2003年, 盛宝怀, 周颂平在文献[1]中研究了正系数代数多项式倒数对非负连续函数在Lp空间内的逼近问题, 得到了以下结果:

定理A  设$ f\left(x \right) \in C\left(S \right), f\left(x \right) \ge 0, f\left(x \right) \ne 0, p > 1, $则存在

$ P_n \left( x \right) \in \Pi _{n, d}^ + = \left\{ {P_n \left( x \right): = P_n \left( x \right) = \sum\limits_{\left| k \right| \le n} {a_k x^k \left( {1 - \left| x \right|} \right)^{n - \left| k \right|} } :x \in S, a_k \ge 0} \right\}, $

使得

$ \left\| {f - \frac{1}{{P_n }}} \right\|_p \le C\left[{\omega _p^2 \left( {f, \frac{1}{{\sqrt[4]{n}}}} \right) + \frac{{\left\| f \right\|}}{{\sqrt n }}} \right], $

其中$S$为$ R^d $中的单纯性, $ d \ge 1 $, $ C\left(S \right) $为$S$上的连续函数类. $C$是与$n$无关的常数, 并且在不同处可以表示不同的值(以下同).

本文在Orlicz空间内研究了类似的问题, 考虑到Orlicz空间是Lp空间的扩充, 本文的研究工作具有一定的拓展意义.

对于$ f\left(x \right) \in C\left[{0, 1} \right] $, 其Bernstein-Durrmeyer算子为

$ M_n \left( {f, x} \right) = \left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)f\left( t \right)dt} }, $

其中$ P_{n, k} \left(x \right) = \left({\begin{array}{*{20}c} n \\ k \\ \end{array}} \right)x^k \left({1 -x} \right)^{n -k}. $

本文将其作为一种逼近工具, 再利用$K$-泛函、光滑模讨论了正系数代数多项式倒数对非负连续函数在Orlicz空间内的逼近问题.

文中用$ M\left(u \right) $和$ N\left(v \right) $表示互余的$N$函数, 关于$N$函数的定义及其性质见文献[2].

由$N$函数$ M\left(u \right) $生成的Orlicz空间$ L_M^* \left[{0, 1} \right] $指具有有限Orlicz范数

$ \left\| u \right\|_M = \mathop {\sup }\limits_{\rho \left( {v, N} \right) \le 1} \left| {\int_0^1 {u\left( x \right)v\left( x \right)dx} } \right| $

的可测函数全体$ \left\{ {u\left(x \right)} \right\} $, 其中$ \rho \left({v, N} \right) = \int_0^1 {N\left({v\left(x \right)} \right)} dx $是$ v\left(x \right) $关于$ N\left(v \right) $的模.

又由文献[2]知, Orlicz范数的等价形式为

$ \left\| u \right\|_M = \mathop {\inf }\limits_{\alpha > 0} \frac{1}{\alpha }\left( {1 + \int_0^1 {M\left( {\alpha u\left( x \right)} \right)dx} } \right). $

对于$ f\left(x \right) \in L_M^* \left[{0, 1} \right], $ $t \ge 0, $定义$K$-泛函, 连续模和二阶光滑模为

$ K_r \left( {f, t} \right)_M = \inf \left\{ {\left\| {f - g} \right\|_M + t^r \left\| {g^{\left( r \right)} } \right\|_M :g^{\left( {r - 1} \right)} \in AC\left[{0, 1} \right], g^{\left( r \right)} \in L_M^* } \right\};\\ \omega _1 \left( {f, t} \right)_M = \mathop {\sup }\limits_{0 \le h \le t} \left\| {f\left( { \cdot + h} \right) - f\left( \cdot \right)} \right\|_M;\\ \omega _2 \left( {f, t} \right)_M = \mathop {\sup }\limits_{0 \le h \le t} \left\| {f\left( { \cdot + h} \right) + f\left( { \cdot - h} \right) - 2f\left( \cdot \right)} \right\|_M. $

由文献[3]可知$ \omega _2 \left({f, t} \right)_M \to 0 $$ \left({t \to 0} \right) $当且仅当$ M\left(u \right) $满足$ \Delta _2$-条件.

定理  设$ f\left(x \right) \in L_M^*, $$ f\left(x \right) \ge 0 $且$ f\left(x \right) $不恒等于0, 则存在$ P_n \left(x \right) \in \prod _n \left(+ \right) $使得

$ \left\| {f - \frac{1}{{P_n }}} \right\|_M \le C\left( {\omega _1 \left( {f, \frac{1}{n}} \right)_M + \omega _2 \left( {f, \frac{1}{{\sqrt n }}} \right)_M } \right), $

$\prod _n \left(+ \right) $表示次数不超过$n$的正系数多项式全体.

