A Study of Weighted Polynomial Approximations with Several Variables (I)

doi:10.4236/am.2017.89095

Applied Mathematics
Vol.08 No.09(2017), Article ID:79144,40 pages
10.4236/am.2017.89095

Ryozi Sakai

●How to Cite this Article

Department of Mathematics, Meijo University, Nagoya, Japan

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

Received: August 3, 2017; Accepted: September 16, 2017; Published: September 19, 2017

ABSTRACT

In this paper, we investigate the weighted polynomial approximations with several variables. Our study relates to the approximation for $W f \in L^{p} (ℝ^{s})$ by weighted polynomials. Then we will estimate the degree of approximation.

Keywords:

Weighted Polynomial Approximations with Several Variables, the Degree of Approximations

1. Introduction

Let $ℝ^{s} : = ℝ \times ℝ \times \dots \times ℝ$ ( $s$ times, $s \geq 1$ integer) be the direct product space, and let $W (x_{1}, x_{2}, \dots, x_{s}) : = w_{1} (x_{1}) w_{2} (x_{2}) \dots w_{s} (x_{s})$ , where $w_{i} (x_{i}) \geq 0$ are even weight functions. We suppose that for every nonnegative integer n,

$\int_{0}^{\infty} x^{n} w_{i} (x) d x < \infty, n = 0, 1, 2, \dots, i = 1, 2, \dots, s .$

In this paper, we will study to approximate the real-valued weighted function $(W f) (x_{1}, x_{2}, \dots, x_{s})$ by weighted polynomials $(W P) (x_{1}, x_{2}, \dots, x_{s})$ , where

$P (x_{1}, x_{2}, \dots, x_{s}) \in P_{n, n, \dots, n} (ℝ^{s})$ . Here, $P_{n, n, \dots, n} (ℝ^{s}) (= : P_{n; s} (ℝ^{s}))$ means a class of

all polynomials with at most n-degree for each variable $x_{i}, i = 1, 2, \dots, s$ . We need to define the norms. Let $0 < p \leq \infty$ , and let $f : ℝ^{s} \to ℝ$ be measurable. Then we define

${‖ W f ‖}_{L^{p} (ℝ^{s})} : = {\begin{cases} {[\int_{- \infty}^{\infty} \dots \int_{- \infty}^{\infty} {| (W f) (x_{1}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}]}^{1 / p}, if 0 < p < \infty; \\ \sup_{(x_{1}, \dots, x_{s}) \in ℝ^{s}} | (W f) (x_{1}, \dots, x_{s}) |, if p = \infty . \end{cases}$

We assume that for $0 < p \leq \infty$ the integral is independent of the order of integration with respect to each $x_{i}, i = 1, 2, \dots, s$ . When ${‖ W f ‖}_{L^{p} (ℝ^{s})} < \infty$ , we write $W f \in L^{p} (ℝ^{s})$ . If $p = \infty$ , we require that $f$ is continuous and

$l i m_{| X | \to \infty} (W f) (X) = 0$ , where $| X | = | (x_{1}, \dots, x_{s}) | = \max | x_{i} |; i = 1, 2, \dots, s$ .

Our purpose in this paper is to approximate the weighted function $W f \in L^{p} (ℝ^{s})$ by weighted polynomials $W P; P \in P_{n; s} (ℝ^{s})$ . The paper is arranged as the following. In Section 2, we give the definition of the weights which are treated in this paper. In Section 3, we consider the approximation for the functions in $L^{p} (ℝ^{s})$ . In Section 4, we consider a property of higher order derivatives. In Section 5, we estimate the degree of approximations. In Section 6, we consider the approximation for the functions with bounded variation. In Section 7, we consider the approximation of the Lipschitz-type functions. In Section 8, we treat the functions with higher order derivatives.

2. Class of Weight Functions and Preliminaries

Throughout the paper $C, C_{1}, C_{2}, \dots$ denote positive constants independent of $n, x, t$ or polynomials $P (x)$ . The same symbol does not necessarily denote the same constant in different occurrences. Let $f (x) ~ g (x)$ mean that there exists a constant $C > 0$ such that $C^{- 1} f (x) \leq g (x) \leq C f (x)$ holds for all $x \in I$ , where $I \subset ℝ$ is a subset.

We say that $f : ℝ \to [0, \infty)$ is quasi-increasing if there exists $C > 0$ such that $f (x) \leq C f (y)$ for $0 < x < y$ . Hereafter we consider following weights.

Definition 2.1. Let $Q : ℝ \to [0, \infty)$ be a continuous and even function, and satisfy the following properties:

(a) $Q^{'} (x)$ is continuous in $ℝ$ , with $Q (0) = 0$ .

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

(d) The function

$T (x) : = \frac{x Q^{'} (x)}{Q (x)}, x \neq 0$

is quasi-increasing in $(0, \infty)$ , with

$T (x) \geq Λ > 1, x \in ℝ \ {0} .$

(e) There exists $C_{1} > 0$ such that

$\frac{Q^{″} (x)}{| Q^{'} (x) |} \leq C_{1} \frac{| Q^{'} (x) |}{Q (x)}, a . e . x \in ℝ .$

Then we write $w = e x p (- Q) \in F (C^{2})$ .

Moreover, if there also exists a compact subinterval $J (∋ 0)$ of $ℝ$ , and $C_{2} > 0$ such that

$\frac{Q^{″} (x)}{| Q^{'} (x) |} \geq C_{2} \frac{| Q^{'} (x) |}{Q (x)}, a . e . x \in ℝ \ J,$

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

For $w (x) = e x p (- Q (x)) \in F (C^{2} +), Q \in C^{3} (ℝ \ {0})$ , if there exists $K > 0$ such that for $| x | \geq K$ ,

$| \frac{Q^{‴} (x)}{Q^{″} (x)} | \leq C | \frac{Q^{″} (x)}{Q^{'} (x)} |,$ (2.1)

and there exist $λ, C > 0$ such that for $0 < λ < \frac{3}{2}$ ,

$\frac{| Q^{'} (x) |}{Q {(x)}^{λ}} \leq C,$ (2.2)

then we write $w \in F_{λ} (C^{3} +)$ . Furthermore, if

$| \frac{Q^{(4)} (x)}{Q^{(3)} (x)} | \leq C | \frac{Q^{‴} (x)}{Q^{″} (x)} | ~ | \frac{Q^{″} (x)}{Q^{'} (x)} |$ (2.3)

and the inequality (2.2) with $0 < λ < \frac{4}{3}$ hold, then we write $w \in F_{λ} (C^{4} +)$ .

We have some examples satisfying Definition 2.1.

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

$1 < Λ_{1} \leq \frac{{(x Q^{'} (x))}^{'}}{Q^{'} (x)} \leq Λ_{2},$

where $Λ_{i}, i = 1, 2$ are constants, then we call $w = \exp (- Q (x))$ the Freud weight. The class $F (C^{2} +)$ contains the Freud weights.

(2) For $α > 1, l \geq 1$ we define

$Q (x) = Q_{l; α} (x) = \exp_{l} ({| x |}^{α}) - \exp_{l} (0),$

where $\exp_{l} (x) = \exp (\exp (\exp \dots \exp x) \dots) (l times)$ . Moreover, we define

$Q_{l; α, m} (x) = {| x |}^{m} {\exp_{l} ({| x |}^{α}) - α^{*} \exp_{l} (0)}, α + m > 1, m \geq 0, α \geq 0,$

where $α^{*} = 0$ if $α = 0$ , and otherwise $α^{*} = 1$ . We note that $Q_{l;0, m}$ gives a Freud-type weight, that is, $T (x)$ is bounded..

(3) We define

$Q_{α} (x) = {(1 + | x |)}^{{| x |}^{α}} - 1, α > 1.$

(4) Let $w = e x p (- Q) \in F (C^{2} +)$ , and let us define

$μ_{+} : = \underset{x \to \infty}{\lim \sup} \frac{Q^{″} (x)}{Q^{'} (x)} / \frac{Q^{'} (x)}{Q (x)}, μ_{-} : = \underset{x \to \infty}{\lim \inf} \frac{Q^{″} (x)}{Q^{'} (x)} / \frac{Q^{'} (x)}{Q (x)} .$

If $μ_{+} = μ_{-}$ , 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 $w \in F_{λ} (C^{3} +)$ . If the weight w is regular and if $Q \in C^{3} (ℝ \ {0})$ satisfies (2.1), then for the regular weights we have $w \in F_{λ} (C^{3} +)$ (see [3] , Corollary 5.5 (5.8)).

The following fact is very important for our study.

Proposition 2.3 ( [3] , Theorem 4.1 and (4.11)). Let $0 < λ < 3 / 2$ and $α \in ℝ$ . Then for $w = e x p (- Q) \in F_{λ} (C^{3} +)$ , we can construct a new weight

$w_{α} \in F (C^{2} +)$ such that

$T_{w} {(x)}^{α} w (x) ~ w_{α} (x) on ℝ,$

and for some $C \geq 1$ ,

$a_{n / C} (w_{α}) \leq a_{n} (w) \leq a_{C n} (w_{α}) and T_{w_{α}} (x) ~ T_{w} (x) = T (x),$

where $a_{n} (w_{α})$ and $a_{n} (w)$ are MRS-numbers for the weight $w_{α}$ and $w$ , respectively, and $T_{w_{α}}$ and $T_{w}$ are correspond for $w_{α}$ or $w$ , respectively.

