{{x}_{k}}-{{x}_{k-1}}}\int_{{{x}_{j-1}}}^{{{x}_{j}}}{\int_{{{x}_{k-1}}}^{{{x}_{k}}}{\frac{\omega \left( x \right)}{\omega \left( {{x}_{k}} \right)}\left| f\left( t \right)\omega \left( t \right) \right|\text{d}t{{r}_{k}}\left( x \right)\text{d}x}}\le }} \\ C\sum\limits_{k=2}^{n-1}{{{\left( \left| j-k \right|+1 \right)}^{2\alpha {{\left( N+1 \right)}_{e}}-{{C}_{M}}\left( N+1 \right)\left| k-j \right|}}}\times \\ \int_{{{x}_{k-1}}}^{{{x}_{k}}}{\left| f\left( t \right)\omega \left( t \right) \right|\text{d}t}\le C{{\left\| f \right\|}_{1,\omega }}. \\ \end{array}$

当p=∞时，由引理1和式(8)得

$ \begin{array}{l} \left| {\omega \left( x \right){L_n}\left( {f,x} \right)} \right| = \\ \omega \left( x \right)\left| {\sum\limits_{k = 2}^{n - 1} {\frac{1}{{{x_k} - {x_{k - 1}}}}\int_{{x_{k - 1}}}^{{x_k}} {f\left( t \right){\rm{d}}t} } } \right|{r_k}\left( x \right) \le \\ C\sum\limits_{k = 2}^{n - 1} {\frac{1}{{{x_k} - {x_{k - 1}}}}} \frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}}\int_{{x_{k - 1}}}^{{x_k}} {\left| {f\left( t \right)\omega \left( t \right)} \right|{\rm{d}}t{r_k}\left( x \right)} \le \\ C{\left\| f \right\|_{\infty ,\omega }}\sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}}{r_k}\left( x \right) \le } \\ C{\left\| f \right\|_{\infty ,\omega }}\sum\limits_{k = 2}^{n - 1} {{{\left( {\left| {j - k} \right| + 1} \right)}^{\alpha {{\left( {N + 1} \right)}_e} - {C_M}\left( {N + 1} \right)\left| {k - j} \right|}} \le } \\ C\left\| f \right\|\infty ,\omega . \end{array} $

根据Riesz-Thorin引理^[14]，即得

$ {\left\| {{L_n}\left( f \right)} \right\|_{p,\omega }} \le C{\left\| f \right\|_{p,\omega }},1 \le p \le \infty . $

2 定理的证明 2.1 定理3的证明

因为L_n(f, x)∈R_n(∧), 所以只要证明：

$ {\left\| {f - {L_n}\left( f \right)} \right\|_{p,\omega }} \le C{\omega _\psi }{\left( {f,\frac{1}{n}} \right)_{p,\omega }}. $

(11)

由引理3知，存在g∈AC_{[0, 1]}使得

$ {\left\| {f - g} \right\|_{p,\omega }} \le C{\omega _\psi }{\left( {f,\frac{1}{n}} \right)_{p,\omega }}, $

(12)

$ \frac{1}{n}{\left\| {\psi g'} \right\|_{p,\omega }} \le C{\omega _\psi }{\left( {f,\frac{1}{n}} \right)_{p,\omega }}, $

(13)

$ \frac{1}{{{n^{N + 1}}}}{\left\| {g'} \right\|_{p,\omega }} \le C{\omega _\psi }{\left( {f,\frac{1}{n}} \right)_{p,\omega }}. $

(14)

利用引理6和式(12)，有

$\begin{array}{l} {\left\| {{L_n}\left( f \right) - f} \right\|_{p,\omega }} \le {\left\| {{L_n}\left( {f - g} \right)} \right\|_{p,\omega }} + \\ \;\;\;\;\;\;{\left\| {{L_n}\left( g \right) - g} \right\|_{p,\omega }} + {\left\| {g - f} \right\|_{p,\omega }} \le \\ \;\;\;\;\;\;C{\left\| {f - g} \right\|_{p,\omega }} + {\left\| {{L_n}\left( g \right) - g} \right\|_{p,\omega }},\\ \;\;\;\;\;\;C{\omega _\psi }{\left( {f,\frac{1}{n}} \right)_{p,\omega }} + {\left\| {{L_n}\left( g \right) - g} \right\|_{p,\omega }}. \end{array} $

因此，只要证明：

$ {\left\| {{L_n}\left( g \right) - g} \right\|_{p,\omega }} \le C{\omega _\psi }{\left( {f,\frac{1}{n}} \right)_{p,\omega }}. $

(15)

