ABSTRACT

In this paper we investigate weighted polynomial approximations with several variables. Our study relates to the approximation for $Wf\in L^p\left({ℝ}^s\right)$ by weighted polynomial. Then we will give some results relating to the Lagrange interpolation, the best approximation, the Markov-Bernstein inequality and the Nikolskii-type inequality.

Keywords:

Weighted Polynomial Approximations, the Lagrange Interpolation, the Best of Approximation, Inequalities

1. Introduction

Let ${ℝ}^s:=ℝ\times ℝ\times \cdots \times ℝ$ ( $s$ times, $s\ge 1$ integer) be the direct product space, and let $W\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_s\right)\mathrm{<mo>\; :\; </mo>}=w_1\left(x_1\right)w_2\left(x_2\right)\cdots w_s\left(x_s\right)$ , where $w_i\left(x_i\right)\ge 0$ be even weight functions. We suppose that for every nonnegative integer n,

${\displaystyle {\int }_0^{\infty }}{x}^n{w}_i\left(x\right)\text{d}x<\infty ,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}n=0,1,2,\cdots ,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}i=1,2,\cdots ,s.$

In this paper we will study to approximate the real-valued weighted function $\left(Wf\right)\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_s\right)$ by weighted polynomials $\left(WP\right)\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_s\right)$ , where $P\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_s\right)\in {\mathcal{P}}_{n\mathrm{,}n\mathrm{,}\cdots \mathrm{,}n}\left({ℝ}^s\right)$ . Here, ${\mathcal{P}}_{n\mathrm{,}n\mathrm{,}\cdots \mathrm{,}n}\left({ℝ}^s\right)\left(=\mathrm{<mo>\; :\; </mo>}{\mathcal{P}}_{n\mathrm{<mo>\; ;\; </mo>}s}\left({ℝ}^s\right)\right)$ means a class of all polynomials with at most n-degree for each variable $x_i,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,s$ . We need to define the norms. Let $0<p\le \infty$ , and let $f\mathrm{<mo>\; :\; </mo>}{ℝ}^s\to ℝ$ be measurable. Then we define

${\Vert Wf\Vert }_{L^p\left({ℝ}^s\right)}:=\{\begin{array}{l}{\left[{\displaystyle {\int }_{-\infty }^{\infty }}\cdots {\displaystyle {\int }_{-\infty }^{\infty }}{\left|\left(Wf\right)\left(x_1,\cdots ,x_s\right)\right|}^p\text{d}{x}_1\cdots \text{d}{x}_s\right]}^{1/p},\text{if}0<p<\infty ;\\ \underset{\left(x_1,\cdots ,x_s\right)\in {ℝ}^s}{\sup }\left|\left(Wf\right)\left(x_1,\cdots ,x_s\right)\right|,\text{if}p=\infty .\end{array}$

We assume that for $0<p\le \infty$ the integral is independent of the order of integration with respect to each $x_i,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,s$ . When ${\Vert Wf\Vert }_{L^p\left({ℝ}^s\right)}<\infty$ , we write $Wf\in L^p\left({ℝ}^s\right)$ . If $p=\infty$ , we require that f is continuous and $li{m}_{\left|X\right|\to \infty }W\left(X\right)f\left(X\right)=0$ , where $\left|X\right|=\left|\left(x_1,\cdots ,x_s\right)\right|=\max \left|x_i\right|;i=1,2,\cdots ,s$ . Then we write $Wf\in C_0\left({ℝ}^s\right)$ .

Our purpose in this paper is to approximate the weighted function $Wf\in L^p\left({ℝ}^s\right)$ by weighted polynomials $WP\mathrm{<mo>\; ;\; </mo>\; <mi>\; </mi>}P\in {\mathcal{P}}_{n\mathrm{<mo>\; ;\; </mo>}s}\left({ℝ}^s\right)$ . In Section 2, we give a class of the weights which are treated in this paper. In Section 3, we state our main theorems. First, we consider the Lagrange interpolation polynomials. Next, we give the necessary and sufficient conditions for the best approximation. In Sections 4 and 5, we will prove theorems.

2. Class of Weight Functions and Preliminaries

Throughout the paper $C\mathrm{,}{C}_1\mathrm{,}{C}_2\mathrm{,}\cdots$ denote positive constants independent of $n\mathrm{,\; <mi>\; </mi>}x\mathrm{,}t$ or polynomials $P\left(x\right)$ . The same symbol does not necessarily denote the same constant in different occurrences. Let $f\left(x\right)~g\left(x\right)$ mean that there exists a constant $C>0$ such that $C^{-1}f\left(x\right)\le g\left(x\right)\le Cf\left(x\right)$ holds for all $x\in I$ , where $I\subset ℝ$ is a subset.

We say that $f\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}ℝ\to \left[\mathrm{0,}\infty \right)$ is quasi-increasing if there exists $C>0$ such that $f\left(x\right)\le Cf\left(y\right)$ for $0<x<y$ . Hereafter we consider following weights.

Definition 2.1. Let $Q\mathrm{<mo>\; :\; </mo>\; <mi>\; </mi>}ℝ\to \left[\mathrm{0,}\infty \right)$ be a continuous and even function, and satisfy the following properties:

(a) $Q^{\prime }\left(x\right)$ is continuous in $ℝ$ , with $Q\left(0\right)=0$ .