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

$\int_{- \infty}^{\infty} p_{n} (x) p_{m} (x) w^{2} (x) d x = δ_{m n} (theKroneckerdelta) .$

For $1 \leq p \leq \infty$ , we denote by $L^{p} (ℝ)$ the usual $L^{p}$ space on $ℝ$ (here for $p = \infty$ , if $w f \in L^{\infty} (ℝ)$ , then we require $f$ to be continuous, and $w f$ to have limit 0 at $\pm \infty$ ). For $w f \in L^{p} (ℝ)$ , we set

$\begin{array}{l} s_{n} (f, x) : = \sum_{k = 0}^{n - 1} b_{k} (f) p_{k} (x), \\ where b_{k} (f) = \int_{- \infty}^{\infty} f (t) p_{k} (t) w^{2} (t) d t \end{array}$ (2.4)

for $n \in ℕ$ (the partial sum of Fourier-type series). The de la Vallée Poussin mean of order n is defined by

$v_{n} (f, x) : = \sum_{j = n + 1}^{2 n} s_{j} (f, x) .$ (2.5)

Let $w \in F (C^{2} +)$ . We need the Mhaskar-Rakhmanov-Saff numbers (MRS-numbers) $a_{x}$ ;

$x = \frac{2}{π} \int_{0}^{1} \frac{a_{x} u Q^{'} (a_{x} u)}{{(1 - u^{2})}^{1 / 2}} d u, x > 0.$

We easily see

$\lim_{x \to \infty} a_{x} = \infty and \lim_{x \to + 0} a_{x} = 0$

and

$\lim_{x \to \infty} \frac{a_{x}}{x} = 0 and \lim_{x \to + 0} \frac{a_{x}}{x} = \infty .$

For $w f \in L_{p} (ℝ) (1 \leq p \leq \infty)$ , the degree of weighted polynomial approximation is defined by

$E_{n, p} (w; f) : = \underset{P \in P_{n}}{i n f} {‖ w (f - P) ‖}_{L^{p} (ℝ)} .$

3. Approximations for L_p-Functions

In this section, we treat the function such as $W f \in L^{p} (ℝ^{s})$ , where $1 \leq p \leq \infty$ ), and if $p = \infty$ , then we suppose that $W f$ is continuous and

$l i m_{| X | \to \infty} (W f) (X) = 0$ . For any multivariate point $X = (x_{1}, \dots, x_{s}) \in ℝ^{s}$ , we consider the weights;

$W (X) : = \prod_{j = 1}^{s} w_{j} (x_{j}) = \prod_{j = 1}^{s} \exp (- Q_{j} (x_{j})) .$

As shown under, we will also use $X (u) : = (u_{1}, u_{2}, \dots, u_{s})$ . Let

$w_{i} = \exp (- Q_{i}) \in F_{λ} (C^{3} +)$ , $0 < λ < 3 / 2$ , $i = 1, 2, \dots, s$ . From Proposition 2.3 we see $T_{i}^{1 / 4} w_{i} ~ w_{i, 1 / 4} \in F (C^{2} +), i = 1, 2, \dots, s$ . Then we admit to write

$T_{i}^{1 / 4} w_{i} \in F (C^{2} +)$ . For the weight $W$ we construct the modulus of continuity of $f$ . It involves the function

$Φ_{t, i} (x_{i}) : = \sqrt{1 - \frac{| x_{i} |}{σ_{i} (t)}} + \frac{1}{\sqrt{T_{i} (σ_{i} (t))}}, i = 1, 2, \dots, s,$

where $σ_{i} (t)$ is defined by

$σ_{i} (t) : = \inf {a_{n}^{(i)} : \frac{a_{n}^{(i)}}{n} \leq t}, t > 0,$

where $a_{n}^{(i)}$ is the MRS-number for the weight $w_{i} (x)$ . If $a_{n}^{(i)} / n = t$ , then we have $σ_{i} (t) = a_{n}^{(i)}$ . In the sequel, if $1 \leq j \leq s$ is an integer, then ${‖ f ‖}_{p; j}$ will denote the $L^{p}$ norm of $f$ taken with respect to the j-th variable. This is a function of the remaining $(s - 1)$ variables. For each fixed

${\hat{X}}_{j} : = (x_{1}, \dots, x_{j - 1}, x_{j + 1}, \dots, x_{s}) \in ℝ_{j}^{s - 1}$ , we write

$f_{{\hat{X}}_{j}} (x) : = f (x_{1}, \dots, x_{j - 1}, x, x_{j + 1}, \dots, x_{s}), j = 1, 2, \dots, s .$ (3.1)

Using

$Δ_{h} f_{{\hat{X}}_{j}} (x) : = f_{{\hat{X}}_{j}} (x + \frac{h}{2}) - f_{{\hat{X}}_{j}} (x - \frac{h}{2}),$

we define the modulus of continuity. For the Freud-type weight, we define

$\begin{array}{l} {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, w_{j}; t) \\ : = {(\frac{1}{t} \int_{0}^{t} {‖ w_{j} (x) (Δ_{h} f_{{\hat{X}}_{j}} (x)) ‖}_{L^{p} (| x | \leq σ_{j} (2 t))}^{p} d h)}^{1 / p} \\ + \inf_{c_{j} (constant)} {‖ (f_{{\hat{X}}_{j}} (x) - c_{j}) w_{j} (x) ‖}_{L^{p} (| x | \leq σ_{j} (4 t))} . \end{array}$

If $w_{j}$ is Erdös-type, then we define

$\begin{array}{l} {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, w_{j}; t) \\ : = {(\frac{1}{t} \int_{0}^{t} {‖ w_{j} (x) (Δ_{h Φ_{t, j} (x)} f_{{\hat{X}}_{j}} (x)) ‖}_{L^{p} (| x | \leq σ_{j} (2 t))}^{p} d h)}^{1 / p} \\ + \inf_{c_{j} (constant)} {‖ (f_{{\hat{X}}_{j}} (x) - c_{j}) w_{j} (x) ‖}_{L^{p} (| x | \leq σ_{j} (4 t))} . \end{array}$

We remark that if $T_{j} (x)$ is bounded, then we see $Φ_{t, j} (x) ~ 1$ , so we do not need the definition for the Freud-type weight.

Let v_n be the de la Vallée Poussin mean opetator, and let $v_{n, j} (f), j = 1, 2, \dots, s$ denote the operation to f with respect to j-th co-ordinate, and $v_{n}^{[j]}$ will denote the operator $v_{n}$ applied to f with respect to each of the first j co-ordinates. Clearly,

$v_{n}^{[1]} (f) = v_{n, 1} (f), v_{n}^{[j]} (f) = v_{n}^{[j - 1]} (v_{n, j} (f)), j = 2, 3, \dots, s .$ (3.2)

Let $a_{n}^{(j)}$ be the MRS-number for the weight $w_{j} = \exp (- Q_{j})$ .

First, we consider the following Proposition.

Proposition 3.1 ( [4] , Theorem 3.14). For $1 \leq p < \infty$ , $C_{c} (ℝ^{s})$ is dence in $L^{p} (ℝ^{s})$ , where $C_{c} (ℝ^{s})$ is a set of all continuous functions with a compact support on $ℝ^{s}$ .

From this proposition, for any $ε > 0$ there exist a constant $K > 0$ and a continuous function $f_{K}$ with a compact support ${[- K, K]}^{s}$ such that

${‖ W (X) (f (X) - f_{K} (X)) ‖}_{L^{p} (| X | \leq K)} < ε .$ (3.3)

Then we give the following assumption:

Assumption 3.2. In (3.3) we suppose that for every co-ordinate $x_{j}, j = 1, 2, \dots, s$

${‖ w_{j} (x) (f_{{\hat{X}}_{j}} (x) - {(f_{K})}_{{\hat{X}}_{j}} (x)) ‖}_{L^{p} (| x | \leq K)} < ε$ (3.4)

holds.

We define a new class of functions $L_{p}^{*} (ℝ^{s}),1 \leq p \leq \infty$ as follows:

$L_{W}^{p *} (ℝ^{s}) : = {f | W f \in L_{p} (ℝ^{s}) holds (3.4)},$ (3.5)

where if $p = \infty$ , then $L_{W}^{p *} (ℝ^{s}) = L_{W}^{p} (ℝ^{s})$ and we suppose that f is continuous and

$\lim_{| X | \to \infty} W (X) f (X) = 0$

(we write this fact as $W f \in C_{0} (ℝ^{s})$ ). We state the theorem in this section.

Theorem 3.3. (1) We suppose

$w_{j} = \exp (- Q_{j}) \in F_{λ} (C^{3} +) (0 < λ < 3 / 2), j = 1, 2, \dots, s$ , and let

$T_{j} (a_{n}^{(j)}) \leq c {(\frac{n}{a_{n}^{(j)}})}^{2 / 3}, j = 1,2, \dots, s .$ (3.6)

Let $n \geq 1,1 \leq p \leq \infty$ . Then we have

$\begin{array}{l} {‖ W (f - v_{n}^{[s]} (f)) ‖}_{L^{p} (ℝ^{s})} \\ \leq \sum_{j = 1}^{s} C_{j} {‖ (\prod_{1 \leq i \leq s, i \neq j} w_{i}) (\prod_{k = 1}^{j - 1} T_{k}^{1 / 4}) {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, T_{j}^{1 / 4} w_{j}; c_{j} \frac{a_{n}^{(j)}}{n}) ‖}_{L^{p} (ℝ_{j}^{s - 1})}, \end{array}$ (3.7)

where

$ℝ_{j}^{s - 1} : = {(x_{1}, \dots, x_{j - 1}, x_{j + 1}, \dots, x_{s})},$ (3.8)

and $\prod_{k = 1}^{0} T_{k}^{1 / 4} = 1$ . Especially $f \in L_{T^{〈 s 〉} W}^{p *} (ℝ^{s})$ , then we have

$\begin{array}{l} \sum_{j = 1}^{s} C_{j} {‖ (\prod_{1 \leq i \leq s, i \neq j} w_{i}) (\prod_{k = 1}^{j - 1} T_{k}^{1 / 4}) {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, T_{j}^{1 / 4} w_{j}; c_{j} \frac{a_{n}^{(j)}}{n}) ‖}_{L^{p} (ℝ_{j}^{s - 1})} \to 0 \\ as n \to \infty . \end{array}$ (3.9)

(2) We suppose $w_{j} = \exp (- Q_{j}) \in F (C^{2} +), j = 1, 2, \dots, s$ , and let (3.6) holds. Let $n \geq 1,1 \leq p \leq \infty$ . Then we have

${‖ \frac{W (f - v_{n}^{[s]} (f))}{\prod_{k = 1}^{s} T_{k}^{1 / 4}} ‖}_{L^{p} (ℝ^{s})} \leq \sum_{j = 1}^{s} C_{j} {‖ (\prod_{1 \leq i \leq s, i \neq j} w_{i}) {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, w_{j}; c_{j} \frac{a_{n}^{(j)}}{n}) ‖}_{L^{p} (ℝ_{j}^{s - 1})} .$ (3.10)

Especially $f \in L_{W}^{p *} (ℝ^{s})$ , then we have

$\sum_{j = 1}^{s} C_{j} {‖ (\prod_{1 \leq i \leq s, i \neq j} w_{i}) {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, w_{j}; c_{j} \frac{a_{n}^{(j)}}{n}) ‖}_{L^{p} (ℝ_{j}^{s - 1})} \to 0 as n \to \infty .$ (3.11)

First we will show (3.7). We need some preliminaries.

Proposition 3.4 ( [5] , Theorem 1). Let $1 \leq p \leq \infty$ .