引理1^[4]  存在两个常数$ C_1 $和$C_2$, 使得$ C_1 \omega \left({f, t} \right)_M \le K\left({f, t} \right)_M \le C_2 \omega \left({f, t} \right)_M. $

引理2^[5]  $ M_n \left({1, x} \right) = 1, $ $ M_n \left({\left({t -x} \right)^i, x} \right) \le \frac{C}{n}, $ $ i = 1, 2. $

引理3^[5]  $ \left\| {M_n \left({f} \right)} \right\|{}_M \le \left\| f \right\|_M. $

引理4  $ \left\| {f^2 } \right\|_M \le C\left\| f \right\|_M ^2.$

证  由Orlicz范数的定义知

$ \left\| {f^2 } \right\|_M = \mathop {\sup }\limits_{\rho (v, N) \le 1} \left| {\int_{\;0}^{\;1} {f^2 (x)v(x)dx} } \right| = \mathop {\sup }\limits_{\rho (v, N) \le 1} \left| {\int_{\;0}^{\;1} {f(x)f(x)v(x)dx} } \right|. $

利用Orlicz空间内的Hölder不等式(见文献[2])得

$ \sup \limits_{\rho (v, N) \le 1} \left| {\int_{\;0}^{\;1} {f(x)f(x)v(x)dx} } \right| \le \sup\limits_{\rho (v, N) \le 1} \left\| f \right\|_M \left\| {fv} \right\|_N \\ =||f||_M \sup \limits_{\rho (u, M) \le 1} |\int_{\;0}^{\;1} {f(x)v(x)u(x)dx|} =||f||_M \sup \limits_{\rho (u, M) \le 1} ||fu||_M ||v||_N. $

由$\displaystyle \rho (v, N) = \int_{\; 0}^{\; 1} {N(v(x))dx \le 1}, \rho (u, M) = \int_{\; 0}^{\; 1} {M(u(x))dx \le 1} $及$N$函数的性质(见文献[2])容易看出, $ u(x) $和$ v(x) $在区间$[0,1]$上一定是几乎处处有界的, 故

$ \mathop {\sup }\limits_{\rho (u, M) \le 1} ||fu||_M ||v||_N \le C||f||_M, $

从而$ \left\| {f^2 } \right\|_M \le C\left\| f \right\|_M^2. $

引理5  对于$ f\left(x \right) \in L_M^*, $记

$ K_{n, 1} \left( {f, x} \right) = \left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {f\left( t \right) - f\left( x \right)} \right)dt} }, \\ K_{n, 2} \left( {f, x} \right) = \left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {f\left( t \right) - f\left( x \right)} \right)^2 dt} }, $

则

$ \left\| {K_{n, 1} \left( {f} \right)} \right\|_M \le C\left( {\omega _1 \left( {f, \frac{1}{n}} \right)_M + \omega _2 \left( {f, \frac{1}{{\sqrt n }}} \right)_M } \right), \\ \left\| {K_{n, 2} \left( {f} \right)} \right\|_M \le C\left( {\omega _1 \left( {f, \frac{1}{n}} \right)^2 _M + \omega _2 \left( {f, \frac{1}{{\sqrt n }}} \right)^2 _M } \right). $

证  对于$ g\left(x \right) $满足$ g^{\left({r -1} \right)} \left(x \right) \in AC\left[{0, 1} \right] $且$ g^{\left(r \right)} \left(x \right) \in L_M^* \left[{0, 1} \right], $

$ K_{n, 1} \left( {f, x} \right) \le \left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {f\left( t \right) - g\left( t \right)} \right)dt} } \\ + \left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {g\left( t \right) - g\left( x \right)} \right)dt} } \\ + \left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {g\left( x \right) - f\left( t \right)} \right)dt} } \\ =: I_1 + I_2 + I_3, $

其中$ I_1 = M_n \left({f -g, x} \right). $由引理3知$ \left\| {I_1 } \right\|_M \le \left\| {f -g} \right\|_M, $ $ I_3 = \left({f -g} \right)M_n \left({1, x} \right), $由引理2知$ \left\| {I_3 } \right\|_M = \left\| {f -g} \right\|_M.$由于

