空间$L_{1, p_2，…，p_n}$中多变量函数的最佳非对称逼近准则

Q4 Mathematics Researches in Mathematics Pub Date : 2021-12-30 DOI:10.15421/242109

M. Tkachenko, V. M. Traktynska

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Traktynska","doi":"10.15421/242109","DOIUrl":null,"url":null,"abstract":"The criterion of the best non-symmetric approximant for $n$-variable functions in the space $L_{1, p_2,...,p_n}$ $(1<p_i<+\\infty , i=2,3,...,n)$ with $(\\alpha ,\\beta )$-norm$$\\|f\\|_{1,p_2,...,p_n;\\alpha,\\beta}=\\left[\\int\\limits_{a_n}^{b_n}\\cdots\\left[\\int\\limits_{a_2}^{b_2}\\left[\\int\\limits_{a_1}^{b_1} |f(x)|_{\\alpha,\\beta} dx_1\\right]^{p_2} dx_2\\right]^{\\frac{p_3}{p_2}}\\cdots dx_n\\right]^{\\frac{1}{p_n}},$$where $0<\\alpha,\\beta<\\infty$, $\\ f_{+}(x)=\\max\\{f(x),0\\},\\ f_{-}(x)=\\max\\{-f(x),0\\},$ $\\mathrm{sgn}_{\\alpha,\\beta}f(x)=\\alpha\\cdot\\mathrm{sgn}f_{+}(x)-\\beta\\cdot\\mathrm{sgn}f_{-}(x),$ $|f|_{\\alpha,\\beta}=\\alpha \\cdot f_{+}+\\beta \\cdot f_{-} =f(x)\\cdot \\mathrm{sgn}_{\\alpha,\\beta}f(x)$, is obtained in the article.It is proved that if $P_m=\\sum\\limits_{k=1}^{m}c_k\\varphi_k$, where $\\{\\varphi_k\\}_{k=1}^m$ is a linearly independent system functions of $L_{1,p_2,...,p_n}$, $c_k$ are real numbers, then the polynomial $P_m^{\\ast}$ is the best $(\\alpha ,\\beta )$-approximant for $f$ in the space $L_{1,p_2,...,p_n}$ $(1<p_i<\\infty $, $i=2,3,...,n)$, if and only if, for any polynomial $P_m$$$\\int \\limits_K P_m\\cdot F_0^{\\ast}dx \\leq \\int \\limits_{a_n}^{b_n}...\\int \\limits_{a_2}^{b_2}\\int \\limits_{e_{x_2,...,x_n}}|P_m|_{\\beta , \\alpha}dx_1 \\cdot \\operatorname *{ess \\,sup}_ {x_1 \\in [a_1,b_1]} |F_0^{\\ast}|_{\\frac{1}{\\alpha },\\frac{1}{\\beta }} dx_2...dx_n,$$where $K=[a_1,b_1]\\times \\ldots\\times [a_n,b_n],$ $e_{x_2,...,x_n}=\\{ x_1\\in [a_1,b_1] : f-P_m^{\\ast}=0\\},$$$F_0^{\\ast}=\\frac{|R_m^{\\ast}|_{1; \\alpha ,\\beta }^{p_2-1}|R_m^{\\ast}|_{1,p_2; \\alpha ,\\beta }^{p_3-p_2}\\cdot ... \\cdot |R_m^{\\ast}|_{1,p_2,...,p_{n-1}; \\alpha ,\\beta }^{p_n-p_{n-1}}\\mathrm{sgn}_{\\alpha ,\\beta} R_m^{\\ast}}{||R_m^{\\ast}||_{1,p_2,...,p_n; \\alpha ,\\beta}^{p_n-1}},$$|f|_{p_k,\\ldots,p_i;\\alpha,\\beta}=\\left[\\int\\limits_{a_i}^{b_i}\\ldots\\left[ \\int\\limits_{a_{k+1}}^{b_{k+1}}\\left[\\int\\limits_{a_k}^{b_k}|f|_{\\alpha,\\beta}^{p_k}dx_k\\right]^{\\frac{p_{k+1}}{p_k}}dx_{k+1} \\right]^{\\frac{p_{k+2}}{p_{k+1}}}\\ldots dx_i \\right]^{\\frac{1}{p_i}},$$($1\\leq k<i\\leq n$), $R_m^{\\ast}=f-P_m^{\\ast}$.This criterion is a generalization of the known Smirnov's criterion for functions of two variables, when $\\alpha =\\beta =1$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Criterion of the best non-symmetric approximant for multivariable functions in space $L_{1, p_2,...,p_n}$\",\"authors\":\"M. Tkachenko, V. M. Traktynska\",\"doi\":\"10.15421/242109\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The criterion of the best non-symmetric approximant for $n$-variable functions in the