(1) We assume that $w \in F (C^{2} +)$ satisfies $T (a_{n}) \leq C {(n / a_{n})}^{2 / 3}$ . Then there exists a constant $C > 0$ such that when $w g \in L_{p} (ℝ)$ , then

${‖ \frac{w v_{n} (g)}{T^{1 / 4}} ‖}_{L^{p} (ℝ)} \leq C {‖ w g ‖}_{L^{p} (ℝ)},$

and so,

${‖ w v_{n} (g) ‖}_{L^{p} (ℝ)} \leq C T {(a_{n})}^{1 / 4} {‖ w g ‖}_{L^{p} (ℝ)} .$

(2) We assume that $w \in F_{λ} (C^{3} +) (0 < λ < 3 / 2)$ satisfies $T (a_{n}) \leq C {(n / a_{n})}^{2 / 3}$ . Then there exists a constant $C > 0$ such that if $T^{1 / 4} w g \in L^{p} (ℝ)$ , then

${‖ w v_{n} (g) ‖}_{L^{p} (ℝ)} \leq C {‖ T^{1 / 4} w g ‖}_{L^{p} (ℝ)} .$

Proposition 3.5 ( [5] , Corollary 6.2 (6.5)). Let $1 \leq p \leq \infty$ .

(1) Let $w \in F (C^{2} +)$ , and $n \geq 1$ be an integer. Then

${‖ \frac{w (g - v_{n} (g))}{T^{1 / 4}} ‖}_{L^{p} (ℝ)} \leq C E_{n, p} (w; g),$

where C do not depend on g and n.

(2) Let $w \in F_{λ} (C^{3} +) (0 < λ < 3 / 2)$ , and $n \geq 1$ be an integer. Then

${‖ w (g - v_{n} (g)) ‖}_{L^{p} (ℝ)} \leq C E_{n, p} (T^{1 / 4} w; g),$

where C do not depend on g and n.

Proposition 3.6. ( [6] ) Let $w = e x p (- Q) \in F (C^{2} +)$ , and let $0 < p \leq \infty$ . Then there exist $n_{0} \in ℕ$ and positive constants C₁, C₂ such that for every $w g \in L^{p} (ℝ)$ (and for $p = \infty$ , we require $g$ to be continuous, and $w f$ to vanish at $\pm \infty$ ) and every $n \geq n_{0}$ ,

$E_{n, p} (g; w) \leq C_{1} {\bar{ω}}_{p} (g, w; C_{2} \frac{a_{n}}{n}),$

where $n_{0}$ and $C_{1}, C_{2}$ do not depend on $g$ and $n$ , and $ω_{p}^{*} (g, w, t)$ will be defined in Section 6.

We set

$T^{〈 j 〉} : = \prod_{i = 1}^{j} T_{i}^{1 / 4}, ℝ_{(j)} : = {x_{j} \in ℝ},$

$ℝ_{\leq j}^{j} : = {(x_{1}, \dots, x_{j}) \in ℝ^{j}}, ℝ_{j \leq}^{s - j + 1} : = {(x_{j}, \dots, x_{s}) \in ℝ^{s - j + 1}},$

$ℝ_{j}^{s - 1} : = {(x_{1}, \dots, x_{j - 1}, x_{j + 1}, \dots, x_{s}) \in ℝ^{s - 1}},$

$W : = \prod_{i = 1}^{s} w_{i}, W_{j} : = \prod_{i = 1, i \neq j}^{s} w_{i}, j = 1, 2, \dots, s .$

We need the infinite-finite inequality.

Theorem 3.7 (Infinite-finite inequality). Let $0 < p \leq \infty, L > 0$ , and let

$P (X) \in P_{n, \dots, n} (ℝ^{s}) (= : P_{n; s} (ℝ^{s}))$ . Then

${‖ W (X) P (X) ‖}_{L^{p} (ℝ^{s})} \leq C {‖ W (X) P (X) ‖}_{L^{p} (| x_{i} | \leq a_{n}^{(i)} (1 - L δ_{n}^{(i)}), i = 1,2, \dots, s)} .$ (3.12)

If $r > 1$ , then there exists $ε > 0$ such that

${‖ W (X) P (X) ‖}_{L^{p} (ℝ_{i, r}^{s})} \leq C e x p (- n^{ε}) {‖ W (X) P (X) ‖}_{L^{p} (ℝ^{s})},$ (3.13)

where $ℝ_{i, r}^{s} : = {x_{i}; | x_{i} | \geq a_{r n}^{(i)}} \times ℝ_{i}^{s - 1}$ .

To prove Theorem 3.7 we use the following proposition with $s = 1$ .

Proposition 3.8 ( [2] , Theorem 1.9). Let $0 < p \leq \infty, L > 0$ , and let $P (x) \in P_{n} (ℝ)$ . Then

${‖ w (x) P (x) ‖}_{L^{p} (ℝ)} \leq C {‖ w (x) P (x) ‖}_{L^{p} (| x | \leq a_{n} (1 - L δ_{n}))} .$ (3.14)

If $r > 1$ , then there exists $ε > 0$ such that

${‖ w (x) P (x) ‖}_{L^{p} (a_{r n} \leq | x |)} \leq C e x p (- n^{ε}) {‖ w (x) P (x) ‖}_{L^{p} (| x | \leq a_{n})} .$ (3.15)

Proof of Theorem 3.7. For the proof of (3.12) we use (3.14). We put A for the left side of the above equation. Let $0 < p < \infty$ . By repeatedly applying Proposition 3.8 (3.14), we have

$\begin{matrix} A^{p} = \int_{- \infty}^{\infty} \dots \int_{- \infty}^{\infty} {| w_{2} (x_{2}) \dots w_{s} (x_{s}) |}^{p} \times {\int_{- \infty}^{\infty} {| w_{1} (x_{1}) P (x_{1}, \dots, x_{s}) |}^{p} d x_{1}} d x_{2} \dots d x_{s} \\ \leq C_{1} \int_{- \infty}^{\infty} \dots \int_{- \infty}^{\infty} {| w_{2} (x_{2}) \dots w_{s} (x_{s}) |}^{p} \int_{- a_{n}^{(1)} (1 - L δ_{n}^{(1)})}^{a_{n}^{(1)} (1 - L δ_{n}^{(1)})} {| w_{1} (x_{1}) P (x_{1}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s} \\ = C_{1} \int_{- a_{n}^{(1)} (1 - L δ_{n}^{(1)})}^{a_{n}^{(1)} (1 - L δ_{n}^{(1)})} w_{1}^{p} (x_{1}) \int_{- \infty}^{\infty} \dots \int_{- \infty}^{\infty} {| w_{2} (x_{2}) \dots w_{s} (x_{s}) P (x_{1}, \dots, x_{s}) |}^{p} d x_{2} \dots d x_{s} d x_{1} \\ \leq C_{2} \int_{- a_{n}^{(1)} (1 - L δ_{n}^{(1)})}^{a_{n}^{(1)} (1 - L δ_{n}^{(1)})} \int_{- a_{n}^{(2)} (1 - L δ_{n}^{(2)})}^{a_{n}^{(2)} (1 - L δ_{n}^{(2)})} w_{1}^{p} (x_{1}) w_{2}^{p} (x_{2}) \\ \times \int_{- \infty}^{\infty} \dots \int_{- \infty}^{\infty} {| w_{3} (x_{2}) \dots w_{s} (x_{s}) P (x_{1}, \dots, x_{s}) |}^{p} d x_{3} \dots d x_{d} d x_{1} d x_{2} \\ \leq \dots \\ \leq C_{s} \int_{- a_{n}^{(1)} (1 - L δ_{n}^{(1)})}^{a_{n}^{(1)} (1 - L δ_{n}^{(1)})} \dots \int_{- a_{n}^{(s)} (1 - L δ_{n}^{(s)})}^{a_{n}^{(s)} (1 - L δ_{n}^{(s)})} {| w_{1} (x_{1}) \dots w_{s} (x_{s}) P (x_{1}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s} \\ = C_{s} {‖ W (X) P (X) ‖}_{L^{p} (| x_{i} | \leq a_{n}^{(i)} (1 - L δ_{n}^{(i)}), i = 1, 2, \dots, s)}^{p} . \end{matrix}$

Next, we show the case of $p = \infty$ .

$\begin{matrix} A = \sup_{x_{s} \in ℝ} \dots \sup_{x_{2} \in ℝ} | w_{2} (x_{2}) \dots w_{s} (x_{s}) | \sup_{x_{1} \in ℝ} | w_{1} (x_{1}) P (x_{1}, \dots, x_{s}) | \\ \leq C_{1} \sup_{x_{s} \in ℝ} \dots \sup_{x_{2} \in ℝ} | w_{2} (x_{2}) \dots w_{s} (x_{s}) | \sup_{| x_{1} | \leq a_{n}^{(1)} (1 - L δ_{n}^{(1)})} | w_{1} (x_{1}) P (x_{1}, \dots, x_{s}) | \\ = C_{1} \sup_{x_{s} \in ℝ} \dots \sup_{x_{3} \in ℝ} | w_{3} (x_{3}) \dots w_{s} (x_{s}) | \sup_{| x_{1} | \leq a_{n}^{(1)} (1 - L δ_{n}^{(1)})} \sup_{x_{2} \in ℝ} | w_{1} (x_{1}) w_{2} (x_{2}) P (x_{1}, \dots, x_{s}) | \\ \leq C_{1} C_{2} \sup_{x_{s} \in ℝ} \dots \sup_{x_{3} \in ℝ} | w_{3} (x_{3}) \dots w_{s} (x_{s}) | \\ \times \sup_{| x_{1} | \leq a_{n}^{(1)} (1 - L δ_{n}^{(1)})} \sup_{| x_{2} | \leq a_{n}^{(2)} (1 - L δ_{n}^{(2)})} | w_{1} (x_{1}) w_{2} (x_{2}) P (x_{1}, \dots, x_{s}) | \\ \leq \dots \\ \leq C_{1} C_{2} \dots C_{s} \sup_{| x_{1} | \leq a_{n}^{(1)} (1 - L δ_{n}^{(1)})} \sup_{| x_{2} | \leq a_{n}^{(2)} (1 - L δ_{n}^{(2)})} \dots \sup_{| x_{2} | \leq a_{n}^{(s)} (1 - L δ_{n}^{(s)})} | w_{1} (x_{1}) w_{2} (x_{2}) \\ \times \dots \times w_{s} (x_{s}) P (x_{1}, \dots, x_{s}) | \\ = C \sup_{| x_{i} | \leq a_{n}^{(i)} (1 - L δ_{n}^{(i)}), i = 1, \dots, s} | w_{1} (x_{1}) w_{2} (x_{2}) \dots w_{s} (x_{s}) P (x_{1}, \dots, x_{s}) | . \end{matrix}$

Similarly, using Proposition 3.8 (3.15), we easily have (3.13). #

Lemma 3.9. Let $1 \leq p \leq \infty$ .