(b) $Q^{″}\left(x\right)$ exists and is positive in $ℝ\backslash \left\{0\right\}$ .

(c) ${\lim }_{x\to \infty }Q\left(x\right)=\infty .$

(d) The function

$T\left(x\right):=\frac{x{Q}^{\prime }\left(x\right)}{Q\left(x\right)},\mathrm{<mi>\; </mi>\; <mi>\; </mi>}x\ne 0$

is quasi-increasing in $\left(\mathrm{0,}\infty \right)$ , with

$T\left(x\right)\ge \Lambda >1,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}x\in ℝ\backslash \left\{0\right\}.$

(e) There exists $C_1>0$ such that

$\frac{Q^{″}\left(x\right)}{\left|Q^{\prime }\left(x\right)\right|}\le C_1\frac{\left|Q^{\prime }\left(x\right)\right|}{Q\left(x\right)},\mathrm{<mi>\; </mi>\; <mi>\; </mi>}a.e\mathrm{.\; <mi>\; </mi>}x\in ℝ.$

Then we write $w=exp\left(-Q\right)\in \mathcal{F}\left(C^2\right)$ .

Moreover, if there also exists a compact subinterval $J\left(\ni 0\right)$ of $ℝ$ , and $C_2>0$ such that

$\frac{Q^{″}\left(x\right)}{\left|Q^{\prime }\left(x\right)\right|}\ge C_2\frac{\left|Q^{\prime }\left(x\right)\right|}{Q\left(x\right)}\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>}a\mathrm{.}e\mathrm{.\; <mi>\; </mi>}x\in ℝ\backslash J\mathrm{,}$

then we write $w=\exp \left(-Q\right)\in \mathcal{F}\left(C^2+\right)$ . If $T\left(x\right)$ is bounded, then the weight $w=exp\left(-Q\right)\in \mathcal{F}\left(C^2+\right)$ is called a Freud-type weight, and if $T\left(x\right)$ is unbounded, then w is called an Erdös-type weight.

Let $w\left(x\right)=exp\left(-Q\left(x\right)\right)\in \mathcal{F}\left(C^2+\right)$ , $0<\lambda <\left(m+2\right)/\left(m+1\right)$ and $m\ge 1$ be an integer. Then we write $w\in {\mathcal{F}}_{\lambda }\left(C^{m+2}+\right)$ if $Q$ is $C^{m+2}$ -class and there exist $C\ge 1$ and $K\ge 1$ such that for all $\left|x\right|\ge K$ ,

$\frac{\left|Q^{\prime }\left(x\right)\right|}{Q{\left(x\right)}^{\lambda }}\le C$ (2.1)

and

$\left|\frac{Q^{″}\left(x\right)}{Q^{\prime }\left(x\right)}\right|~\left|\frac{Q^{\left(k+1\right)}\left(x\right)}{Q^{\left(k\right)}\left(x\right)}\right|$

for every $k=2,\cdots ,m$ and also

$\left|\frac{Q^{\left(m+2\right)}\left(x\right)}{Q^{\left(m+1\right)}\left(x\right)}\right|\le \left|\frac{Q^{\left(m+1\right)}\left(x\right)}{Q^{\left(m\right)}\left(x\right)}\right|\mathrm{.}$

Specific examples are shown in the following:

Example 2.2 (cf. [1] [2] ). (1) If an exponential $Q\left(x\right)$ satisfies

$1<{\Lambda }_1\le \frac{{\left(x{Q}^{\prime }\left(x\right)\right)}^{\prime }}{Q^{\prime }\left(x\right)}\le {\Lambda }_2,$

where ${\Lambda }_i,\mathrm{<mi>\; </mi>}i=1,2$ are constants, then we call $w=\exp \left(-Q\left(x\right)\right)$ the Freud weight. The class $\mathcal{F}\left(C^2+\right)$ contains the Freud weights.

(2) For $\alpha >1,\mathrm{<mi>\; </mi>}l\ge 1$ we define

$Q\left(x\right)=Q_{l;\alpha }\left(x\right)={\exp }_l\left({\left|x\right|}^{\alpha }\right)-{\exp }_l\left(0\right),$

where ${\exp }_l\left(x\right)=\exp \left(\exp \left(\exp \cdots \exp x\right)\cdots \right)\mathrm{<mi>\; </mi>}\left(l\mathrm{<mi>\; </mi>}\text{times}\right)$ . Moreover, we define

$Q_{l;\alpha ,m}\left(x\right)={\left|x\right|}^m\left\{{\exp }_l\left({\left|x\right|}^{\alpha }\right)-{\alpha }^{*}{\exp }_l\left(0\right)\right\},\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\alpha +m>1,\mathrm{<mi>\; </mi>}m\ge 0,\mathrm{<mi>\; </mi>}\alpha \ge 0,$

where ${\alpha }^{*}=0$ if $\alpha =0$ , and otherwise ${\alpha }^{*}=1$ . We note that $Q_{l\mathrm{<mo>\; ;\; </mo>0,}m}$ gives a Freud-type weight, that is, $T\left(x\right)$ is bounded..

