Criterion of the best non-symmetric approximant for multivariable functions in space $L_{1, p_2,...,p_n}$

Abstract

The criterion of the best non-symmetric approximant for $n$-variable functions in the space $L_{1, p_2,...,p_n}$ $(1

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空间$L_{1, p_2，…，p_n}$中多变量函数的最佳非对称逼近准则

空间$L_{1, p_2，…中$n$变量函数的最佳非对称逼近准则, p_n} $ $ (1 < p_i < + \ infty, i = 2、3、…,n)与(\α,β\)美元美元规范$ $ f \ \ | | _ {1 p_2…,p_n; \α,β\}=左\ [int \ \ limits_ {an} ^ {b_n} \ cdots \离开[int \ \ limits_ {a₂}^ {b_2} \离开[int \ \ limits_ {a_1} ^ {b_1} | f (x) | _{\α,β\}dx_1 \右]^ {p_2} dx_2 \右]^{\压裂{p_3} {p_2}} \ cdots dx_n \右]^{\压裂{1}{p_n}}, $ $, $ 0 < \α,β\ < \ infty $,$ f {+} (x) = \ \马克斯\ \}{f (x), 0, f {-} (x) = \ \马克斯\ {- f (x), 0 \}, $ $ \ mathrm{胡志明市}_{\α,β\}f (x) = \α\ cdot \ mathrm f{+}{胡志明市}(x) -β\ \ cdot \ mathrm f{-}{胡志明市}(x) f $ $ | | _{\α,β\}= f{+} \α\ cdot +β\ \ cdot f {-} = f (x) \ cdot \ mathrm{胡志明市}_{\α,β\}f (x),美元了。证明了如果$P_m=\sum\limits_{k=1}^{m}c_k\varphi_k$，其中$\{\varphi_k\}_{k=1}^m$是$L_{1,p_2，…，p_n}$， $c_k$都是实数，那么多项式$P_m^{\ast}$是$f$在空间$L_{1,p_2，…中最好的$(\alpha，\beta)$-逼近。, p_n} $ $ (1 < p_i < \ infty $, $ i = 2, 3,…,n)美元,当且仅当,对于任何多项式P_m $ $美元\ int \ limits_K P_m \ cdot F_0 ^ {\ ast} dx \ leq \ int \ limits_ {an} ^ {b_n}…\int \limits_{a_2}^{b_2}\int \limits_{e_{x_2，…, x_n}} | P_m | _{\β\α}dx_1 \ cdot \ operatorname * {ess \,一口}_ {x_1 \在(a_1、b_1)} | F_0 ^ {\ ast} | _{\压裂{1}{\α}\压裂{1}{\β}}dx_2……dx_n, $ $ $ K = (a_1、b_1) \ \ ldots \乘以(an, b_n) $ $ e_ {x_2,……x_n} = \ {x_1 \ [a_1、b_1): f-P_m ^ {\ ast} = 0 \}, F_0美元$ $ ^ {\ ast} = \压裂{| R_m ^ {\ ast} | _ {1;\alpha，\beta}^{p_2-1}|R_m^{\ast}|_{1,p_2;\alpha，\beta}^{p_3-p_2}\cdot…\ cdot | R_m ^ {\ ast} | _ {1 p_2…,p_ {n};\α,β\}^ {p_n-p_ {n}} \ mathrm{胡志明市}_{\α,β\}R_m ^ {\ ast}} {| | R_m ^ {\ ast} | | _ {1 p_2…,p_n;\α,β\}^ {p_n-1}}, f $ $ | | _ {\ ldots p_k, p_i; \α,β\}=左\ [int \ \ limits_ {ai} ^ {b_i}左\ ldots \ [int \ \ limits_{现代{k + 1}} ^ {b_ {k + 1}}左\ [int \ \ limits_ {a_k} ^ {b_k} |女| _{\α,β\}^ {p_k} dx_k \右]^{\压裂{p_ {k + 1}} {p_k}} dx_ {k + 1} \右]^{\压裂{p_ {k + 2}} {p_ {k + 1}}} \ ldots dx_i正确\]^{\压裂{1}{p_i}}, $ $(1美元\ leq k <我\ leq n),美元R_m ^ {\ ast} = f-P_m ^ {\ ast} $。当$\ α =\ β =1$时，此准则是对已知的二元函数的Smirnov准则的推广。

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