因为$\sum\limits_{k = 2}^{n - 1} {{r_k}\left( x \right)} = 1$, 所以

$ \begin{array}{l} {L_n}\left( {g,x} \right) - g\left( x \right) = \\ \;\;\;\;\;\;\;\sum\limits_{k = 2}^{n - 1} {\left( {\frac{1}{{{x_k} - {x_{k - 1}}}}\int_{{x_{k - 1}}}^{{x_k}} {g\left( u \right){\rm{d}}u - g\left( x \right)} } \right)r\left( x \right)} = \\ \;\;\;\;\;\;\;\sum\limits_{k = 2}^{n - 1} {\frac{1}{{{x_k} - {x_{k - 1}}}}\int_{{x_{k - 1}}}^{{x_k}} {\int_x^u {g'\left( t \right){\rm{d}}t{\rm{d}}u{r_k}\left( x \right)} } } . \end{array} $

(16)

利用ω(x_k-1)~ω(x_k), 2≤k≤n-1, 有(当p＞1时要利用Hölder不等式，当p=1时直接讨论可得)：

$ \begin{array}{l} \left\| {{L_n}\left( g \right) - g} \right\|_{p,\omega }^p \le \\ \int_0^1 {{\omega ^p}\left( x \right)} \sum\limits_{k = 2}^{n - 1} {{{\left| {\frac{1}{{{x_k} - {x_{k - 1}}}}\int_{{x_{k - 1}}}^{{x_k}} {\int_x^u {g'\left( t \right){\rm{d}}t{\rm{d}}u} } } \right|}^p}} {r_k}\left( x \right){\rm{d}}x = \\ C\int_0^1 {{\omega ^p}\left( x \right)} \sum\limits_{k = 2}^{n - 1} {{{\left| {\frac{1}{{{x_k} - {x_{k - 1}}}}\int_{{x_{k - 1}}}^{{x_k}} {\int_x^u {\frac{{\omega \left( t \right)g'\left( t \right)}}{{\omega \left( t \right)}}{\rm{d}}t{\rm{d}}u} } } \right|}^p}} {r_k}\left( x \right){\rm{d}}x \le \\ C\int_0^1 {{\omega ^p}\left( x \right)} \sum\limits_{k = 2}^{n - 1} {\left| {\frac{1}{{{x_k} - {x_{k - 1}}}}\int_{{x_{k - 1}}}^{{x_k}} {\left( {\frac{1}{{\omega \left( x \right)}} + \frac{1}{{\omega \left( u \right)}}} \right)} } \right|} \times \\ \int_x^u {\left| {\omega \left( t \right)g'\left( t \right)} \right|{\rm{d}}t{{\left| {{\rm{d}}u} \right|}^p}{r_k}\left( x \right)} {\rm{d}}x \le \\ C\int_0^1 {{\omega ^p}\left( x \right)} \sum\limits_{k = 2}^{n - 1} {{{\left| {\frac{1}{{{x_k} - {x_{k - 1}}}}\left( {\frac{1}{{\omega \left( x \right)}} + \frac{1}{{\omega \left( {{x_k}} \right)}}} \right)} \right|}^p}} \times \\ \left| {\int_{{x_{k - 1}}}^{{x_k}} {} } \right|\int_x^u {\left| {\omega \left( t \right)g'\left( t \right)} \right|{\rm{d}}t{{\left| {{\rm{d}}u} \right|}^p}{r_k}\left( x \right)} {\rm{d}}x \le \\ C\sum\limits_{j = 1}^n {\sum\limits_{k = 2}^{n - 1} {\int_{{x_{j - 1}}}^{{x_j}} {{{\left| {\frac{1}{{{x_k} - {x_{k - 1}}}}\left( {1 + \frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}}} \right)} \right|}^p}} } } \times \\ {\left| {\int_{{x_{k - 1}}}^{{x_k}} {\int_{x * }^{{t^ * }} {\left| {\omega \left( t \right)g'\left( t \right)} \right|{\rm{d}}t{\rm{d}}u} } } \right|^p}{r_k}\left( x \right){\rm{d}}x, \end{array} $

(17)

这里，当j＜k时，x^*=x_j-1，t^*=x_k；而当j≥k时，x^*=x_j，t^*=x_k-1.换而言之，使得积分$\int {_x^u} $尽可能大.