space $L_{1, p_2,...,p_n}$ $(1<p_i<+\\\\infty , i=2,3,...,n)$ with $(\\\\alpha ,\\\\beta )$-norm$$\\\\|f\\\\|_{1,p_2,...,p_n;\\\\alpha,\\\\beta}=\\\\left[\\\\int\\\\limits_{a_n}^{b_n}\\\\cdots\\\\left[\\\\int\\\\limits_{a_2}^{b_2}\\\\left[\\\\int\\\\limits_{a_1}^{b_1} |f(x)|_{\\\\alpha,\\\\beta} dx_1\\\\right]^{p_2} dx_2\\\\right]^{\\\\frac{p_3}{p_2}}\\\\cdots dx_n\\\\right]^{\\\\frac{1}{p_n}},$$where $0<\\\\alpha,\\\\beta<\\\\infty$, $\\\\ f_{+}(x)=\\\\max\\\\{f(x),0\\\\},\\\\ f_{-}(x)=\\\\max\\\\{-f(x),0\\\\},$ $\\\\mathrm{sgn}_{\\\\alpha,\\\\beta}f(x)=\\\\alpha\\\\cdot\\\\mathrm{sgn}f_{+}(x)-\\\\beta\\\\cdot\\\\mathrm{sgn}f_{-}(x),$ $|f|_{\\\\alpha,\\\\beta}=\\\\alpha \\\\cdot f_{+}+\\\\beta \\\\cdot f_{-} =f(x)\\\\cdot \\\\mathrm{sgn}_{\\\\alpha,\\\\beta}f(x)$, is obtained in the article.It is proved that if $P_m=\\\\sum\\\\limits_{k=1}^{m}c_k\\\\varphi_k$, where $\\\\{\\\\varphi_k\\\\}_{k=1}^m$ is a linearly independent system functions of $L_{1,p_2,...,p_n}$, $c_k$ are real numbers, then the polynomial $P_m^{\\\\ast}$ is the best $(\\\\alpha ,\\\\beta )$-approximant for $f$ in the space $L_{1,p_2,...,p_n}$ $(1<p_i<\\\\infty $, $i=2,3,...,n)$, if and only if, for any polynomial $P_m$$$\\\\int \\\\limits_K P_m\\\\cdot F_0^{\\\\ast}dx \\\\leq \\\\int \\\\limits_{a_n}^{b_n}...\\\\int \\\\limits_{a_2}^{b_2}\\\\int \\\\limits_{e_{x_2,...,x_n}}|P_m|_{\\\\beta , \\\\alpha}dx_1 \\\\cdot \\\\operatorname *{ess \\\\,sup}_ {x_1 \\\\in [a_1,b_1]} |F_0^{\\\\ast}|_{\\\\frac{1}{\\\\alpha },\\\\frac{1}{\\\\beta }} dx_2...dx_n,$$where $K=[a_1,b_1]\\\\times \\\\ldots\\\\times [a_n,b_n],$ $e_{x_2,...,x_n}=\\\\{ x_1\\\\in [a_1,b_1] : f-P_m^{\\\\ast}=0\\\\},$$$F_0^{\\\\ast}=\\\\frac{|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-1}; \\\\alpha ,\\\\beta }^{p_n-p_{n-1}}\\\\mathrm{sgn}_{\\\\alpha ,\\\\beta} R_m^{\\\\ast}}{||R_m^{\\\\ast}||_{1,p_2,...,p_n; \\\\alpha ,\\\\beta}^{p_n-1}},$$|f|_{p_k,\\\\ldots,p_i;\\\\alpha,\\\\beta}=\\\\left[\\\\int\\\\limits_{a_i}^{b_i}\\\\ldots\\\\left[ \\\\int\\\\limits_{a_{k+1}}^{b_{k+1}}\\\\left[\\\\int\\\\limits_{a_k}^{b_k}|f|_{\\\\alpha,\\\\beta}^{p_k}dx_k\\\\right]^{\\\\frac{p_{k+1}}{p_k}}dx_{k+1} \\\\right]^{\\\\frac{p_{k+2}}{p_{k+1}}}\\\\ldots dx_i \\\\right]^{\\\\frac{1}{p_i}},$$($1\\\\leq k<i\\\\leq n$), $R_m^{\\\\ast}=f-P_m^{\\\\ast}$.This criterion is a generalization of the known Smirnov's criterion for functions of two variables, when $\\\\alpha =\\\\beta =1$.\",\"PeriodicalId\":52827,\"journal\":{\"name\":\"Researches in Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-12-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Researches in Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.15421/242109\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Researches in Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.15421/242109","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}

引用次数: 0

摘要

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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