$ g\left( t \right) - g\left( x \right) = g'\left( x \right)\left( {t - x} \right) + \int_x^t {\left( {t - x} \right)g''\left( u \right)du}. $

故

$ I_2 \le \left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {g'\left( x \right)\left( {t - x} \right)} \right)dt} } \\ + \left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {\int_x^t {\left( {t - x} \right)g''\left( u \right)du} } \right)dt} } \\ \le \left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {g'\left( x \right)\left( {t - x} \right)} \right)dt} }\\ + \left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {\left( {t - x} \right)^2 \theta _{g''} \left( x \right)} \right)dt} } \\ = M_n \left( {t - x, x} \right)g'\left( x \right) + M_n \left( {\left( {t - x} \right)^2, x} \right)\theta _{g''} \left( x \right) \\ \le \frac{C}{n}g'\left( x \right) + \frac{C}{n}\theta _{g''} \left( x \right), $

其中$ \theta _{g''} \left(x \right) $是$ g''\left(x \right) $的Hardy-Littlewood极大函数.由文献[6]知$ \left\| {\theta _{g'} \left(x \right)} \right\|_M \le C\left\| {g'} \right\|_M. $故$ \left\| {I_2 } \right\|_M \le \frac{C}{n}\left\| {g'} \right\|_M + \frac{C}{n}\left\| {g''} \right\|_M. $因此

$ \left\| {K_{n, 1} \left( {f} \right)} \right\|_M \le \left\| {I_1 } \right\|_M + \left\| {I_2 } \right\|_M + \left\| {I_3 } \right\|_M \\ \le \left\| {f - g} \right\|_M + \frac{C}{n}\left\| {g'} \right\|_M + \frac{C}{n}\left\| {g''} \right\|_M + \left\| {f - g} \right\|_M. $

故

$ \left\| {K_{n, 1} \left( {f} \right)} \right\|_M \le C\left( {K_1 \left( {f, \frac{1}{n}} \right)_M + K_2 \left( {f, \frac{1}{{\sqrt n }}} \right)_M } \right) \\ \le C\left( {\omega _1 \left( {f, \frac{1}{n}} \right)_M + \omega _2 \left( {f, \frac{1}{{\sqrt n }}} \right)_M } \right), \\ K_{n, 2} \left( {f, x} \right) \le 3\left[{\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {f\left( t \right)-g\left( t \right)} \right)^2 dt} } } \right. \\ + \left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {g\left( t \right) -g\left( x \right)} \right)^2 dt} } \\ +\left. {\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {g\left( x \right)-f\left( t \right)^2 } \right)dt} } } \right] \\ = : I'_1 + I'_2 + I'_3, $

其中$ I'_1 = M_n \left({\left({f -g} \right)^2, x} \right). $由引理2和引理4知$ \left\| {I'_1 } \right\|_M \le \left\| {\left({f -g} \right)^2 } \right\|_M \le C\left\| {f -g} \right\|^2 _M, $ $ I'_3 = \left({f -g} \right)^2 M_n \left({1, x} \right), $由引理1和引理4知$ \left\| {I'_3 } \right\|_M = \left\| {\left({f -g} \right)^2 } \right\|_M \le C\left\| {f -g} \right\|^2 _M. $利用

$ g\left( t \right) - g\left( x \right) = g'\left( x \right)\left( {t - x} \right) + \int_x^t {\left( {t - x} \right)g''\left( u \right)du}, $

可得出

$ \left( {g\left( t \right) - g\left( x \right)} \right)^2 \le 2\left[{g'\left( x \right)\left( {t-x} \right)} \right]^2 + 2\left[{\int_x^t {\left( {t-x} \right)g''\left( u \right)du} } \right]^2, $

从而

$ I'_2 \le 2\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left[{g'\left( x \right)\left( {t-x} \right)} \right]^2 dt} } \\ + 2\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {\int_x^t {\left( {t - x} \right)g''\left( u \right)du} } \right)^2 dt} } \\ \le 2\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left[{g'\left( x \right)\left( {t-x} \right)} \right]^2 dt} }\\ + 2\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {\left( {t - x} \right)^2 \theta _{g''} \left( x \right)} \right)^2 dt} } \\ = 2\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {g'\left( x \right)} \right)^2 \left( {t - x} \right)^2 dt} } \\ + 2\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {t - x} \right)^4 \theta ^2 _{g''} \left( x \right)dt} } \\ \le 2M_n \left( {\left( {t - x} \right)^2, x} \right)\left( {g'\left( x \right)} \right)^2 \\ + 2\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {t - x} \right)^2 \theta ^2 _{g''} \left( x \right)dt} } \\ \le 2M_n \left( {\left( {t - x} \right)^2, x} \right)\left( {g'\left( x \right)} \right)^2 + 2M_n \left( {\left( {t - x} \right)^2, x} \right)\theta ^2 _{g''} \left( x \right) \\ \le \frac{C}{n}\left( {g'\left( x \right)} \right)^2 + \frac{C}{n}\theta ^2 _{g''} \left( x \right), $