(3) We define

$Q_{\alpha }\left(x\right)={\left(1+\left|x\right|\right)}^{{\left|x\right|}^{\alpha }}-1,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\alpha >1.$

(4) Let $w=exp\left(-Q\right)\in \mathcal{F}\left(C^2+\right)$ , and let us define

${\mu }_{+}:=\underset{x\to \infty }{\lim \sup }\frac{Q^{″}\left(x\right)}{Q^{\prime }\left(x\right)}/\frac{Q^{\prime }\left(x\right)}{Q\left(x\right)},\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{\mu }_{-}:=\underset{x\to \infty }{\lim \inf }\frac{Q^{″}\left(x\right)}{Q^{\prime }\left(x\right)}/\frac{Q^{\prime }\left(x\right)}{Q\left(x\right)}.$

If ${\mu }_{+}={\mu }_{-}$ , then we say that the weight w is regular. All weights in examples (1), (2) and (3) are regular.

(5) More generally we can give the examples of weights $w\in {\mathcal{F}}_{\lambda }\left(C^{m+2}+\right)$ . If the weight w is regular and if $Q\in C^{m+2}\left(ℝ\backslash \left\{0\right\}\right)$ satisfies definition (2.1), then for the regular weights we have $w\in {\mathcal{F}}_{\lambda }\left(C^{m+2}+\right)$ (see [3] , Corollary 5.5 (5.8)).

Proposition 2.3 ( [3] , Theorem 4.2 and (4.11)). Let m be a positive integer, $0<\lambda <\left(m+2\right)/\left(m+1\right)$ and let $w=exp\left(-Q\right)\in {\mathcal{F}}_{\lambda }\left(C^{m+2}+\right)$ . Then for $\mu \mathrm{,}\nu \mathrm{,}\alpha \mathrm{,}\beta \in ℝ$ , we can construct a new weight $w_{\mu \mathrm{,}\nu \mathrm{,}\alpha \mathrm{,}\beta }\in {\mathcal{F}}_{\lambda }\left(C^{m+1}+\right)\subset \mathcal{F}\left(C^2+\right)$ such that

$T_w{\left(x\right)}^{\mu }{\left(1+x^2\right)}^{\nu }{\left(1+Q\left(x\right)\right)}^{\alpha }{\left(1+\left|Q^{\prime }\left(x\right)\right|\right)}^{\beta }w\left(x\right)~w_{\mu \mathrm{,}\nu \mathrm{,}\alpha \mathrm{,}\beta }\left(x\right)\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\text{on}\mathrm{<mi>\; </mi>}ℝ\mathrm{,}$

and for some $C\ge 1$ ,

$a_{n\mathrm{<mo>\; /\; </mo>}C}\left(w_{\mu \mathrm{,}\nu \mathrm{,}\alpha \mathrm{,}\beta }\right)\le a_n\left(w\right)\le a_{Cn}\left(w_{\mu \mathrm{,}\nu \mathrm{,}\alpha \mathrm{,}\beta }\right)\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\text{and}\mathrm{<mi>\; </mi>}{T}_{w_{\mu \mathrm{,}\nu \mathrm{,}\alpha \mathrm{,}\beta }}\left(x\right)~T_w\left(x\right)=T\left(x\right)\mathrm{,}$

where $a_n\left(w_{\mu \mathrm{,}\nu \mathrm{,}\alpha \mathrm{,}\beta }\right)$ and $a_n\left(w\right)$ are MRS-numbers for the weight $w_{\mu \mathrm{,}\nu \mathrm{,}\alpha \mathrm{,}\beta }$ or $w$ , respectively, and $T_{w_{\mu \mathrm{,}\nu \mathrm{,}\alpha \mathrm{,}\beta }}\mathrm{,\; <mi>\; </mi>}{T}_w$ are correspond for $w_{\mu \mathrm{,}\nu \mathrm{,}\alpha \mathrm{,}\beta }$ or w, respectively.

Let $\left\{p_n\right\}$ be orthonormal polynomials with respect to a weight w, that is, $p_n$ is the polynomial of degree n such that

${\displaystyle {\int }_{-\infty }^{\infty }}{p}_n\left(x\right)p_m\left(x\right)w^2\left(x\right)\text{d}x={\delta }_{mn}\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\left(\text{theKroneckerdelta}\right)\mathrm{.}$

For $1\le p\le \infty$ , we denote by $L^p\left(ℝ\right)$ the usual $L^p$ space on $ℝ$ (here for $p=\infty$ , if $wf\in L^{\infty }\left(ℝ\right)$ then we require $f$ to be continuous, and $fw$ to have limit 0 at $\pm \infty$ ). Let $w\in \mathcal{F}\left(C^2+\right)$ . We need the Mhaskar-Rakhmanov-Saff numbers (MRS numbers) $a_x$ ;

$x=\frac{2}{\text{π}}{\displaystyle {\int }_0^1}\frac{a_{x}u{Q}^{\prime }\left(a_{x}u\right)}{{\left(1-u^2\right)}^{1/2}}\text{d}u,\mathrm{<mi>\; </mi>}x>0.$

we see easily

$\underset{x\to \infty }{\lim }{a}_x=\infty \mathrm{<mi>\; </mi>\; <mi>\; </mi>}\text{and}\mathrm{<mi>\; </mi>}\underset{x\to +0}{\lim }{a}_x=0$

and

$\underset{x\to \infty }{\lim }\frac{a_x}{x}=\mathrm{0\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\text{and}\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\underset{x\to +0}{\lim }\frac{a_x}{x}=\infty .$

For $wf\in L_p\left(ℝ\right)\left(1\le p\le \infty \right)$ the degree of weighted polynomial approximation is defined by

$E_{n\mathrm{,}p}\left(w\mathrm{<mo>\; ;\; </mo>}f\right)\mathrm{<mo>\; :\; </mo>}=\underset{P\in {\mathcal{P}}_n}{inf}{\Vert w\left(f-P\right)\Vert }_{L^p\left(ℝ\right)}\mathrm{.}$

3. Main Results

Let $w_i\in \mathcal{F}\left(C^2+\right)\mathrm{,\; <mi>\; </mi>}i=\mathrm{1,2,}\cdots \mathrm{,}s$ , and let $W\left(X\right)={\displaystyle {\prod }_{i=1}^s{w}_i\left(x_i\right)}$ , where $X=\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_s\right)\in {ℝ}^s$ . Then we have the following theorem.