(1) We assume that $w_{i} \in F (C^{2} +), i = 1,2, \dots, s$ satisfies (3.6). Then there exists a constant $C > 0$ such that when $W h \in L_{p} (ℝ^{s})$ ,

${‖ \frac{W v_{n}^{[j]} (h)}{T^{〈 j 〉}} ‖}_{L^{p} (ℝ^{s})} \leq C {‖ W h ‖}_{L^{p} (ℝ^{s})}, j = 1, 2, \dots, s,$

and

${‖ W v_{n}^{[s]} (f) ‖}_{L^{p} (ℝ^{s})} \leq C \prod_{i = 1}^{s} T_{i}^{1 / 4} (a_{n}^{(i)}) {‖ W f ‖}_{L^{p} (ℝ^{s})} .$

(2) We assume that $w_{i} \in F_{λ} (C^{3} +) (0 < λ < 3 / 2), i = 1, 2, \dots, s$ . Let

$T^{〈 s 〉} W h \in L^{p} (ℝ^{s})$ , then

${‖ W v_{n}^{[j]} (h) ‖}_{L^{p} (ℝ^{s})} \leq C {‖ T^{〈 j 〉} W h ‖}_{L^{p} (ℝ^{s})}, j = 1,2, \dots, s .$

Proof. (1) From Theorem 3.4 (1), for $j = 1$

${‖ \frac{W v_{n}^{[1]} (h)}{T^{〈 1 〉}} ‖}_{L^{p} (ℝ^{s})} = {‖ {‖ \frac{W v_{n, 1} (h_{{\hat{X}}_{1}})}{T_{1}^{1 / 4}} ‖}_{L^{p} (ℝ_{\leq 1}^{1})} ‖}_{L^{p} (ℝ_{2 \leq}^{s - 1})} \leq C {‖ W h ‖}_{L^{p} (ℝ^{s})} .$

Inductively,

$\begin{matrix} {‖ \frac{W v_{n}^{[j]} (h)}{T^{〈 j 〉}} ‖}_{L^{p} (ℝ^{s})} = {‖ {‖ \frac{W v_{n}^{[j - 1]} (v_{n, j} (h))}{T^{〈 j 〉}} ‖}_{L^{p} (ℝ_{\leq j - 1}^{j - 1})} ‖}_{L^{p} (ℝ_{j \leq}^{s - j + 1})} \\ \leq C {‖ \frac{W v_{n, j} (h_{{\hat{X}}_{j}})}{T_{j}^{1 / 4}} ‖}_{L^{p} (ℝ^{s})} \leq C {‖ W h ‖}_{L^{p} (ℝ^{s})} . \end{matrix}$

For the second formula, using Theorem 3.7 and the above inequality, we have

$\begin{array}{l} {‖ W v_{n}^{[j]} (h) ‖}_{L^{p} (ℝ^{s})} \\ \leq C \prod_{i = 1}^{j} T_{i}^{1 / 4} (a_{n}^{(i)}) {‖ (\prod_{k = j + 1}^{s} w_{k}) {‖ \frac{(\prod_{i = 1}^{j} w_{i}) v_{n}^{[j]} (h)}{T^{〈 j 〉}} ‖}_{L^{p} (| x_{i} | \leq a_{2 n}^{(i)}), 1 \leq i \leq j} ‖}_{L^{p} (ℝ_{j + 1 \leq}^{s - j})} \\ \leq C \prod_{i = 1}^{j} T_{i}^{1 / 4} (a_{n}^{(i)}) {‖ W h ‖}_{L^{p} (ℝ^{s})}, \\ j = 1, 2, \dots, s . \end{array}$

Similarly we have the following:

(2) From Theorem 3.4 (2) for $j = 1$ ,

$\begin{array}{l} {‖ W v_{n}^{[1]} (h) ‖}_{L^{p} (ℝ^{s})} = {‖ W v_{n, 1} (h) ‖}_{L^{p} (ℝ^{s})} \leq C {‖ W T_{1}^{1 / 4} h ‖}_{L^{p} (ℝ^{s})}, \\ j = 1, 2, \dots, s . \end{array}$

Inductively,

$\begin{matrix} {‖ W v_{n}^{[j]} (h) ‖}_{L^{p} (ℝ^{s})} = {‖ W v_{n}^{[j - 1]} (v_{n, j} (h)) ‖}_{L^{p} (ℝ^{s})} \leq C {‖ W T^{〈 j - 1 〉} v_{n, j} (h) ‖}_{L^{p} (ℝ^{s})} \\ \leq C {‖ W T^{〈 j 〉} h ‖}_{L^{p} (ℝ^{s})} . \end{matrix}$ #

Proof of (3.7) in Theorem 3.3. By Proposition 3.5 (2) and Proposition 3.6, we get

${‖ w_{1} (x) (f_{{\hat{X}}_{1}} - v_{n, 1} (f_{{\hat{X}}_{1}})) ‖}_{L^{p} (ℝ)} \leq C E_{n, p} (f_{{\hat{X}}_{1}}; T_{1}^{1 / 4} w_{1}) \leq C_{1} {\bar{ω}}_{p, 1} (f_{{\hat{X}}_{1}}, T_{1}^{1 / 4} w_{1}; c_{1} \frac{a_{n}^{(1)}}{n}),$

where the constant C₁ and c₁ is independent of ${\hat{X}}_{1}$ . Similarly, for $j = 1, 2, \dots, s$ ,

${‖ w_{j} (x) (f_{{\hat{X}}_{j}} - v_{n, j} (f_{{\hat{X}}_{j}})) ‖}_{L^{p} (ℝ)} \leq C_{j} {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, T_{j}^{1 / 4} w_{j}; c_{j} \frac{a_{n}^{(j)}}{n}) .$ (3.16)

Using $f - v_{n}^{[s]} = (f - v_{n}^{[1]} (f)) + (v_{n}^{[1]} (f) - v_{n}^{[2]} (f)) + \dots + (v_{n}^{[s - 1]} (f) - v_{n}^{[s]} (f))$ , we get from Lemma 3.9 (2) and (3.2),

$\begin{array}{l} {‖ W (f - v_{n}^{[s]} (f)) ‖}_{L^{p} (ℝ^{s})} \\ \leq {‖ W (f - v_{n}^{[1]} (f)) ‖}_{L^{p} (ℝ^{s})} + \sum_{j = 2}^{s} {‖ W (v_{n}^{[j - 1]} (f) - v_{n}^{[j]} (f)) ‖}_{L^{p} (ℝ^{s})} \\ \leq C [{‖ W (f - v_{n, 1} (f)) ‖}_{L^{p} (ℝ^{s})} + \sum_{j = 2}^{s} {‖ W v_{n}^{[j - 1]} (f - v_{n, j} (f)) ‖}_{L^{p} (ℝ^{s})}] \\ \leq C [{‖ W (f - v_{n, 1} (f)) ‖}_{L^{p} (ℝ^{s})} + \sum_{j = 2}^{s} {‖ W (\prod_{k = 1}^{j - 1} T_{k}^{1 / 4}) (f - v_{n, j} (f)) ‖}_{L^{p} (ℝ^{s})}] \\ \leq C_{1} {‖ W_{1} {\bar{ω}}_{p, 1} (f_{{\hat{X}}_{1}}, T_{1}^{1 / 4} w_{1}; c_{1} \frac{a_{n}^{(1)}}{n}) ‖}_{L^{p} (ℝ_{1}^{s - 1})} \\ + \sum_{j = 2}^{s} C_{j} {‖ W_{j} (\prod_{k = 1}^{j - 1} T_{k}^{1 / 4}) {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, T_{j}^{1 / 4} w_{j}; c_{j} \frac{a_{n}^{(j)}}{n}) ‖}_{L^{p} (ℝ_{j}^{s - 1})} \end{array}$

by (3.16)

$\leq \sum_{j = 1}^{s} C_{j} {‖ W_{j} (\prod_{k = 1}^{j - 1} T_{k}^{1 / 4}) {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, T_{j}^{1 / 4} w_{j}; c_{j} \frac{a_{n}^{(j)}}{n}) ‖}_{L^{p} (ℝ_{j}^{s - 1})},$

where $\prod_{k = 1}^{j - 1} T_{k}^{1 / 4} = 1$ for $j = 1$ . Hence we obtain (3.7).

Proof of (3.10) in Theorem 3.3. By Proposition 3.5 (1) and Proposition 3.6, we get

${‖ \frac{w_{1} (x) (f_{{\hat{X}}_{1}} - v_{n, 1} (f_{{\hat{X}}_{1}}))}{T_{1}^{1 / 4} (x)} ‖}_{L^{p} (ℝ)} \leq C E_{n, p} (f_{{\hat{X}}_{1}}; w_{1}) \leq C_{1} {\bar{ω}}_{p, 1} (f_{{\hat{X}}_{1}}, w_{1}; c_{1} \frac{a_{n}^{(1)}}{n}),$

where the constant C₁ and c₁ is independent of ${\hat{X}}_{1}$ . Similarly, for $j = 1, 2, \dots, s$ ,

${‖ \frac{w_{j} (x) (f_{{\hat{X}}_{j}} - v_{n, j} (f_{{\hat{X}}_{j}}))}{T_{j}^{1 / 4} (x)} ‖}_{L^{p} (ℝ)} \leq C_{j} {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, w_{j}; c_{j} \frac{a_{n}^{(j)}}{n}) .$ (3.17)

Using $f - v_{n}^{[s]} = (f - v_{n}^{[1]} (f)) + (v_{n}^{[1]} (f) - v_{n}^{[2]} (f)) + \dots + (v_{n}^{[s - 1]} (f) - v_{n}^{[s]} (f))$ , we get from Lemma 3.9 (1) and (3.2),

