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 $L_{1, p_2,...,p_n}$ $(1<p_i<+\infty , i=2,3,...,n)$ 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 $\alpha =\beta =1$.