Theorem 3.1 ( [4] , Theorem 3.3). We suppose $w_j=\exp \left(-Q_j\right)\in {\mathcal{F}}_{\lambda }\left(C^3+\right)\mathrm{<mi>\; </mi>}\left(0<\lambda <3/2\right)$ , $j=1,2,\cdots ,s$ and let

$T_j\left(a_n^{\left(j\right)}\right)\le c{\left(\frac{n}{a_n^{\left(j\right)}}\right)}^{2/3},\mathrm{<mi>\; </mi>}j=1,2,\cdots ,s.$

If ${\displaystyle {\prod }_{i=1}^s\left\{T_i^{1/4}{w}_i\right\}}f\in C_0\left({ℝ}^s\right)$ , then there exist $P_n\in {\mathcal{P}}_{n;s}\left({ℝ}^s\right),\mathrm{<mi>\; </mi>}n=1,2,3,\cdots$ such that we have

${\Vert W\left(f-P_n\right)\Vert }_{L^{\infty }\left({ℝ}^s\right)}\to \mathrm{0\; <mi>\; </mi>\; <mi>\; </mi>}\text{as}\mathrm{<mi>\; </mi>}n\to \infty \mathrm{.}$

First, we consider the Lagrange interpolation operators. We construct the orthonormal polynomials $p_{n\mathrm{,}i}$ with respect to the weight $w_i$ for each $i=1,2,\cdots ,s$ . Let $x_{n,n,i}<x_{n-1,n,i}<\cdots <x_{1,n,i},\mathrm{<mi>\; </mi>}i=1,2,\cdots ,s$ are zeros of the orthonormal polynomial $p_{n\mathrm{,}i}$ , that is, $p_{n,i}\left(x_{k_i,n,i}\right)=0,\mathrm{<mi>\; </mi>}{k}_i=1,2,\cdots ,n$ and put $S_n=\left\{\left(x_{k_1,n,1},\cdots ,x_{k_s,n,s}\right);\mathrm{<mi>\; </mi>1}\le k_i\le n,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,s\right\}$ . Then for $Wf\in C_0\left(ℝ\right)$ we define the Lagrange interpolation polynomial on $S_n$ as

$L_n\left(f;x_1,\cdots ,x_s\right)={\displaystyle \underset{k_s=1}{\overset{n}{\sum }}}\cdots {\displaystyle \underset{k_1=1}{\overset{n}{\sum }}}f\left(x_{k_1,n,1},\cdots ,x_{k_s,n,s}\right){\displaystyle {\prod }_{i=1}^s{l}_{k_i,n,i}\left(x_i\right)},$ (3.1)

where

$l_{k_i,n,i}\left(x_i\right)=\frac{p_{n,i}\left(x_i\right)}{\left(x_i-x_{k_i,n,i}\right){p^{\prime }}_{n,i}\left(x_{k_i,n,i}\right)}.$ (3.2)

In the rest of this paper, if $w_i,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,s$ are the Freud-type weights then we suppose $a_n^{\left(i\right)}=o\left(1\right)n^{2/3}$ .

Theorem 3.2. Let $w_i\in \mathcal{F}\left(C^2+\right)\mathrm{,\; <mi>\; </mi>}i=\mathrm{1,2,}\cdots \mathrm{,}s$ , and let $f\mathrm{<mo>\; :\; </mo>}{ℝ}^s\to ℝ$ be continuous. If

${\displaystyle {\prod }_{i\mathrm{<mo>\; =\; </mo>1}}^s\left\{{\left(1+x_i^2\right)}^{\beta /2}{w}_i\left(x_i\right)\right\}}\left|f\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)\right|\in C_0\left({ℝ}^s\right)$ (3.3)

holds, then there exists $n_0>0$ such that for $n\ge n_0$

$\begin{array}{l}\left|{\displaystyle \underset{k_s=1}{\overset{n}{\sum }}}\cdots {\displaystyle \underset{k_1=1}{\overset{n}{\sum }}}{\displaystyle {\prod }_{i=1}^s{\lambda }_{k_i,n,i}f\left(x_{k_1,n,1},\cdots ,x_{k_s,n,s}\right)}\right|\\ \le C{\Vert {\displaystyle {\prod }_{i=1}^s\left\{{\left(1+x_i^2\right)}^{\beta }{w}_i^2\left(x_i\right)\right\}f\left(x_1,\cdots ,x_s\right)}\Vert }_{L^{\infty }\left({ℝ}^s\right)},\end{array}$ (3.4)

where for each $i=1,2,\cdots ,s$ , $x_{k_i,n,i}\mathrm{<mi>\; </mi>}\left(k_i=1,2,\cdots ,n\right)$ are zeros of $p_{n\mathrm{,}i}\left(x_i\right)$ . In particular,