$\begin{array}{l} {‖ \frac{W (f - v_{n}^{[s]} (f))}{T^{〈 s 〉}} ‖}_{L^{p} (ℝ^{s})} \\ \leq {‖ \frac{W (f - v_{n}^{[1]} (f))}{T^{〈 1 〉}} ‖}_{L^{p} (ℝ^{s})} + \sum_{j = 2}^{s} {‖ \frac{W (v_{n}^{[j - 1]} (f) - v_{n}^{[j]} (f))}{T^{〈 j 〉}} ‖}_{L^{p} (ℝ^{s})} \\ \leq C [{‖ \frac{W (f - v_{n, 1} (f))}{T_{1}^{1 / 4}} ‖}_{L^{p} (ℝ^{s})} + \sum_{j = 2}^{s} {‖ \frac{W v_{n}^{[j - 1]} ({f - v_{n, j} (f)}) / T_{j}^{1 / 4}}{T^{〈 j - 1 〉}} ‖}_{L^{p} (ℝ^{s})}] \\ \leq C [{‖ \frac{W (f - v_{n, 1} (f))}{T_{1}^{1 / 4}} ‖}_{L^{p} (ℝ^{s})} + \sum_{j = 2}^{s} {‖ \frac{W (f - v_{n, j} (f))}{T_{j}^{1 / 4}} ‖}_{L^{p} (ℝ^{s})}] \\ \leq C_{1} {‖ (\prod_{2 \leq i \leq s} w_{i}) {\bar{ω}}_{p, 1} (f_{{\hat{X}}_{1}}, w_{1}; c_{1} \frac{a_{n}^{(1)}}{n}) ‖}_{L^{p} (ℝ_{1}^{s - 1})} \\ + \sum_{j = 2}^{s} C_{j} {‖ (\prod_{1 \leq i \leq s, i \neq j} w_{i}) {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, w_{j}; c_{j} \frac{a_{n}^{(j)}}{n}) ‖}_{L^{p} (ℝ_{j}^{s - 1})} \end{array}$

by (3.17)

$\leq \sum_{j = 1}^{s} C_{j} {‖ (\prod_{1 \leq i \leq s, i \neq j} w_{i}) {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, w_{j}; c_{j} \frac{a_{n}^{(j)}}{n}) ‖}_{L^{p} (ℝ_{j}^{s - 1})} .$

Hence we obtain (3.10).

To prove (3.9) and (3.11) we need a lemma:

Lemma 3.10. Let $0 < h \leq 1$ , $σ (t) \geq 1$ and $| x | \leq σ (2 t)$ . We have

${‖ (w f) (x \pm h Φ_{t} (x)) ‖}_{L^{p} (ℝ)} ~ {‖ (w f) (x) ‖}_{L^{p} (ℝ)} .$ (3.18)

Proof. We may show

${‖ (w f) (x \pm h \sqrt{1 - \frac{| x |}{σ (t)}}) ‖}_{L^{p} (ℝ)} ~ {‖ (w f) (x) ‖}_{L^{p} (ℝ)} .$ (3.19)

Let $x > 0$ . If we put

$x \pm h \sqrt{1 - \frac{x}{σ (t)}} = : y, \frac{a_{2 u}}{2 u} = t, \frac{a_{v}}{v} = 2 t,$ (3.20)

we will see

$\frac{1}{2} \leq \frac{d y}{d x} \leq \frac{3}{2} .$ (3.21)

Then we conclude (3.19). Now, from (3.20)

$\frac{d y}{d x} = 1 \mp \frac{h}{2 \sqrt{σ (t)} \sqrt{σ (t) - x}} .$

Since $a_{2 u} / u = 2 t$ , we see

$\frac{a_{v}}{v} = \frac{a_{2 u}}{u} > \frac{a_{u}}{u},$

that is, we have

$σ (2 t) = a_{v} < a_{u} < σ (t) = a_{2 u} .$

Then, using ( [2] , Lemmas 3.6, 3.7), we see

$\begin{matrix} \frac{h}{2 \sqrt{σ (t)} \sqrt{σ (t) - x}} \leq \frac{t}{2 \sqrt{σ (t)} \sqrt{σ (t) - σ (2 t)}} \leq \frac{\sqrt{a_{2 u}}}{4 u} \frac{1}{\sqrt{a_{2 u} - a_{u}}} \\ \leq \frac{\sqrt{a_{2 u}}}{4 u} \sqrt{C_{1} \frac{T (a_{2 u})}{a_{2 u}}} \leq \frac{1}{4 u} \sqrt{C_{1} C_{2}} u^{1 - δ} = \frac{\sqrt{C_{1} C_{2}}}{4} \frac{1}{u^{δ}} \leq \frac{1}{2} \end{matrix}$

for some $0 < δ \leq 1$ and $u$ large enough. We have (3.21). So we conclude (3.19). #

Proof of (3.9). We will estimate

${‖ W_{j} T^{〈 j - 1 〉} {\frac{1}{t} \int_{0}^{t} {‖ T_{j}^{1 / 4} (x) w_{j} (x) (Δ_{h Φ_{t, j} (x)} f_{{\hat{X}}_{j}} (x)) ‖}_{L^{p} (| x | \leq σ_{j} (2 t))}^{p} d h}^{1 / p} ‖}_{L^{p} (ℝ_{j}^{s - 1})} .$

To do so we may estimate

$I_{1} : = {\frac{1}{t} \int_{0}^{t} {‖ T_{j}^{1 / 4} (x) w_{j} (x) (Δ_{h Φ_{t, j} (x)} f_{{\hat{X}}_{j}} (x)) ‖}_{L^{p} (| x | \leq σ_{j} (2 t))}^{p} d h}^{1 / p} .$

For $ε > 0$ we take $K > 0$ large enough, and then by $f \in L_{T^{〈 s 〉} w}^{p *} (ℝ)$ we can select a continuous function $f_{K}$ such that

$\begin{matrix} I_{1} \leq {\frac{1}{t} \int_{0}^{t} {‖ T_{j}^{1 / 4} (x) w_{j} (x) (Δ_{h Φ_{t, j} (x)} {(f - f_{K})}_{{\hat{X}}_{j}} (x)) ‖}_{L^{p} (| x | \leq σ_{j} (2 t))}^{p} d h}^{1 / p} \\ + {\frac{1}{t} \int_{0}^{t} {‖ T_{j}^{1 / 4} (x) w_{j} (x) (Δ_{h Φ_{t, j} (x)} {(f_{K})}_{{\hat{X}}_{j}} (x)) ‖}_{L^{p} (| x | \leq σ_{j} (2 t))}^{p} d h}^{1 / p} \\ = A + B . \end{matrix}$

We note $w_{j} (x) ~ w (x + (h / 2) Φ_{t, j} (x)) ~ w (x - (h / 2) Φ_{t, j} (x))$ (see [7] , Lemma 7). If $σ_{j} (2 t) \leq K$ from our assumption and Lemma 3.10 we have

$\begin{matrix} A \leq C {\frac{1}{t} \int_{0}^{t} {‖ T_{j}^{1 / 4} (x) w_{j} (x) {(f - f_{K})}_{{\hat{X}}_{j}} (x) ‖}_{L^{p} (| x | \leq σ_{j} (2 t))}^{p} d h}^{1 / p} \\ \leq C ε {\frac{1}{t} \int_{0}^{t} d h}^{1 / p} \leq C ε . \end{matrix}$

If $σ_{j} (2 t) > K$ , then by Lemma 3.10 we see

$\begin{matrix} A \leq C {\frac{1}{t} \int_{0}^{t} {‖ T_{j}^{1 / 4} (x) w_{j} (x) {(f - f_{K})}_{{\hat{X}}_{j}} (x) ‖}_{L^{p} (| x | \leq K)}^{p} d h}^{1 / p} \\ + {\frac{1}{t} \int_{0}^{t} {‖ T_{j}^{1 / 4} (x) w_{j} (x) f_{{\hat{X}}_{j}} (x) ‖}_{L^{p} (K < | x | \leq σ_{j} (2 t))}^{p} d h}^{1 / p} \\ \leq C ε + C_{1} ε \leq C_{2} ε . \end{matrix}$

When we take $t > 0$ small enough, we see

$\begin{matrix} B = {\frac{1}{t} \int_{0}^{t} {‖ T_{j}^{1 / 4} (x) w_{j} (x) (Δ_{h Φ_{t, j} (x)} {(f_{K})}_{{\hat{X}}_{j}} (x)) ‖}_{L^{p} (| x | \leq K)}^{p} d h}^{1 / p} \\ \leq {\frac{1}{t} \int_{0}^{t} ε d h}^{1 / p} \leq C ε, \end{matrix}$

because of the continuity of $f_{K}$ . Therefore we have $I_{1} < ε$ . Consequently, we have

$\begin{array}{l} {‖ W_{j} T^{〈 j - 1 〉} {\frac{1}{t} \int_{0}^{t} {‖ T_{j}^{1 / 4} (x) w_{j} (x) (Δ_{h Φ_{t, j} (x)} f_{{\hat{X}}_{j}} (x)) ‖}_{L^{p} (| x | \leq σ_{j} (2 t))}^{p} d h}^{1 / p} ‖}_{L^{p} (ℝ_{j}^{s - 1})} \\ \leq C ε {‖ W_{j} T^{〈 j - 1 〉} ‖}_{L^{p} (ℝ_{j}^{s - 1})} \leq C ε . \end{array}$ (3.22)

Finally, we see

$\begin{array}{l} {‖ W_{j} T^{〈 j - 1 〉} \underset{c_{j} (constant)}{i n f} {‖ (f_{{\hat{X}}_{j}} (x) - c_{j}) T_{j}^{1 / 4} (x) w_{j} (x) ‖}_{L^{p} (ℝ \ [- σ_{j} (4 t), σ_{j} (4 t)])} ‖}_{L^{p} (ℝ_{j}^{s - 1})} \\ \leq C {‖ W_{j} T^{〈 j - 1 〉} {‖ f_{{\hat{X}}_{j}} (x) T_{j}^{1 / 4} (x) w_{j} (x) ‖}_{L^{p} (ℝ \ [- σ_{j} (4 t), σ_{j} (4 t)])} ‖}_{L^{p} (ℝ_{j}^{s - 1})} \\ \leq C w_{j}^{1 / 4} (σ_{j} (4 t)) {‖ W f ‖}_{L^{p} (ℝ_{j}^{s})} . \end{array}$

Here, if we set $4 t = a_{u} / u$ , then we see

$w_{j}^{1 / 4} (σ_{j} (4 t)) = e x p (- \frac{1}{4} Q_{j} (a_{u})) ~ e x p (- \frac{u}{2 \sqrt{T (a_{u})}}) \leq e^{- u^{δ}}$

for some $0 < δ < 1$ , that is,

$w_{j}^{1 / 4} (σ_{j} (4 t)) \leq C e^{- u^{δ}} \leq C \frac{a_{u}}{4 u} = C t .$

Therefore

For given $ε > 0$ if we take $K > 0$ large enough and then $t > 0$ small enough, then by (3.22) and (3.23) we have

${‖ W_{j} T^{〈 j - 1 〉} {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, T_{j}^{1 / 4} w_{j}, t) ‖}_{L^{p} (ℝ_{j}^{s - 1})} < ε .$

Consequently, we have (3.9).