当j=1时，利用引理1和式(8)，得

$\begin{array}{*{35}{l}} \int_{0}^{{{x}_{1}}}{\sum\limits_{k=2}^{n-1}{{{\left| \frac{1}{{{x}_{k}}-{{x}_{k-1}}}\left( 1+\frac{\omega \left( x \right)}{\omega \left( {{x}_{k}} \right)} \right) \right|}^{p}}}}\times \\ \ \ \ \ \ \ {{\left| \int_{{{x}_{k-1}}}^{{{x}_{k}}}{\int_{x*}^{{{t}^{*}}}{\left| \omega \left( t \right){g}'\left( t \right) \right|\text{d}t\text{d}u}} \right|}^{p}}{{r}_{k}}\left( x \right)\text{d}x\le \\ \ \ \ \ \ \ C{{x}_{1}}\sum\limits_{k=2}^{n-1}{{{k}^{\alpha \left( N+1 \right){{p}_{e}}-{{C}_{M}}\left( N+1 \right)k}}}x_{k}^{p-1}\int_{0}^{{{x}_{k}}}{{{\left| \omega \left( t \right){g}'\left( t \right) \right|}^{p}}\text{d}t}\le \\ \ \ \ \ \ \ C\frac{1}{{{n}^{N+1}}}\sum\limits_{k=2}^{n-1}{{{k}^{\alpha p{{\left( N+1 \right)}_{e}}-{{C}_{M}}\left( N+1 \right)k}}{{\left( \frac{k}{n} \right)}^{\left( p-1 \right)\left( N+1 \right)}}}\times \\ \ \ \ \ \ \ \int_{0}^{{{x}_{k}}}{{{\left| \omega \left( t \right){g}'\left( t \right) \right|}^{p}}\text{d}t}\le \\ \ \ \ \ \ \ C\frac{1}{{{n}^{p\left( N+1 \right)}}}\left\| {{g}'} \right\|_{p,\omega }^{p}\sum\limits_{k=2}^{n-1}{{{k}^{\alpha p\left( N+1 \right)+\left( p-1 \right){{\left( N+1 \right)}_{e}}-{{C}_{M}}\left( N+1 \right)k}}}\le \\ \ \ \ \ \ \ C\frac{1}{{{n}^{p\left( N+1 \right)}}}\left\| {{g}'} \right\|_{p,\omega }^{p}. \\ \end{array}$

(18)

需要下列不等式：

$\frac{\psi \left( {{x}_{j}} \right)}{\psi \left( t \right)}\le {{\left( \left| j-k \right|+1 \right)}^{N}},\ \ \ \ t\in \left[ {{x}^{*}},{{t}^{*}} \right]$

或

$t\in \left[ {{t}^{*}},{{x}^{*}} \right],\ \ \ j\ge 2.$

(19)

事实上，当j＜k时, 有x^*=x_j-1, t^*=x_k.因此，

$ \begin{array}{*{20}{c}} {\frac{{\psi \left( {{x_j}} \right)}}{{\psi \left( t \right)}} \le \frac{{\psi \left( {{x_j}} \right)}}{{\psi \left( {{x_{j - 1}}} \right)}} \le {{\left( {1 + \frac{1}{{j - 1}}} \right)}^N} \le }\\ {{{\left( {\left| {j - k} \right| + 1} \right)}^N}.} \end{array} $

当j≥k时，有x^*=x_j, t^*=x_k-1.因此，

$ \begin{array}{l} \frac{{\psi \left( {{x_j}} \right)}}{{\psi \left( t \right)}} \le \frac{{\psi \left( {{x_j}} \right)}}{{\psi \left( {{x_{k - 1}}} \right)}} = {\left( {\frac{j}{{k - 1}}} \right)^N} = \\ \;\;\;\;\;\;\;\;{\left( {1 + \frac{{j - k}}{{k - 1}}} \right)^N} \le {\left( {\left| {j - k} \right| + 1} \right)^N}. \end{array} $