$\begin{array}{l}\underset{n\to \infty }{\lim }{\displaystyle \underset{k_s=1}{\overset{n}{\sum }}}\cdots {\displaystyle \underset{k_1=1}{\overset{n}{\sum }}}{\displaystyle {\prod }_{i=1}^s{\lambda }_{k_i,n,i}f\left(x_{k_1,n,1},\cdots ,x_{k_s,n,s}\right)}\\ ={\displaystyle {\int }_{-\infty }^{\infty }}\cdots {\displaystyle {\int }_{-\infty }^{\infty }}{\displaystyle {\prod }_{i=1}^s{w}_i^2\left(x_i\right)f\left(x_1,\cdots ,x_s\right)\text{d}{x}_1\cdots \text{d}{x}_s}.\end{array}$ (3.5)

Theorem 3.3. Let $w_i\in {\mathcal{F}}_{\lambda }\left(C^3+\right)\left(0<\lambda <3/2\right),\mathrm{<mi>\; </mi>}i=1,2,3,\cdots ,s$ . Let $\beta >1/2$ , and let $f\mathrm{<mo>\; :\; </mo>}{ℝ}^s\to ℝ$ satisfy (3.3), then we have for $n=1,2,\cdots$ ,

${\Vert {\displaystyle {\prod }_{i=1}^s{w}_i{L}_n\left(f\right)}\Vert }_{L^{\infty }\left({ℝ}^s\right)}\le C{\Vert {\displaystyle {\prod }_{i=1}^s\left\{{\left(1+x_i^2\right)}^{\left(\beta /2\right)}{w}_i\left(x_i\right)\right\}f\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)}\Vert }_{L^{\infty }\left({ℝ}^s\right)}\mathrm{,}$ (3.6)

where for each $i=1,2,\cdots ,s$ , $x_{k_i,n,i},\mathrm{<mi>\; </mi>}{k}_i=1,2,\cdots ,n$ are zeros of $p_{n\mathrm{,}i}\left(x_i\right)$ . In

particular, if ${\displaystyle {\prod }_{i=1}^s\left\{{\left(1+x_i^2\right)}^{\left(\beta /2\right)}{T}_i^{1/4}\left(x_i\right)w_i\left(x_i\right)\right\}f\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)}\in C_0\left({ℝ}^s\right)$ , then we

have

$\underset{n\to \infty }{\lim }{\Vert {\displaystyle {\prod }_{i=1}^s{w}_i\left(f-L_n\left(f\right)\right)}\Vert }_{L^2\left({ℝ}^s\right)}=0.$

For $p\ne 2$ we also obtain the similar results. We need a function as follows:

$\Psi \left(x\right)\mathrm{<mo>\; :\; </mo>}=\frac{1}{{\left(1+Q\left(x\right)\right)}^{2/3}T\left(x\right)}\mathrm{.}$ (3.7)

Theorem 3.4. Let $w_i\in {\mathcal{F}}_{\lambda }\left(C^3+\right),\mathrm{<mi>\; </mi>0}<\lambda <3/2\mathrm{<mi>\; </mi>}\left(i=1,2,\cdots ,s\right)$ . Let $1<p\le 2$ and $\beta >1/p$ , and let $\Psi \left(x\right)$ be defined by (3.7) for each $w_i,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,s$ . If $f\mathrm{<mo>\; :\; </mo>}ℝ\to ℝ$ is continuous, and satisfies

$\begin{array}{l}\left|{\displaystyle {\prod }_{i=1}^s\left\{{\left(1+x_i^2\right)}^{\beta /2}{T}_i^{1/2}\left(x_i\right){\Psi }_i^{-1/4}\left(x_i\right)\right\}W\left(X\right)f\left(X\right)}\right|<\infty ,\mathrm{<mi>\; </mi>}\\ X\in {ℝ}^s,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}i=1,2,\cdots ,s,\end{array}$ (3.8)

then we have

$\begin{array}{l}{\Vert W{L}_n\left(f\right)\Vert }_{L^p\left({ℝ}^s\right)}\le C{\Vert {\displaystyle {\prod }_{i=1}^s\left\{{\left(1+x_i^2\right)}^{\beta /2}{T}_i^{1/2}\left(x_i\right){\Psi }_i^{-1/4}\left(x_i\right)\right\}Wf}\Vert }_{L^{\infty }\left({ℝ}^s\right)}\mathrm{,\; <mi>\; </mi>}\\ \mathrm{<mi>\; </mi>}n=1,2,\cdots \mathrm{<mi>\; </mi>.}\end{array}$ (3.9)

Especially, if $f$ satisfies

$\left|{\displaystyle {\prod }_{i=1}^s\left\{{\left(1+x_i^2\right)}^{\beta /2}{T}_i^{3/4}\left(x_i\right){\Psi }_i^{-1/4}\left(x_i\right)\right\}W\left(X\right)f\left(X\right)}\right|\in C_o\left({ℝ}^s\right)\mathrm{,}$ (3.10)

then we have

$\underset{n\to \infty }{\lim }{\Vert W\left(f-L_n\left(f\right)\right)\Vert }_{L^p\left({ℝ}^s\right)}=0.$ (3.11)

For $2<p\le \infty$ we have the following:

Theorem 3.5. Let $w_i\in {\mathcal{F}}_{\lambda }\left(C^3+\right),\mathrm{<mi>\; </mi>0}<\lambda <3/2\mathrm{<mi>\; </mi>}\left(i=1,2,\cdots ,s\right)$ , and let satisfy $T_i\left(a_n^{\left(i\right)}\right)\le C{n}^{1/2}$ . Let $2<p\le \infty$ and $\beta >1/p$ . Furthermore we assume

$Q_i\left(\frac{a_n^{\left(i\right)}}{4}\right)\ge C{\left(log\left(Q_i\left(a_n^{\left(i\right)}\right)\right)\right)}^4\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>}i=\mathrm{1,2,}\cdots \mathrm{,}s\mathrm{.}$ (3.12)

If $f\mathrm{<mo>\; :\; </mo>}{ℝ}^s\to ℝ$ is continuous, and satisfies

$\left|{\displaystyle {\prod }_{i=1}^s\left\{{\left(1+x_i^2\right)}^{\beta /2}{T}_i^{1/2}\left(x_i\right){\Psi }_i^{-1}\left(x_i\right)\right\}W\left(X\right)f\left(X\right)}\right|<\infty \mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>}X\in {ℝ}^s\mathrm{,}$ (3.13)

then we have

$\begin{array}{l}{\Vert \left\{{\displaystyle {\prod }_{i=1}^s{\Psi }_i^{3/4}\left(x_i\right)}\right\}W{L}_n\left(f\right)\Vert }_{L^p\left({ℝ}^s\right)}\\ \le C{\Vert {\displaystyle {\prod }_{i=1}^s\left\{{\left(1+x_i^2\right)}^{\beta /2}{T}_i^{1/2}\left(x_i\right){\Psi }_i^{-1/4}\left(x_i\right)\right\}Wf}\Vert }_{L^{\infty }\left({ℝ}^s\right)}\mathrm{.}\end{array}$

Especially, by (3.13) we have

$\underset{n\to \infty }{\lim }{\Vert \left\{{\displaystyle {\prod }_{i=1}^s{\Psi }_i^{3/4}\left(x_i\right)}\right\}W\left(f-L_n\left(f\right)\right)\Vert }_{L^p\left({ℝ}^s\right)}=0.$

Remark 3.6. (1) We note that (3.13) means

$\left({\displaystyle {\prod }_{i=1}^s{T}_i^{1/4}}\right)W^{\mathrm{<mo>\; *\; </mo>}}f\in C_0\left({ℝ}^s\right)\mathrm{,}$

where ${\displaystyle {\prod }_{i=1}^s\left\{{\left(1+x_i^2\right)}^{\beta \mathrm{<mo>\; /\; </mo>2}}{T}_i^{1/2}\left(x_i\right){\Psi }_i^{-1/4}\left(x_i\right)\right\}W}~W^{\mathrm{<mo>\; *\; </mo>}}\in \mathcal{F}\left(C^2+\right)$ (see Theorem 2.3).

(2) All examples in Example 2.2 hold (3.12).

(3) To prove Theorem 3.5 we use Proposition 4.5. Then Assumption (3.12) plays an important role.

Next, we characterize the best approximation polynomial (cf. [5] ).

Theorem 3.7. Let $0<p\le \infty$ . There is a best approximation polynomial $P_f\in {\mathcal{P}}_{n\mathrm{<mo>\; ;\; </mo>}s}$ such that

$E_{p\mathrm{,}n\mathrm{<mo>\; ;\; </mo>}s}\left(W\mathrm{,}f\right)\mathrm{<mo>\; :\; </mo>}=\underset{P\in {\mathcal{P}}_{n\mathrm{<mo>\; ;\; </mo>}s}}{inf}{\Vert W\left(f-P\right)\Vert }_{L^p\left({ℝ}^s\right)}={\Vert W\left(f-P_{n\mathrm{<mo>\; ;\; </mo>}f}\right)\Vert }_{L^p\left({ℝ}^s\right)}\mathrm{.}$

Theorem 3.8 (Kolmogorov-type theorem). $P\in {\mathcal{P}}_{n\mathrm{<mo>\; ;\; </mo>}s}$ is a best of approximation for a continuous function $f$ with $li{m}_{\left|\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)\right|\to \infty }W\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)f\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)=0$ , if and only if for each polynomial $Q\in {\mathcal{P}}_{n\mathrm{<mo>\; ;\; </mo>}s}$ ,

$\underset{\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)\in A}{max}\left[W\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)\left(f\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)-P\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)\right)\right]Q\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)\ge \mathrm{0,}$ (3.14)

where $A$ denotes the set (which depends on $f$ and $P$ ) of all points $\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)\in A$ for which

$\left|W\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)\left(f\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)-P\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)\right)\right|={\Vert W\left(f-P\right)\Vert }_{L^{\infty }\left({ℝ}^s\right)}$ .