Proof of (3.11). If in the proof of (3.9) we set as $T = 1$ (constant), then we obtain (3.11). #

Corollary 3.11. We suppose that $w_{j} = e x p (- Q_{j}) \in F (C^{2} +), j = 1,2, \dots, s$ are the Freud-type weights. Let $1 \leq p \leq \infty$ . Then we have

$\begin{array}{l} {‖ W (f - v_{n}^{[s]} (f)) ‖}_{L^{p} (ℝ^{s})} \\ \leq \sum_{j = 1}^{s} C_{j} {‖ (\prod_{1 \leq i \leq s, i \neq j} w_{i}) {\bar{ω}}_{p, j} (f_{{\hat{X}}_{j}}, w_{j}; c_{j} \frac{a_{n}^{(j)}}{n}) ‖}_{L^{p} (ℝ_{j}^{s - 1})} . \end{array}$

Especially $f \in L_{W}^{p *} (ℝ^{s})$ , then we have

Corollary 3.12. We suppose

$w_{j} = e x p (- Q_{j}) \in F_{λ} (C^{3} +) (0 < λ < 3 / 2), j = 1, 2, \dots, s$ , and let (3.6) hold. Let $1 \leq p \leq \infty$ . If $T^{〈 s 〉} W f \in C_{0} (ℝ^{s}) \cap L^{p} (ℝ^{s})$ ), then we have

${‖ W (f - v_{n}^{[s]} (f)) ‖}_{L^{p} (ℝ^{s})} \to 0 as n \to \infty .$

Moreover, we suppose $w_{j} = e x p (- Q_{j}) \in F (C^{2} +), j = 1,2, \dots, s$ , and let (3.6) hold. If $W f \in C_{0} (ℝ^{s}) \cap L^{p} (ℝ^{s})$ , then we have

${‖ \frac{W (f - v_{n}^{[s]} (f))}{T^{〈 s 〉}} ‖}_{L^{p} (ℝ^{s})} \to 0 as n \to \infty .$

4. A Property of Higher Order Derivatives

In this section we show an important theorem which is useful in approximation theory. We use the following notations for

$w_{i} (x_{i}) = \exp (- Q_{i} (x_{i})) \in F (C^{2} +), i = 1, 2, \dots, s$ . Let r be a positive integer, and $| x_{i} | \geq γ > 0$ .

$W_{0} : = \prod_{i = 1}^{s} w_{i, 0}; {Q^{'}}_{i} {(x_{i})}^{r} w_{i} (x_{i}) ~ w_{i, 0} (x_{i}) = \exp (- Q_{i, 0} (x_{i})) \in F (C^{2} +),$

$\begin{array}{l} W_{ν} : = \prod_{i = 1}^{s} w_{i, ν}; {Q^{'}}_{i} {(x_{i})}^{r - ν} w_{i} (x_{i}) ~ w_{i, ν} (x_{i}) = \exp (- Q_{i, ν} (x_{i})) \in F (C^{2} +), \\ ν = 1, 2, \dots, r . \end{array}$

Then we see

${Q^{'}}_{i} {(x_{i})}^{r - ν + 1} w_{i} (x_{i}) ~ {Q^{'}}_{i, ν} (x_{i}) w_{i, ν} (x_{i}) .$

Especially, if $ν = r$ , then

$w_{i, r} (x_{i}) = w_{i} (x_{i}) .$

Theorem 4.1. Let $w_{i} = \exp (- Q_{i}) \in F (C^{2} +), i = 1, 2, \dots, s$ , and let $1 \leq p \leq \infty$ . Let a constant $γ \geq 0$ be fixed. We suppose that $g : = g (x_{1}, x_{2}, \dots, x_{s})$ is absolutely continuous on $ℝ^{s}$ and $W g^{(1,1, \dots,1)} \in L^{p} (ℝ^{s})$ . Then we have

$\begin{array}{l} {‖ \prod_{i = 1}^{s} ({Q^{'}}_{i} w_{i}) g ‖}_{L^{p} (| x_{i} | \leq γ, i = 1, \dots, s)} \\ \leq C {\int_{| x_{1} | \leq γ} \dots \int_{| x_{s} | \leq γ} \sum_{0 \leq j_{1} \leq 1} {| w_{1, j_{1}} (x_{1}) |}^{p} \dots \sum_{0 \leq j_{s} \leq 1} {| w_{s, j_{s}} (x_{s}) |}^{p} \\ \times {| g^{(j_{1}, j_{2}, \dots, j_{s})} (y_{1}, y_{2}, \dots, y_{s}) |}^{p} d x_{s} \dots d x_{1}}^{1 / p}, \end{array}$ (4.1)

where we set for each $j_{i} = 0$ or 1, $i = 1, 2, \dots, s$ ,

$y_{i} = {\begin{cases} γ, j_{i} = 0; \\ x_{i}, j_{i} = 1. \end{cases}$

Furthermore, let r be a positive integer, and let for each $i = 1, 2, \dots, s$ , $w_{i} = \exp (- Q_{i}) \in F_{λ} (C^{3} +) \subset F (C^{2} +)$ $(0 < λ < 3 / 2)$ . We suppose that $g^{(r - 1, r - 1, \dots, r - 1)}$ is absolutely continuous and $W g^{(r, r, \dots, r)} \in L^{p} (ℝ^{s})$ . Then we have

${\int_{ℝ^{s}} {| w_{1, j_{1}} (x_{1}) |}^{p} {| w_{s, j_{s}} (x_{s}) |}^{p} {| g^{(j_{1}, j_{2}, \dots, j_{s})} (y_{1}, y_{2}, \dots, y_{s}) |}^{p} d x_{s} \dots d x_{1}}^{1 / p} < \infty,$ (4.2)

for each $0 \leq j_{i} \leq r, i = 1,2, \dots, s$ with

$y_{i} = {\begin{cases} γ, 0 \leq j_{i} \leq r - 1; \\ x_{i}, j_{i} = r, \end{cases}$ (4.3)

and

$\begin{array}{l} {‖ {(\prod_{i = 1}^{s} {Q^{'}}_{i})}^{r} W g ‖}_{L^{p} (| x_{i} | \geq γ, i = 1, \dots, s)} \\ \leq C {\int_{| x_{1} | \geq γ} \dots \int_{| x_{s} | \geq γ} \sum_{0 \leq j_{1} \leq r} {| w_{1, j_{1}} (x_{1}) |}^{p} \dots \sum_{0 \leq j_{s} \leq r} {| w_{s, j_{s}} (x_{s}) |}^{p} \\ \times {| g^{(j_{1}, j_{2}, \dots, j_{s})} (y_{1}, y_{2}, \dots, y_{s}) |}^{p} d x_{s} \dots d x_{1}}^{1 / p} < \infty . \end{array}$ (4.4)

Proposition 4.2 ( [8] , Theorem 9, cf. [9] , Lemma 3.4.4). Let

$w = e x p (- Q) \in F (C^{2} +)$ and a constant $γ \geq 0$ be fixed.

(a) We have

$| Q^{'} (x) w (x) \int_{γ}^{x} w^{- 1} (t) d t | \leq C, | x | \geq γ .$

(b) Let $1 \leq p \leq \infty$ , and let r be a positive integer. If g is absolutely continuous, $g (γ) = 0$ and $w g^{'} \in L^{p} (ℝ)$ , then

${‖ Q^{'} w g ‖}_{L^{p} (| x | \geq γ)} \leq C {‖ w g^{'} ‖}_{L^{p} (| x | \geq γ)} .$

When $w = e x p (- Q) \in F_{λ} (C^{r + 1} +) \subset F (C^{2} +)$ ( $0 < λ < (r + 1) / r$ ), and $g^{(r - 1)}$ is absolutely continuous, $g^{(j)} (γ) = 0, j = 0, 1, \dots, r - 1$ with $w g^{(r)} \in L^{p} (ℝ)$ , we see

${‖ {(Q^{'})}^{r} w g ‖}_{L^{p} (| x | \geq γ)} \leq C {‖ w g^{(r)} ‖}_{L^{p} (| x | \geq γ)} .$

Proposition 4.3 ( [3] , Theorem 4.2). Let $w = e x p (- Q) \in F_{λ} (C^{3} +) \subset F (C^{2} +)$ . Then for $α \in ℝ$ , we can construct a new weight $w_{α} \in F (C^{2} +)$ such that

${(1 + | Q^{'} (x) |)}^{α} w (x) ~ w_{α} (x) = e x p (- Q_{α})$

on $ℝ$ , $(1 / c) a_{n} (w) \leq a_{n} (w_{α}) \leq c a_{n} (w)$ (c is an absolutely constant) on $ℕ$ and $T_{w_{α}} (x) ~ T_{w} (x)$ hold on $ℝ$ . Furthermore, we see

$Q_{α}^{(j)} (x) ~ Q^{(j)} (x) (j = 0, 1) for | x | \geq γ > 0.$

Proof of Theorem 4.1. For the proof of (4.1) we may put $r = 1$ with $w_{i} = \exp (- Q_{i}) \in F (C^{2} +), i = 1, 2, \dots, s$ in the proof of (4.4) below. So we prove only (4.4). We use Proposition 4.2 and 4.3 repeatedly.