由引理1以及式(3)(8)和(19)，得

$ \begin{array}{l} \sum\limits_{j = 1}^n {\sum\limits_{k = 2}^{n - 1} {\int_{{x_{j - 1}}}^{{x_j}} {{{\left| {\frac{1}{{{x_k} - {x_{k - 1}}}}\left( {1 + \frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}}} \right)} \right|}^p}} } } \times \\ {\left| {\int_{{x_{k - 1}}}^{{x_k}} {\left| {\int_{x * }^{{t^ * }} {\left| {\omega \left( t \right)g'\left( t \right)} \right|{\rm{d}}t} } \right|{\rm{d}}u} } \right|^p}{r_k}\left( x \right){\rm{d}}x \le \\ C\sum\limits_{j = 2}^n {\sum\limits_{k = 2}^{n - 1} {\int_{{x_{j - 1}}}^{{x_j}} {{{\left( {1 + \frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}}} \right)}^p}} } } \times \\ {\left| {\int_{x * }^{{t^ * }} {\left| {\omega \left( t \right)\psi \left( t \right)g'\left( t \right)} \right|\frac{{\psi \left( {{x_j}} \right)}}{{\psi \left( t \right)}}{\rm{d}}t} } \right|^p}\psi {\left( {{x_j}} \right)^{ - p}}{r_k}\left( x \right){\rm{d}}x \le \\ C\sum\limits_{j = 2}^n {\sum\limits_{k = 2}^{n - 1} {{{\left( {\left| {j - k} \right| + 1} \right)}^{\alpha p\left( {N + 1} \right) + p{N_e} - {C_M}\left( {N + 1} \right)\left| {k - j} \right|}}{{\left| {\frac{{{t^ * } - {x^ * }}}{{\psi \left( {{x_j}} \right)}}} \right|}^{p - 1}}} } \times \\ \frac{{\left| {{x_j} - {x_{j - 1}}} \right|}}{{\psi \left( {{x_j}} \right)}}\left| {\int_{x * }^{{t^ * }} {{{\left| {\omega \left( t \right)\psi \left( t \right)g'\left( t \right)} \right|}^p}{\rm{d}}t} } \right| \le \\ \frac{C}{{{n^p}}}\sum\limits_{j = 2}^n {\sum\limits_{k = 2,k \ne j}^{n - 1} {{{\left( {\left| {j - k} \right| + 1} \right)}^{\left( {\alpha p + p - 1} \right)\left( {N + 1} \right) + p{N_e} - {C_M}\left( {N + 1} \right)\left| {k - j} \right|}}} } \times \\ \left| {\int_{x * }^{{t^ * }} {{{\left| {\omega \left( t \right)\psi \left( t \right)g'\left( t \right)} \right|}^p}{\rm{d}}t} } \right| + \\ \frac{C}{{{n^p}}}\sum\limits_{j = 2}^n {\int_{{x_{j - 1}}}^{{x_j}} {{{\left| {\omega \left( t \right)\psi \left( t \right)g'\left( t \right)} \right|}^p}{\rm{d}}t} } \le \\ \frac{C}{{{n^p}}}\sum\limits_{m = 1}^n {{m^{\left( {\alpha p + p - 1} \right)\left( {N + 1} \right) + p{N_e} - {C_M}\left( {N + 1} \right)m}}} \times \\ \sum\limits_{\left| {j - k} \right| = m} {\left| {\int_{x * }^{{t^ * }} {{{\left| {\omega \left( t \right)\psi \left( t \right)g'\left( t \right)} \right|}^p}{\rm{d}}t} } \right|} + \frac{C}{{{n^p}}}\left\| {\psi g'} \right\|_{p,\omega }^p \le \\ \frac{C}{{{n^p}}}\sum\limits_{m = 1}^n {{m^{\left( {\alpha p + p - 1} \right)\left( {N + 1} \right) + pN + {1_e} - {C_M}\left( {N + 1} \right)m}}} \times \\ \int_0^1 {{{\left| {\omega \left( t \right)\psi \left( t \right)g'\left( t \right)} \right|}^p}{\rm{d}}t} + \frac{C}{{{n^p}}}\left\| {\psi g'} \right\|_{p,\omega }^p \le \\ \frac{C}{{{n^p}}}\left\| {\psi g'} \right\|_{p,\omega }^p. \end{array} $

(20)

由式(13)，(14)，(17)，(18)和(20)，证得式(15)，从而定理3得证.

2.2 定理4的证明

显然只要证明：

$ \begin{array}{*{20}{c}} {{{\left\| {f - {L_n}\left( f \right)} \right\|}_{p,\omega }} \le C\sum\limits_{\mu = 1}^r {\frac{{{{\left\| {{f^{\left( \mu \right)}}} \right\|}_{p,\omega }}}}{{{n^\mu }\mu !}}} + }\\ {\frac{C}{{{n^r}}}{\omega _\psi }{{\left( {{f^{\left( r \right)}},\frac{1}{n}} \right)}_{p,\omega }},\;\;\;{f^{\left( r \right)}} \in L_\omega ^p.} \end{array} $

(21)

由引理3知，存在g∈AC_{[0, 1]}使得

$ {\left\| {{f^{\left( r \right)}} - g} \right\|_{p,\omega }} \le C{\omega _\psi }{\left( {{f^{\left( r \right)}},\frac{1}{n}} \right)_{p,\omega }}, $

(22)

$ \frac{1}{n}{\left\| {\psi g'} \right\|_{p,\omega }} \le C{\omega _\psi }{\left( {{f^{\left( r \right)}},\frac{1}{n}} \right)_{p,\omega }}, $

(23)

$ \frac{1}{{{n^{N + 1}}}}{\left\| {g'} \right\|_{p,\omega }} \le C{\omega _\psi }{\left( {{f^{\left( r \right)}},\frac{1}{n}} \right)_{p,\omega }}. $