Theorem 3.9. Let $W={\displaystyle {\prod }_{i=1}^s{w}_i},\mathrm{<mi>\; </mi>}{w}_i\in \mathcal{F}\left(C^2+\right),\mathrm{<mi>\; </mi>}i=1,\cdots ,s$ and $1\le p<\infty$ . Let ${\varphi }_K$ , where $K=\left(k_1,\cdots ,k_s\right)\mathrm{<mi>\; </mi>}\left(0\le k_j\le n\right)\mathrm{<mi>\; </mi>}\left(j=1,\cdots ,s\right)$ be a linearly independent system satisfying $W{\varphi }_K\in L_p\left({ℝ}^s\right)$ , and we consider polynomials

$Q\left(X\right)={\displaystyle \underset{k_s=1}{\overset{n}{\sum }}}\cdots {\displaystyle \underset{k_1=1}{\overset{n}{\sum }}}{c}_{\left(k_1,\cdots ,k_s\right)}{\varphi }_{\left(k_1,\cdots ,k_s\right)}\left(X\right).$ (3.15)

Let $Wf\in L_p\left({ℝ}^s\right)$ . The polynomial

$P\left(X\right)={\displaystyle \underset{k_s=1}{\overset{n}{\sum }}}\cdots {\displaystyle \underset{k_1=1}{\overset{n}{\sum }}}{c}_{\left(k_1,\cdots ,k_s\right)}^{\left[0\right]}{\varphi }_{\left(k_1,\cdots ,k_s\right)}\left(X\right)$ (3.16)

is a polynomial of the best approximation for $f$ if and only if for every polynomial (3.15) the following equality (3.17) holds.

$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^s}}Q\left(X\right){\left|f\left(X\right)-P\left(X\right)\right|}^{p-1}\left[\text{sign}\left(f\left(X\right)-P\left(X\right)\right)\right]W^p\left(X\right)DX\\ \mathrm{<mo>\; :\; </mo>}={\displaystyle {\int }_{ℝ}}\cdots {\displaystyle {\int }_{ℝ}}Q\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right){\left|f\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)-P\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)\right|}^{p-1}\\ \mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\times \left[\text{sign}\left(f\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)-P\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)\right)\right]W^p\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_s\right)\text{d}{x}_1\cdots \text{d}{x}_s=0\end{array}$ (3.17)

holds. If $p=1$ , we also assume that $f\left(X\right)-P\left(X\right)$ vanish only on a set of measure zero.

4. Proofs of Theorems 3.2, 3.3, 3.4 and 3.5

Lemma 4.1. Let $x_{n,n,i}<x_{n-1,n,i}<\cdots <x_{1,n,i}$ be zeros of the orthonormal polynomial $p_{n\mathrm{,}i}$ , and let

$P_{n-1}\left(x_1,x_2,\cdots ,x_s\right)={\displaystyle \underset{0\le j_i\le n-1,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,s}{\sum }}{a}_{j_1,\cdots ,j_s}{x}_1^{j_1}\cdots x_s^{j_s},$

where $a_{j_1\mathrm{,}\cdots \mathrm{,}{j}_s}$ are coefficients. If for every $\left(x_{k_1\mathrm{,}n\mathrm{,1}}\mathrm{,}\cdots ,x_{k_s\mathrm{,}n\mathrm{,}s}\right)$ ,

$P_{n-1}\left(x_{k_1,n,1},\cdots ,x_{k_s,n,s}\right)=0,\mathrm{<mi>\; </mi>\; <mi>\; </mi>1}\le k_i\le n,\mathrm{<mi>\; </mi>}i=1,\cdots ,s,$ (4.1)

then we have $P_{n-1}=0$ . Therefore, for $P_{n-1}\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_s\right)\in {\mathcal{P}}_{n-1}\left({ℝ}^s\right)$ we have

$P_{n-1}=L_n\left(P_{n-1}\right).$ (4.2)

Proof. Now we fix any $\left(x_{k_2,n,2},\cdots ,x_{k_s,n,s}\right)\in {ℝ}^{s-1}$ , and then we consider the polynomial $Q_{n-1}\left(x_1\right)$ in ${\mathcal{P}}_{n-1}\left(ℝ\right)$ such that

$Q_{n-1}\left(x_1\right)={\displaystyle \underset{j_1=0}{\overset{n-1}{\sum }}}\left({\displaystyle \underset{0\le j_i\le n-1,\mathrm{<mi>\; </mi>}i=2,3,\cdots ,s}{\sum }}{a}_{j_1,\cdots ,j_s}{x}_{k_2,n,2}^{j_2}\cdots x_{k_s,n,s}^{j_s}\right)x_1^{j_1}.$

Since

$Q_{n-1}\left(x_{k_1,n,1}\right)=0,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{k}_1=1,2,\cdots ,n$

(see (4.1)), all coefficients of $Q_{n-1}\left(x_1\right)$ equal to zero, that is,

${\displaystyle \underset{0\le j_i\le n-1,\mathrm{<mi>\; </mi>}i=2,3,\cdots ,s}{\sum }}{a}_{j_1,\cdots ,j_s}{x}_{k_2,n,2}^{j_2}\cdots x_{k_s,n,s}^{j_s}=0,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\text{for}\text{each}{j}_1=0,1,\cdots ,n-1.$ (4.3)

Next, we fix any

$j_1\left(0\le j_1\le n-1\right)\mathrm{<mi>\; </mi>}\text{and}\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\left(x_{k_3\mathrm{,}n\mathrm{,3}}\mathrm{,}\cdots ,x_{k_s\mathrm{,}n\mathrm{,}s}\right)\in {ℝ}^{s-2}\mathrm{,}$

and we consider $R_{n-1}\in {\mathcal{P}}_{n-1}\left(ℝ\right)$ such that

$R_{n-1}\left(x_2\right)={\displaystyle \underset{j_2=0}{\overset{n-1}{\sum }}}\left({\displaystyle \underset{0\le j_i\le n-1,\mathrm{<mi>\; </mi>}i=3,4,\cdots ,s}{\sum }}{a}_{j_1,\cdots ,j_s}{x}_{k_3,n,3}^{j_3}\cdots x_{k_s,n,s}^{j_s}\right)x_2^{j_2}.$