此处用了$ t -x \le 1. $故$ \left\| {I'_2 } \right\|_M \le \frac{C}{n}\left\| {g'} \right\|_M^2 + \frac{C}{n}\left\| {g''} \right\|_M^2. $从而

$ \left\| {K_{n, 2} \left( {f} \right)} \right\|_M \le C\left\| {f - g} \right\|_M^2 + \frac{C}{n}\left\| {g'} \right\|_M^2 + \frac{C}{n}\left\| {g''} \right\|_M^2 + C\left\| {f - g} \right\|_M^2, $

由此可得

$ \left\| {K_{n, 2} \left( {f} \right)} \right\|_M \le C\left( {K_1 \left( {f, \frac{1}{n}} \right)_M^2 + K_2 \left( {f, \frac{1}{{\sqrt n }}} \right)_M^2 } \right) \\ \le C\left( {\omega _1 \left( {f, \frac{1}{n}} \right)_M^2 + \omega _2 \left( {f, \frac{1}{{\sqrt n }}} \right)_M^2 } \right). $

定理的证明  对于给定的$ \varepsilon > 0, $对任意的$ f\left(x \right) \in L_M^*, f\left(x \right) \ge 0 $且不恒为0.令$ f_\varepsilon \left(x \right) = f\left(x \right) + \varepsilon, $则$ f_\varepsilon \left(x \right) \ge \varepsilon. $取$ P_n \left(x \right) = M_n \left({\frac{1}{{f_\varepsilon }}, x} \right), $显然$ P_n \left(x \right) \in \prod _n \left(+ \right). $由Cauchy-Schwarz不等式知$ \frac{1}{{P_n \left(x \right)}} \le M_n \left({f_\varepsilon, x} \right). $令$ T_1 = \{ {x \in [{0, 1}], \frac{1}{{P_n (x)}} \ge f_\varepsilon (x)} \}, T_2 = [{0, 1}]\backslash T_1. $对任意的$ x \in T_1 $, 有

$ 0 \le \left\| {\frac{1}{{P_n }} - f_\varepsilon } \right\|_{M\left[{T_1 } \right]} \le \left\| {M_n \left( {f_\varepsilon } \right) - f_\varepsilon } \right\|_{M\left[{T_1 } \right]} \\ = \left\| {\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {f_\varepsilon \left( t \right) - f_\varepsilon \left( x \right)} \right)dt} } } \right\|_{M\left[{T_1 } \right]} \\ = \left\| {K_{n, 1} \left( {f_\varepsilon } \right)} \right\|_{M\left[{T_1 } \right]} \le C\left( {\omega _1 \left( {f_\varepsilon, \frac{1}{n}} \right)_{M\left[{T_1 } \right]} + \omega _2 \left( {f_\varepsilon, \frac{1}{{\sqrt n }}} \right)_{_{M\left[{T_1 } \right]} } } \right) \\ \le C\left( {\omega _1 \left( {f_\varepsilon, \frac{1}{n}} \right)_M + \omega _2 \left( {f_\varepsilon, \frac{1}{{\sqrt n }}} \right)_M } \right). $