(24)

根据Taylor展开式

$ \begin{array}{l} f\left( t \right) = \sum\limits_{\mu = 0}^{r - 1} {\frac{{{f^{\left( \mu \right)}}\left( x \right)}}{{\mu !}}{{\left( {t - x} \right)}^\mu }} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{{\left( {r - 1} \right)!}}\int_x^t {{f^{\left( r \right)}}\left( \theta \right){{\left( {t - \theta } \right)}^{r - 1}}{\rm{d}}\theta } , \end{array} $

有

$ \begin{array}{l} \omega \left( x \right)\left| {{L_n}\left( {f,x} \right) - f\left( x \right)} \right| = \\ \sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}\int_{{x_{k - 1}}}^{{x_k}} {\left| {\left( {\sum\limits_{\mu = 1}^{r - 1} {\frac{{{f^{\left( \mu \right)}}\left( x \right)}}{{\mu !}}{{\left( {t - x} \right)}^\mu }} + } \right.} \right.} } \\ \left. {\frac{1}{{\left( {r - 1} \right)!}}\int_x^t {{f^{\left( r \right)}}\left( \theta \right){{\left( {t - \theta } \right)}^{r - 1}}{\rm{d}}\theta } } \right) \le \\ \sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}\int_{{x_{k - 1}}}^{{x_k}} {\left| {\sum\limits_{\mu = 1}^{r - 1} {\frac{{{f^{\left( \mu \right)}}\left( x \right)}}{{\mu !}}{{\left( {t - x} \right)}^\mu }} + } \right.} } \\ \left. {\frac{1}{{\left( {r - 1} \right)!}}\int_x^t {\left( {{f^{\left( r \right)}}\left( \theta \right) - {f^{\left( r \right)}}\left( x \right)} \right){{\left( {t - \theta } \right)}^{r - 1}}{\rm{d}}\theta } } \right|{\rm{d}}t \le \\ \sum\limits_{k = 2}^{n - 1} {\sum\limits_{\mu = 1}^r {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}\frac{{\left| {{f^{\left( \mu \right)}}\left( x \right)} \right|}}{{\mu !}}\int_{{x_{k - 1}}}^{{x_k}} {{{\left| {t - x} \right|}^\mu }{\rm{d}}t} } } + \\ \sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}\frac{1}{{\left( {r - 1} \right)!}} \times } \\ \int_{{x_{k - 1}}}^{{x_k}} {\left| {\int_x^t {\left( {{f^{\left( r \right)}}\left( \theta \right) - {f^{\left( r \right)}}\left( x \right)} \right){{\left( {t - \theta } \right)}^{r - 1}}{\rm{d}}\theta } } \right|{\rm{d}}t} = :{K_1} + {K_2}. \end{array} $

(25)

根据式(6)，有

$ \begin{array}{l} \left\| {{K_1}} \right\|_{p,\omega }^p = \int_0^1 {\left| {\sum\limits_{k = 2}^{n - 1} {\sum\limits_{\mu = 1}^r {\frac{{\omega \left( x \right)}}{{{x_k} - {x_{k - 1}}}}\frac{{{f^{\left( \mu \right)}}\left( x \right)}}{{\mu !}} \times } } } \right.} \\ \;\;\;\;{\left. {\int_{{x_{k - 1}}}^{{x_k}} {{{\left( {t - x} \right)}^\mu }{\rm{d}}t{r_k}\left( x \right)} } \right|^p}{\rm{d}}x \le \\ \;\;\;\;\sum\limits_{\mu = 1}^r {\int_0^1 {\left| {\frac{{\omega \left( x \right){f^{\left( \mu \right)}}\left( x \right)}}{{\mu !}} \times } \right.} } \\ \;\;\;\;{\left. {\sum\limits_{k = 2}^{n - 1} {\max \left( {{{\left| {{x_k} - x} \right|}^\mu },{{\left| {{x_{k - 1}} - x} \right|}^\mu }} \right){r_k}\left( x \right)} } \right|^p}{\rm{d}}x \le \\ \;\;\;\;C\sum\limits_{\mu = 1}^r {\frac{{\left\| {{f^{\left( \mu \right)}}} \right\|_{p,\omega }^p}}{{{{\left( {{n^\mu }\mu !} \right)}^p}}}} . \end{array} $

(26)