Then by (4.3), we see

$R_{n-1}\left(x_{k_2,n,2}\right)=0,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{k}_2=1,2,\cdots ,n.$

Hence,

${\displaystyle \underset{0\le j_i\le n-1,\mathrm{<mi>\; </mi>}i=3,4,\cdots ,s}{\sum }}{a}_{j_1,\cdots ,j_s}{x}_{3,k_3}^{j_3}\cdots x_{s,k_s}^{j_s}=0,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\text{for}\text{each}{j}_1,j_2=0,1,\cdots ,n-1.$

If we continue this method inductively, then we have

${\displaystyle \underset{j_s=0}{\overset{n-1}{\sum }}}{a}_{j_1,\cdots ,j_s}{x}_{k_s,n,s}^{j_s}=0,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\text{for}\text{each}{j}_1,j_2,\cdots ,j_{s-1}=0,1,\cdots ,n-1.$ (4.4)

We put $H_{n-1}\in {\mathcal{P}}_{n-1}\left(ℝ\right)$ as

$H_{n-1}\left(x_s\right)={\displaystyle \underset{j_s=0}{\overset{n-1}{\sum }}}{a}_{j_1,\cdots ,j_s}{x}_s^{j_s},$

then from (4.4) we have $H_{n-1}\left(x_{k_s,n,s}\right)=0,\mathrm{<mi>\; </mi>}{k}_s=1,2,\cdots ,n$ . Therefore, we conclude

$a_{j_1,\cdots ,j_s}=0,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{j}_i=0,1,\cdots ,n-\mathrm{1\; <mi>\; </mi>}\left(i=1,2,\cdots ,s\right),$

that is, $P_{n-1}=0$ . #

In the rest of this paper, we use the following notations:

$W={\displaystyle {\prod }_{i=1}^s{w}_i},\mathrm{<mi>\; </mi>\; <mi>\; </mi>}X=\left(x_1,\cdots ,x_s\right),\mathrm{<mi>\; </mi>\; <mi>\; </mi>}X\left(u\right)=\left(u_1,\cdots ,u_s\right)$

$D\left(X\right)=\text{d}{x}_1\cdots \text{d}{x}_s,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}D\left(X\left(u\right)\right)=\text{d}{u}_1\cdots \text{d}{u}_s.$

We also use

$\begin{array}{l}{\mathcal{P}}_{n;s}\left({ℝ}^s\right):={\mathcal{P}}_{n,\cdots ,n}\left({ℝ}^s\right):=\{P_n|P_n\left(x_1,\cdots ,x_s\right)\text{arepolynomialswithdegree}\le n\\ \text{for}\text{each}{x}_i,\mathrm{<mi>\; </mi>}i=1,\cdots ,s\}.\end{array}$

Proposition 4.2 (cf. [6] , Theorem 1.2.2). Let $w_i\in \mathcal{F}\left(C^2+\right)\mathrm{,\; <mi>\; </mi>}i=\mathrm{1,2,}\cdots \mathrm{,}s$ , and let $n\ge 1$ be an integer. Then for all $P\in {\mathcal{P}}_{2n-\mathrm{1\; <mo>\; ;\; </mo>}s}\left({ℝ}^s\right)$ , we have

$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^s}}P\left(X\left(u\right)\right)W^2\left(X\left(u\right)\right)D\left(X\left(u\right)\right)\\ ={\displaystyle \underset{k_s\mathrm{<mo>\; =\; </mo>1}}{\overset{n}{\sum }}}\cdots {\displaystyle \underset{k_1\mathrm{<mo>\; =\; </mo>1}}{\overset{n}{\sum }}}{\lambda }_{k_1\mathrm{,}n\mathrm{,1}}\cdots {\lambda }_{k_s\mathrm{,}n\mathrm{,}s}P\left(x_{k_1\mathrm{,}n\mathrm{,1}}\mathrm{,}\cdots \mathrm{,}{x}_{k_s\mathrm{,}n\mathrm{,}s}\right)\mathrm{,}\end{array}$ (4.5)

where

${\lambda }_{k_i,n,i}={\displaystyle {\int }_{-\infty }^{\infty }}{l}_{k_i,n,i}{w}_i^2\left(x_i\right)\text{d}{x}_i,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}i=1,2,\cdots ,s.$ (4.6)

Proof. (see [6] , pp.12-13). Let $P\in {\mathcal{P}}_{n-\mathrm{1\; <mo>\; ;\; </mo>}s}\left({ℝ}^s\right)$ . From (3.1) and (4.2) we see

$\begin{array}{c}P\left(X\right)=L_n\left(P;X\right)\\ ={\displaystyle \underset{k_s=1}{\overset{n}{\sum }}}\cdots {\displaystyle \underset{k_1=1}{\overset{n}{\sum }}}{l}_{k_1,n,1}\left(x_1\right)\cdots l_{k_s,n,s}\left(x_s\right)P\left(x_{k_1,n,1},\cdots ,x_{k_s,n,s}\right).\end{array}$