Criterion of the best non-symmetric approximant for multivariable functions in space L1,p2,...,pn
Abstract
‖
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.
Keywords
MSC 2020
Full Text:
PDFReferences
Smirnov G.S. "General form of linear functional and criterion of polynomial of the best approximation in spaces with mixed integral metric", Ukrainian Math. J., 1973; 25(1): pp. 134-138. (in Russian) doi:10.1007/BF01085405
Smirnov G.S. "Criterion of polynomial of the best approximation in spaces L_{p;1}, L_{1;q}", Ukrainian Math. J., 1973; 25(3): pp. 415-419. (in Russian) doi:10.1007/BF01091890
Traktynska V.M. "Characterization of the best integral approximant of multivariable functions", Res. Math., 2007; 15: pp. 134-136. (in Russian) doi:10.15421/240719
Traktynska V.M., Tkachenko M.Ye. "Criterion of the best non-symmetric approximant for multivariable functions in spaces L_{p_1,...,p_n}", Res. Math., 2015; 23: pp. 90-97. (in Ukrainian) doi:10.15421/241511
Kostyuk O.D., Traktynska V.M., Tkachenko M.Ye. "Criterion of the best approximant for multivariable functions in spaces L_{1,p_2,...,p_n} and L_{p_1,...,p_{n-1},1}", Res. Math., 2016; 24: pp. 44-51. (in Ukrainian) doi:10.15421/241608
DOI: https://doi.org/10.15421/242109
Refbacks
- There are currently no refbacks.
Copyright (c) 2021 M.Ye. Tkachenko, V.M. Traktynska

This work is licensed under a Creative Commons Attribution 4.0 International License.