对K₂，有

$ \begin{array}{l} {K_2} \le \frac{1}{{\left( {r - 1} \right)!}}\sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}} \times \\ \int_{{x_{k - 1}}}^{{x_k}} {\left| {\int_x^t {\left( {{f^{\left( r \right)}}\left( \theta \right) - {g^{\left( \theta \right)}}} \right){{\left( {t - \theta } \right)}^{r - 1}}{\rm{d}}\theta } } \right|{\rm{d}}t} + \\ \frac{1}{{\left( {r - 1} \right)!}}\sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}} \times \\ \int_{{x_{k - 1}}}^{{x_k}} {\left| {\int_x^t {\left( {g\left( x \right) - {f^{\left( r \right)}}\left( x \right)} \right){{\left( {t - \theta } \right)}^{r - 1}}{\rm{d}}\theta } } \right|{\rm{d}}t} + \\ \frac{1}{{\left( {r - 1} \right)!}}\sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}} \times \\ \int_{{x_{k - 1}}}^{{x_k}} {\left| {\int_x^t {\int_\theta ^x {g'\left( \mu \right){\rm{d}}\mu } {{\left( {t - \theta } \right)}^{r - 1}}{\rm{d}}\theta } } \right|{\rm{d}}t} = :{K_{21}} + {K_{22}} + {K_{23}}. \end{array} $

(27)

利用式(7)，有

$ \left| {{x^ * } - {t^ * }} \right| \le C\frac{{\left| {j - k} \right| + 1}}{n}. $

(28)

利用式(28)以及ω(x_k)~ωx_k-1, 2≤k≤n-1，类似于式(20)的讨论，可得

$ \begin{array}{l} \left\| {{K_{21}}} \right\|_{p,\omega }^p = \\ \int_0^1 {\left| {\frac{1}{{\left( {r - 1} \right)!}}\sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}} \int_{{x_{k - 1}}}^{{x_k}} {\left| {\int_x^t {\frac{{\omega \left( \theta \right)}}{{\omega \left( \theta \right)}}\left( {{f^{\left( r \right)}}\left( \theta \right) - } \right.} } \right.} } \right.} \\ {\left. {\left. {g\left( \theta \right){{\left( {t - \theta } \right)}^{r - 1}}{\rm{d}}\theta } \right|{\rm{d}}t} \right|^p}{\rm{d}}x \le \\ \int_0^1 {\left| {\frac{1}{{\left( {r - 1} \right)!}}\sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}} \times } \right.} \\ \left. {\int_{{x_{k - 1}}}^{{x_k}} {\left( {\frac{1}{{\omega \left( t \right)}} + \frac{1}{{\omega \left( x \right)}}} \right){{\left| {{t^ * } - {x^ * }} \right|}^{r - 1}}} } \right| \times \\ \int_{{x^ * }}^{{t^ * }} {\omega \left( \theta \right)\left( {{f^{\left( r \right)}}\left( \theta \right) - g\left( \theta \right)} \right){\rm{d}}\theta {{\left| {{\rm{d}}t} \right|}^p}{\rm{d}}x} \le \\ C\sum\limits_{j = 1}^n {\int_{{x_{j - 1}}}^{{x_j}} {\left| {\sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}} \times } } \right.} } \\ \left. {\int_{{x_{k - 1}}}^{{x_k}} {\left( {\frac{1}{{\omega \left( t \right)}} + \frac{1}{{\omega \left( x \right)}}} \right){{\left| {{t^ * } - {x^ * }} \right|}^{r - 1}}} } \right| \times \\ \int_{{x^ * }}^{{t^ * }} {\omega \left( \theta \right)\left( {{f^{\left( r \right)}}\left( \theta \right) - g\left( \theta \right)} \right){\rm{d}}\theta {{\left| {{\rm{d}}t} \right|}^p}{\rm{d}}x} \le \\ C\sum\limits_{j = 1}^n {\int_{{x_{j - 1}}}^{{x_j}} {\left| {\sum\limits_{k = 2}^{n - 1} {\left( {\frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}} + 1} \right){{\left| {{t^ * } - {x^ * }} \right|}^{r - 1}}} } \right|} } \times \\ \int_{{x^ * }}^{{t^ * }} {\omega \left( \theta \right)\left( {{f^{\left( r \right)}}\left( \theta \right) - g\left( \theta \right)} \right){\rm{d}}\theta {{\left| {{r_k}\left( x \right)} \right|}^p}{\rm{d}}x} \le \\ C\sum\limits_{j = 1}^n {\sum\limits_{k = 2}^{n - 1} {\int_{{x_{j - 1}}}^{{x_j}} {{{\left( {\frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}} + 1} \right)}^p}{{\left| {{t^ * } - {x^ * }} \right|}^{pr - 1}}} } } \times \\ \left| {\int_{{x^ * }}^{{t^ * }} {{{\left| {\omega \left( \theta \right)\left( {{f^{\left( r \right)}}\left( \theta \right) - g\left( \theta \right)} \right)} \right|}^p}{\rm{d}}\theta } } \right|{r_k}\left( x \right){\rm{d}}x \le \\ \frac{C}{{{n^{rp}}}}\sum\limits_{j = 1}^n {\sum\limits_{k = 2}^{n - 1} {{{\left( {\left| {j - k} \right| + 1} \right)}^{\left( {\alpha p + p - 1} \right){{\left( {N + 1} \right)}_e} - {C_M}\left| {j - k} \right|}}} } \times \\ \left| {\int_{{x^ * }}^{{t^ * }} {{{\left| {\omega \left( \theta \right)\left( {{f^{\left( r \right)}}\left( \theta \right) - g\left( \theta \right)} \right)} \right|}^p}{\rm{d}}\theta } } \right| \le \\ \frac{C}{{{n^{rp}}}}\left\| {{f^{\left( r \right)}} - g} \right\|_{p,\omega }^p \le \frac{C}{{{n^{rp}}}}{\omega _\psi }\left( {{f^{\left( r \right)}},\frac{1}{n}} \right)_{p,\omega }^p. \end{array} $