对于$ x \in T_2 $, 有

$ 0 \le \left\| {f_\varepsilon - \frac{1}{{P_n }}} \right\|_{M\left[{T_2 } \right]} = \left\| {\frac{{f_\varepsilon \left( x \right)}}{{P_n \left( x \right)}}\left( {P_n \left( x \right) - \frac{1}{{f_\varepsilon \left( x \right)}}} \right)} \right\|_{M\left[{T_2 } \right]}\\ \le \left\| {f_\varepsilon \left( x \right)f_\varepsilon \left( x \right)\left[{M_n \left( {\frac{1}{{f_\varepsilon }}, x} \right)-\frac{1}{{f_\varepsilon \left( x \right)}}} \right]} \right\|_{M\left[{T_2 } \right]} \\ \le \left\| {f_\varepsilon \left( x \right)f_\varepsilon \left( x \right)\left[{\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {\frac{1}{{f_\varepsilon \left( t \right)}}- \frac{1}{{f_\varepsilon \left( x \right)}}} \right)dt} } } \right]} \right\|_{M\left[{T_2 } \right]}\\ \le \left\| {\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\frac{{f_\varepsilon \left( x \right)}}{{f_\varepsilon \left( t \right)}}\left( {f_\varepsilon \left( x \right) - f_\varepsilon \left( t \right)} \right)dt} } } \right\|_{M\left[{T_2 } \right]} \\ = \left\| {\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\frac{{f_\varepsilon ^2 \left( x \right)}}{{f_\varepsilon \left( t \right)}} - f_\varepsilon \left( x \right) + f_\varepsilon \left( x \right) - f_\varepsilon \left( x \right) + f_\varepsilon \left( t \right) - f_\varepsilon \left( t \right)dt} } } \right\|_{M\left[{T_2 } \right]} \\ \le \left\| {\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\frac{{\left( {f_\varepsilon \left( x \right) - f_\varepsilon \left( t \right)} \right)^2 }}{{f_\varepsilon \left( t \right)}} + \left( {f_\varepsilon \left( x \right) - f_\varepsilon \left( t \right)} \right)dt} } } \right\|_{M\left[ {T_2 } \right]}\\ \le \frac{1}{\varepsilon }\left\| {\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {f_\varepsilon \left( x \right) - f_\varepsilon \left( t \right)} \right)^2 dt} } } \right\|_{M\left[{T_2 } \right]} \\ + \left\| {\left( {n + 1} \right)\sum\limits_{k = 0}^n {P_{nk} \left( x \right)\int_0^1 {P_{nk} \left( t \right)\left( {f_\varepsilon \left( x \right) - f_\varepsilon \left( t \right)} \right)dt} } } \right\|_{M\left[{T_2 } \right]}\\ \le \frac{1}{\varepsilon }K_{n, 2} \left( {f_\varepsilon, x} \right) + K_{n, 1} \left( {f_\varepsilon, x} \right) \\ \le \frac{1}{\varepsilon }C\left( {\omega _1 \left( {f_\varepsilon, \frac{1}{n}} \right)_M^2 + \omega _2 \left( {f_\varepsilon, \frac{1}{{\sqrt n }}} \right)_M^2 } \right) + C\left( {\omega _1 \left( {f_\varepsilon, \frac{1}{n}} \right)_M + \omega _2 \left( {f_\varepsilon, \frac{1}{{\sqrt n }}} \right)_M } \right) . $

取$ \varepsilon = \left({\omega _1 \left({f_\varepsilon, \frac{1}{n}} \right)_M + \omega _2 \left({f_\varepsilon, \frac{1}{{\sqrt n }}} \right)_M } \right). $则

$ \left\| {f_\varepsilon - \frac{1}{{P_n }}} \right\|_{M\left[{T_2 } \right]} \le C\left( {\omega _1 \left( {f_\varepsilon, \frac{1}{n}} \right)_M + \omega _2 \left( {f_\varepsilon, \frac{1}{{\sqrt n }}} \right)_M } \right). $

从而

$ \left\| {f_\varepsilon - \frac{1}{{P_n }}} \right\|_M \le \left\| {f_\varepsilon - \frac{1}{{P_n }}} \right\|_{M\left[ {T_1 } \right]} + \left\| {f_\varepsilon - \frac{1}{{P_n }}} \right\|_{M\left[{T_2 } \right]} \le C\left( {\omega _1 \left( {f_\varepsilon, \frac{1}{n}} \right)_M + \omega _2 \left( {f_\varepsilon, \frac{1}{{\sqrt n }}} \right)_M } \right). $

因此

$ \left\| {f - \frac{1}{{P_n }}} \right\|_M \le \left\| {f - f_\varepsilon } \right\|_M + \left\| {f_\varepsilon - \frac{1}{{P_n }}} \right\|_M \\ \le \varepsilon + C\left( {\omega _1 \left( {f_\varepsilon, \frac{1}{n}} \right)_M + \omega _2 \left( {f_\varepsilon, \frac{1}{{\sqrt n }}} \right)_M } \right) \le C\left( {\omega _1 \left( {f, \frac{1}{n}} \right)_M + \omega _2 \left( {f, \frac{1}{{\sqrt n }}} \right)_M } \right). $