(29)

利用式(22)，由引理4得

$ \begin{array}{l} \left\| {{K_{22}}} \right\|_{p,\omega }^p = \\ \int_0^1 {\left| {\frac{1}{{\left( {r - 1} \right)!}}\sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}} \times } \right.} \\ {\left. {\int_{{x_{k - 1}}}^{{x_k}} {\int_x^t {\left( {{f^{\left( r \right)}}\left( x \right) - g\left( x \right)} \right){{\left( {t - \theta } \right)}^{r - 1}}{\rm{d}}\theta {\rm{d}}t} } } \right|^p}{\rm{d}}x = \\ \frac{1}{{r!}}\int_0^1 {{\omega ^p}\left( x \right){{\left| {{f^{\left( r \right)}}\left( x \right) - g\left( x \right)} \right|}^p}} \times \\ {\left| {\sum\limits_{k = 2}^{n - 1} {\frac{{{r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}} \int_{{x_{k - 1}}}^{{x_k}} {{{\left( {t - x} \right)}^r}{\rm{d}}t} } \right|^p}{\rm{d}}x \le \\ \frac{1}{{r!}}\int_0^1 {\omega \left( x \right)\left| {{f^{\left( r \right)}}\left( x \right) - g\left( x \right)} \right|} \times \\ {\left| {\sum\limits_{k = 2}^{n - 1} {\left( {{{\left| {x - {x_k}} \right|}^r} + {{\left| {x - {x_{k - 1}}} \right|}^r}} \right){r_k}\left( x \right)} } \right|^p}{\rm{d}}x \le \\ \frac{C}{{{n^{rp}}}}\left\| {{f^{\left( r \right)}} - g} \right\|_{p,\omega }^p \le \frac{C}{{{n^{rp}}}}{\omega _\psi }\left( {{f^{\left( r \right)}},\frac{1}{n}} \right)_{p,\omega }^p. \end{array} $

(30)

对K₂₃，有

$ \begin{array}{l} \left\| {{K_{23}}} \right\|_{p,\omega }^p = \int_0^1 {\left| {\frac{1}{{\left( {r - 1} \right)!}}\sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}} \times } \right.} \\ {\left. {\int_{{x_{k - 1}}}^{{x_k}} {\int_x^t {\int_x^\theta {g'\left( \mu \right){{\left( {t - \theta } \right)}^{r - 1}}{\rm{d}}\mu {\rm{d}}\theta {\rm{d}}t} } } } \right|^p}{\rm{d}}x = \\ \sum\limits_{j = 1}^n {\int_{{x_{j - 1}}}^{{x_j}} {{{\left| {\frac{1}{{r!}}\sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}} \int_{{x_{k - 1}}}^{{x_k}} {\int_x^t {g'\left( \mu \right){{\left( {t - \mu } \right)}^r}{\rm{d}}\mu {\rm{d}}t} } } \right|}^p}} } {\rm{d}}x. \end{array} $

(31)

类似于式(18)的讨论，推得

$ \begin{array}{l} \int_0^{{x_1}} {{{\left| {\frac{1}{{r!}}\sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}} \int_{{x_{k - 1}}}^{{x_k}} {\int_x^t {g'\left( \mu \right){{\left( {t - \mu } \right)}^r}{\rm{d}}\mu {\rm{d}}t} } } \right|}^p}} {\rm{d}}x \le \\ C\sum\limits_{k = 2}^{n - 1} {\int_0^{{x_1}} {{{\left( {\frac{{{{\left| {x - {x_k}} \right|}^r}}}{{{x_k} - {x_{k - 1}}}}\left( {1 + \frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}}} \right)} \right)}^p} \times } } \\ {\left( {\int_{{x_{k - 1}}}^{{x_k}} {\int_0^{{x_k}} {\left| {\omega \left( t \right)g'\left( t \right)} \right|{\rm{d}}t{\rm{d}}u} } } \right)^p}{r_k}\left( x \right){\rm{d}}x \le \\ C\sum\limits_{K = 2}^{n - 1} {\int_0^{{x_1}} {{{\left( {{{\left| {x - {x_k}} \right|}^r}\left( {1 + \frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}}} \right)} \right)}^p}x_k^{p - 1}{r_k}\left( x \right) \times } } \\ \int_0^{{x_k}} {{{\left| {\omega \left( t \right)g'\left( t \right)} \right|}^p}{\rm{d}}t{\rm{d}}x} \le \\ \frac{C}{{{n^{\left( {N + 1} \right)\left( {rp + p} \right)}}}}\sum\limits_{k = 2}^{n - 1} {^{\left( {rp + p - 1 + \alpha p} \right){{\left( {N + 1} \right)}_e} - {C_M}k}\left\| {\omega g'} \right\|_{p,\omega }^p} \le \\ \frac{C}{{{n^{rp}}}}{\left( {\frac{1}{{{n^{N + 1}}}}{{\left\| {\omega g'} \right\|}_{p,\omega }}} \right)^p} \le \frac{C}{{{n^{rp}}}}{\omega _\psi }\left( {{f^{\left( r \right)}},\frac{1}{n}} \right)_{p,\omega }^p. \end{array} $

(32)

最后一步利用了式(24).

根据式(28)，类似于式(20)的讨论可得

$ \begin{array}{l} \sum\limits_{j = 2}^n {\int_{{x_{j - 1}}}^{{x_j}} {{{\left| {\frac{1}{{r!}}\sum\limits_{k = 2}^{n - 1} {\frac{{\omega \left( x \right){r_k}\left( x \right)}}{{{x_k} - {x_{k - 1}}}}} \int_{{x_{k - 1}}}^{{x_k}} {\int_x^t {g'\left( \mu \right){{\left( {t - \mu } \right)}^r}{\rm{d}}\mu {\rm{d}}t} } } \right|}^p}{\rm{d}}x} } \le \\ C\sum\limits_{j = 2}^n {\sum\limits_{k = 2}^{n - 1} {{{\left( {1 + \frac{{\omega \left( x \right)}}{{\omega \left( {{x_k}} \right)}}} \right)}^p}{{\left| {{t^ * } - {x^ * }} \right|}^{rp}}} } \times \\ {\left( {\frac{{\left| {{t^ * } - {x^ * }} \right|}}{{\psi \left( {{x_j}} \right)}}} \right)^{p - 1}} \times \frac{{\left| {{x_j} - {x_{j - 1}}} \right|}}{{\psi \left( {{x_j}} \right)}} \times \\ \left| {\int_{{x^ * }}^{{t^ * }} {{{\left| {\omega \left( t \right)\psi \left( t \right)g'\left( t \right)} \right|}^p}{\rm{d}}t} } \right|{r_k}\left( x \right){\rm{d}}x \le \\ \frac{C}{{{n^{rp + p}}}}\sum\limits_{j = 2}^n {\sum\limits_{k = 2}^{n - 1} {{{\left( {\left| {j - k} \right| + 1} \right)}^{\left( {\alpha p + rp + p - 1} \right){{\left( {N + 1} \right)}_e} - {C_M}\left( {N + 1} \right)\left| {j - k} \right|}}} } \times \\ \left| {\int_{{x^ * }}^{{t^ * }} {{{\left| {\omega \left( t \right)\psi \left( t \right)g'\left( t \right)} \right|}^p}{\rm{d}}t} } \right|{r_k}\left( x \right){\rm{d}}x \le \\ \frac{C}{{{n^{rp}}}}{\left( {\frac{1}{n}{{\left\| {\psi g'} \right\|}_{p,\omega }}} \right)^p} \le \frac{C}{{{n^{rp}}}}{\omega _\psi }\left( {{f^{\left( r \right)}},\frac{1}{n}} \right)_{p,\omega }^p. \end{array} $

(33)

利用式(23)，由式(31)~(33)，有

$ \left\| {{K_{23}}} \right\|_{p,\omega }^p = \frac{C}{{{n^{rp}}}}{\omega _\psi }\left( {{f^{\left( r \right)}},\frac{1}{n}} \right)_{p,\omega }^p. $

(34)

综合式(25)~(30)和(34)式(21)得证.