$\begin{array}{l} {\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{1} ‖ \geq γ} {| \prod_{i = 1}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) g (x_{1}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}^{1 / p} \\ = {\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{2} ‖ \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {\int_{‖ x_{1} ‖ \geq γ} {| {Q^{'}}_{1}^{r} (x_{1}) w_{1} (x_{1}) g (x_{1}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}}^{1 / p} \\ \leq C_{1, 0} [{\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{2} ‖ \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {\int_{‖ x_{1} ‖ \geq γ} {| {Q^{'}}_{1, 1} (x_{1}) w_{1, 1} (x_{1}) g (x_{1}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}}^{1 / p}], \end{array}$

where ${({Q^{'}}_{1})}^{r} w_{1} = {Q^{'}}_{1} {({Q^{'}}_{1})}^{r - 1} w_{1} ~ {Q^{'}}_{1} w_{1, 1} ~ {Q^{'}}_{1, 1} w_{1, 1}, w_{1, 1} = e^{- Q_{1, 1}},$

$\begin{array}{l} \leq C_{1, 0} [{\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{2} ‖ \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {\int_{‖ x_{1} ‖ \geq γ} {| {Q^{'}}_{1, 1} (x_{1}) w_{1, 1} (x_{1}) (g (x_{1}, \dots, x_{s}) - g (γ, x_{2}, \dots, x_{s})) |}^{p} d x_{1} \dots d x_{s}}}^{1 / p} \\ + {\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{2} ‖ \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {\int_{‖ x_{1} ‖ \geq γ} {| {Q^{'}}_{1, 1} (x_{1}) w_{1, 1} (x_{1}) g (γ, x_{2}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}}^{1 / p}] \\ \leq C_{1, 1} [{\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{2} ‖ \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {\int_{‖ x_{1} ‖ \geq γ} {| w_{1, 1} (x_{1}) g^{(1, 0, \dots, 0)} (x_{1}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}}^{1 / p} \\ + {\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{1} ‖ \geq γ} {| \prod_{i = 1}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) g (γ, x_{2}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}^{1 / p}] \end{array}$

by ${Q^{'}}_{1,1} w_{1,1} ~ {({Q^{'}}_{1})}^{r} w_{1},$

$\begin{array}{l} \leq {C^{'}}_{1, 1} [{\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{2} ‖ \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {\int_{‖ x_{1} ‖ \geq γ} {| {Q^{'}}_{1, 2} (x_{1}) w_{1, 2} (x_{1}) g^{(1, 0, \dots, 0)} (x_{1}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}}^{1 / p} \\ where w_{1, 1} ~ {({Q^{'}}_{1})}^{r - 1} w ~ {Q^{'}}_{1, 2} w_{1, 2}, w_{1, 2} = e^{- Q_{1, 2}}, \\ + {\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{1} ‖ \geq γ} {| \prod_{i = 1}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) g (γ, x_{2}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}^{1 / p}] \end{array}$

$\begin{array}{l} \leq C_{1, 2} [{\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{2} ‖ \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {\int_{‖ x_{1} ‖ \geq γ} {| w_{1, 2} (x_{1}) g^{(2, 0, \dots, 0)} (x_{1}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}}^{1 / p} \\ + {\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{1} ‖ \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {| {Q^{'}}_{1}^{r - 1} (x_{1}) w_{1} (x_{1}) g^{(1, 0, \dots, 0)} (γ, x_{2}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}^{1 / p} \\ + {\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{1} ‖ \geq γ} {| \prod_{i = 1}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) g (γ, x_{2}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}^{1 / p}] \end{array}$

$\begin{array}{l} \leq \dots \\ \leq C_{1, r} [{\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{2} ‖ \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {\int_{‖ x_{1} ‖ \geq γ} {| w_{1} (x_{1}) g^{(r, 0, \dots, 0)} (x_{1}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}}^{1 / p} \\ + {\int_{‖ x_{s} ‖ \geq γ} \dots \int_{‖ x_{1} ‖ \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {| {Q^{'}}_{1} (x_{1}) w_{1} (x_{1}) g^{(1, 0, \dots, 0)} (γ, x_{2}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}^{1 / p} \end{array}$

$\begin{array}{l} + \dots \\ + {\int_{‖ x_{s} ‖ \geq γ} ... \int_{‖ x_{1} ‖ \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {| {Q^{'}}_{1}^{r - 1} (x_{1}) w_{1} (x_{1}) g^{(1, 0, \dots, 0)} (γ, x_{2}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}^{1 / p} \\ + {\int_{‖ x_{s} ‖ \geq γ} ... \int_{‖ x_{1} ‖ \geq γ} {| \prod_{i = 1}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) g (γ, x_{2}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}^{1 / p}] \end{array}$

$\begin{array}{l} = C_{1, r} {\int_{‖ x_{s} ‖ \geq γ} ... \int_{‖ x_{2} ‖ \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {\int_{‖ x_{1} ‖ \geq γ} \sum_{0 \leq j_{1} \leq r} {| w_{1, j_{1}} (x_{1}) g^{(j_{1}, 0, \dots, 0)} (y_{1}, x_{2}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}}^{1 / p}, \end{array}$

where

$y_{1} = {\begin{cases} γ, 0 \leq j_{1} \leq r - 1; \\ x_{1}, j_{1} = r . \end{cases}$

We continue this manner with respect to $x_{2}, x_{3}, \dots, x_{s}$ . Then we can easily obtain as follows:

$\begin{array}{l} {\int_{| x_{s} | \geq γ} \dots \int_{| x_{1} | \geq γ} {| \prod_{i = 1}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) g (x_{1}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}^{1 / p} \\ \leq C_{1, r} {\int_{| x_{s} | \geq γ} \dots \int_{| x_{2} | \geq γ} {| \prod_{i = 2}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {\int_{| x_{1} | \geq γ} \sum_{0 \leq j_{1} \leq r} {| w_{1, j_{1}} (x_{1}) g^{(j_{1}, 0, \dots, 0)} (y_{1}, x_{2}, \dots, x_{s}) |}^{p} d x_{1} \dots d x_{s}}}^{1 / p} \end{array}$

$\begin{array}{l} \leq C_{2} {\int_{| x_{1} | \geq γ} \sum_{0 \leq j_{1} \leq r} {| w_{1, j_{1}} (x_{1}) |}^{p} \int_{| x_{s} | \geq γ} \dots \int_{| x_{3} | \geq γ} {| \prod_{i = 3}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \\ \times {\int_{| x_{2} | \geq γ} \sum_{0 \leq j_{2} \leq r} {| w_{2, j_{2}} (x_{2}) g^{(j_{1}, j_{2}, 0, \dots, 0)} (y_{1}, y_{2}, x_{3}, \dots, x_{s}) |}^{p} d x_{2} \dots d x_{s} d x_{1}}}^{1 / p} \end{array}$

$\begin{array}{l} = C_{2} {\begin{cases} \int_{| x_{1} | \geq γ} \int_{| x_{2} | \geq γ} \sum_{0 \leq j_{1} \leq r} {| w_{1, j_{1}} (x_{1}) |}^{p} \sum_{0 \leq j_{2} \leq r} {| w_{2, j_{2}} (x_{2}) |}^{p} \int_{| x_{s} | \geq γ} \\ \dots \int_{| x_{4} | \geq γ} {| \prod_{i = 4}^{s} ({Q^{'}}_{i}^{r} (x_{i}) w_{i} (x_{i})) |}^{p} \end{cases} \\ \times {\int_{| x_{3} | \geq γ} {| w_{3, j_{3}} (x_{3}) g^{(j_{1}, j_{2}, j_{3}, 0, \dots, 0)} (y_{1}, y_{2}, y_{3}, x_{4}, \dots, x_{s}) |}^{p} \times d x_{3} \dots d x_{s} d x_{2} d x_{1}}}^{1 / p} \end{array}$

$\begin{array}{l} \leq \dots \\ \leq C_{r} {\int_{| x_{1} | \geq γ} \dots \int_{| x_{s} | \geq γ} \sum_{0 \leq j_{1} \leq r} {| w_{1, j_{1}} (x_{1}) |}^{p} \dots \sum_{0 \leq j_{s} \leq r} {| w_{s, j_{s}} (x_{s}) |}^{p} \\ {\underset{}{_{}^{}} \times g^{(j_{1}, j_{2}, \dots, j_{s})} (y_{1}, y_{2}, \dots, y_{s}) |^{p} d x_{s} \dots d x_{1}}}^{1 / p}, \end{array}$

where for each $0 \leq j_{i} \leq r, i = 1, 2, \dots, s$ . We set (4.3).

Let $W g^{(r, \dots, r)} \in L^{p} (ℝ^{s})$ . Then we need to show

$\begin{array}{l} A : = {\int_{| x_{1} | \geq γ} \dots \int_{| x_{s} | \geq γ} \sum_{0 \leq j_{1} \leq r} {| w_{1, j_{1}} (x_{1}) |}^{p} \dots \sum_{0 \leq j_{s} \leq r} {| w_{s, j_{s}} (x_{s}) |}^{p} \\ \times {| g^{(j_{1}, j_{2}, \dots, j_{s})} (y_{1}, y_{2}, \dots, y_{s}) |}^{p} d x_{s} \dots d x_{1}}^{1 / p} < \infty . \end{array}$

We rearrange $(x_{1}, x_{2}, \dots, x_{s})$ as $x_{k_{i}} = y_{i} = x_{i}, t + 1 \leq k_{i} \leq s$ if $j_{i} = r$ , and as $x_{k_{i}} = y_{i} = γ < 1 \leq k_{i} \leq t$ if $0 \leq j_{i} \leq r - 1$ , where $0 \leq t \leq s$ . Then we set

$f (x_{k_{1}}, x_{k_{2}}, \dots, x_{k_{s}}) : = g (x_{1}, x_{2}, \dots, x_{s})$ . We see

$g^{(j_{1}, j_{2}, \dots, j_{s})} (y_{1}, y_{2}, \dots, y_{s}) = f^{(j_{k_{1}}, j_{k_{2}}, \dots, j_{k_{s}})} (γ, \dots, γ, x_{t + 1}, x_{t + 2}, \dots, x_{s}) .$

Then we have

$\begin{matrix} A = {‖ \prod_{i = 1}^{t} w_{k_{i}} {‖ \prod_{i = t + 1}^{s} w_{k_{i}} f^{(j_{k_{1}}, \dots, j_{k_{s}})} (γ, \dots, γ, x_{t + 1}, \dots, x_{s}) ‖}_{L^{p} (ℝ_{t + 1 \leq}^{s - t})} ‖}_{L^{p} (ℝ_{\leq t}^{t})} \\ = : {‖ \prod_{i = 1}^{t} w_{k_{i}} | h^{(j_{k_{1}}, \dots, j_{k_{t}})} (γ, \dots, γ) | ‖}_{L^{p} (ℝ_{\leq t}^{t})} \\ \leq C | h^{(j_{k_{1}}, \dots, j_{k_{t}})} (γ, \dots, γ) | {‖ \prod_{i = 1}^{t} w_{k_{i}} ‖}_{L^{p} (ℝ_{\leq t}^{t})} < \infty . \end{matrix}$ #

We can generalize Theorem 4.1 easily. We give a class of nonnegative integers $(j_{1}, j_{2}, \dots, j_{s})$ , and set $J_{s} : = (j_{1}, j_{2}, \dots, j_{s})$ . For $r_{i} \geq 1, i = 1, 2, \dots, s$ we set $R_{s} : = (r_{1}, r_{2}, \dots, r_{s})$ . Then we consider the order as follows:

$K_{s} : = (k_{1}, k_{2}, \dots, k_{s}) \leq R_{s} : = (r_{1}, r_{2}, \dots, r_{s})$

means

$k_{i} \leq r_{i} (i = 1, 2, \dots, s) .$

Corollary 4.4. Let $K_{s} = (k_{1}, k_{2}, \dots, k_{s}) \leq R_{s} = (r_{1}, r_{2}, \dots, r_{s})$ be classes of nonnegative integers, where $r_{i} \geq 1, i = 1, 2, \dots, s$ . For each $i = 1, 2, \dots, s$ , we

suppose $w_{i} = \exp (- Q_{i}) \in F_{λ} (C^{r + 1} +) \subset F (C^{2} +)$ ( $0 < λ < (r + 1) / r$ ). If

$g^{(r_{1} - 1, r_{2} - 1, \dots, r_{s} - 1)}$ is absolutely continuous, and $W g^{(r_{1}, r_{2}, \dots, r_{s})} \in L^{p} (ℝ^{s})$ , then we see

$\begin{array}{l} {‖ {(\prod_{i = 1}^{s} {Q^{'}}_{i})}^{r_{i} - k_{i}} W g^{(k_{1}, \dots, k_{s})} ‖}_{L^{p} (| x_{i} | \geq γ, i = 1, \dots, s)} \\ \leq C {\int_{| x_{1} | \geq γ} \dots \int_{| x_{s} | \geq γ} \sum_{k_{1} \leq j_{1} \leq r_{1}} {| w_{1, j_{1} - k_{1}} (x_{1}) |}^{p} \dots \sum_{k_{s} \leq j_{s} \leq r_{s}} {| w_{s, j_{s} - k_{s}} (x_{s}) |}^{p} \\ \times {| g^{(j_{1}, j_{2}, \dots, j_{s})} (y_{1}, y_{2}, \dots, y_{s}) |}^{p} d x_{s} \dots d x_{1}}^{1 / p} < \infty, \end{array}$

where for each $i = 1, 2, \dots, s$ we set

$y_{i} = {\begin{cases} γ, k_{i} \leq j_{i} \leq r_{i} - 1; \\ x_{i}, j_{i} = r_{i} . \end{cases}$

We remark that $W g^{(r_{1}, \dots, r_{s})} \in L^{p} (ℝ^{s})$ means $W g^{(k_{1}, \dots, k_{s})} \in L^{p} (ℝ^{s})$ for

$0 \leq K_{s} = (k_{1}, \dots, k_{s}) \leq R_{s} = (r_{1}, \dots, r_{s})$ .

5. Degree of Approximation

We define the degree of approximation for $W f \in L^{p} (ℝ^{s})$ as follows:

$E_{n, p; s} (W, f) : = \inf_{P \in P_{n; s} (ℝ^{s})} {‖ W (f - P) ‖}_{L^{p} (ℝ^{s})} .$

Using this $E_{n, p; s} (W, f)$ , we can estimate the degree of approximation of $W f \in L^{p} (ℝ^{s})$ from $P_{n; s} (ℝ^{s})$ .

Theorem 5.1. (1) Let $w_{i} \in F (C^{2} +) (i = 1, 2, \dots, s)$ and let

$1 \leq p \leq \infty, W f \in L^{p} (ℝ^{s})$ . Furthermore, we suppose (3.3). Then we have

${‖ \frac{W}{\prod_{i = 1}^{s} T_{i} {(x_{i})}^{1 / 4}} (f - v_{n}^{[s]} (f)) ‖}_{L^{p} (ℝ^{s})} \leq C E_{n, p; s} (W, f) .$

(2) If $w_{i} \in F_{λ} (C^{3} +) (i = 1, 2, \dots, s), 0 < λ < 3 / 2$ , and let

${‖ (\prod_{i = 1}^{s} T_{i}^{1 / 4}) W f ‖}_{L^{p} (ℝ^{s})} < \infty$ , then we have

${‖ W (f - v_{n}^{[s]} (f)) ‖}_{L^{p} (ℝ^{s})} \leq C E_{n, p; s} ((\prod_{i = 1}^{s} T_{i}^{1 / 4}) W, f) .$

(3) Let $w_{i} \in F (C^{2} +) (i = 1,2, \dots, s)$ and let $1 \leq p \leq \infty, W f \in L^{p} (ℝ^{s})$ . Then we have

${‖ W (f - v_{n}^{[s]} (f)) ‖}_{L^{p} (ℝ^{s})} \leq C (\prod_{i = 1}^{s} T_{i}^{1 / 4} (a_{n}^{(i)})) E_{n, p; s} (W, f) .$

(4) Furthermore, let $w_{i} \in F_{λ} (C^{3} +) (0 < λ < 3 / 2), i = 1, 2, \dots, s$ . If $f \in L_{W^{δ}}^{p *} (ℝ^{s})$ for some $0 < δ < 1$ , then we have

$E_{n, p; s} (W, f) \to 0 as n \to \infty .$

Proof. (1) There exists $P \in P_{n}$ such that ${‖ W (f - P) ‖}_{L^{p} (ℝ^{s})} \leq C E_{n, p; s} (W, f)$ . Therefore, by Lemma 3.9 (1)

$\begin{array}{l} {‖ \frac{W}{\prod_{i = 1}^{s} T_{i}^{1 / 4}} (f - v_{n}^{[s]} (f)) ‖}_{L^{p} (ℝ^{s})} \\ = {‖ \frac{W}{\prod_{i = 1}^{s} T_{i}^{1 / 4}} (f - P) ‖}_{L^{p} (ℝ^{s})} + {‖ \frac{W}{\prod_{i = 1}^{s} T_{i}^{1 / 4}} v_{n}^{[s]} (f - P) ‖}_{L^{p} (ℝ^{s})} \\ \leq C {‖ W (f - P) ‖}_{L^{p} (ℝ^{s})} \leq C E_{n, p; s} (W, f) . \end{array}$

(2) We see $W (X) \prod_{i = 1}^{s} T_{i}^{1 / 4} (x_{i}) ~ \tilde{W} (X) = \prod_{i = 1}^{s} {\tilde{w}}_{i} (x_{i})$ ,

$T_{i}^{1 / 4} w ~ {\tilde{w}}_{i} \in F (C^{2} +), i = 1, 2, \dots, s$ . Then, there exists $P \in P_{n}$ such that

${‖ \tilde{W} (f - P) ‖}_{L^{p} (ℝ^{s})} \leq C E_{n - 1, p; s} (\tilde{W}, f)$ . Therefore, by Lemma 3.9 (2)

$\begin{array}{l} {‖ W (f - v_{n}^{[s]} (f)) ‖}_{L^{p} (ℝ^{s})} \\ = {‖ W (f - P) ‖}_{L^{p} (ℝ^{s})} + {‖ W v_{n}^{[s]} (f - P) ‖}_{L^{p} (ℝ^{s})} \\ \leq C E_{n, p; s} (\tilde{W}, f) \leq C E_{n, p; s} ((\prod_{i = 1}^{s} T_{i}^{1 / 4}) W, f) . \end{array}$

(3) Similarly, we have (3).

(4) It follows from Theorem 3.3. #

Theorem 5.2. Let $w_{i} \in F_{λ} (C^{3} +) (i = 1, 2, \dots, s), 0 < λ < 3 / 2$ , and let $1 \leq p \leq \infty$ . Then if $W f \in L_{p} (ℝ^{s})$ , we have

${‖ \frac{W}{\prod_{i = 1}^{s} T_{i}^{(2 j_{i} + 1) / 4}} v_{n}^{[s]} {(f)}^{(j_{1}, \dots, j_{s})} ‖}_{L^{p} (ℝ^{s})} \leq C \prod_{i = 1}^{s} {(\frac{n}{a_{n}^{(i)}})}^{j_{i}} {‖ W f ‖}_{L^{p} (ℝ^{s})},$

and

${‖ W v_{n}^{[s]} {(f)}^{(j_{1}, \dots, j_{s})} ‖}_{L^{p} (ℝ^{s})} \leq C \prod_{i = 1}^{s} {(\frac{n}{a_{n}^{(i)}})}^{j_{i}} (\prod_{i = 1}^{s} T_{i}^{(2 j_{i} + 1) / 4} (a_{n}^{(i)})) {‖ W f ‖}_{L^{p} (ℝ^{s})} .$

Proposition 5.3 ( [10] , Lemma 2.5, [2] , Corollary 10.2). Let $1 \leq p \leq \infty$ and $w \in F_{λ} (C^{3} +) (0 < λ < 3 / 2)$ . Then there exists a constant $C_{1} = C_{1} (w, p) > 0$ such that, if $P \in P_{n} (ℝ)$ ( $n \in ℕ$ ),

${‖ \frac{w}{T^{j / 2}} P^{(j)} ‖}_{L^{p} (ℝ^{s})} \leq C_{1} {(\frac{n}{a_{n}})}^{j} {‖ w P ‖}_{L^{p} (ℝ^{s})}, j \in ℕ,$

and

${‖ w P^{(j)} ‖}_{L^{p} (ℝ^{s})} \leq C_{1} {(\frac{n T {(a_{n})}^{1 / 2}}{a_{n}})}^{j} {‖ w P ‖}_{L^{p} (ℝ^{s})}, j \in ℕ .$

Proof of Theorem 5.2. We use Proposition 5.3 and Lemma 3.7 (1).

$\begin{array}{l} {‖ \frac{W}{\prod_{i = 1}^{s} T_{i}^{(2 j_{i} + 1) / 4}} v_{n}^{[s]} {(f)}^{(j_{1}, \dots, j_{s})} ‖}_{L^{p} (ℝ^{s})} \\ \leq C \prod_{i = 1}^{s} {(\frac{n}{a_{n}^{(i)}})}^{j_{i}} {‖ \frac{W}{\prod_{i = 1}^{s} T_{i}^{- 1 / 4}} v_{n}^{[s]} (f) ‖}_{L^{p} (ℝ^{s})} \\ \leq C \prod_{i = 1}^{s} {(\frac{n}{a_{n}^{(i)}})}^{j_{i}} {‖ W f ‖}_{L^{p} (ℝ^{s})}, \end{array}$

and further, using Theorem A₁ (the Markov-Bernstein inequality) in Appendix,

$\begin{array}{l} {‖ W v_{n}^{[s]} {(f)}^{(j_{1}, \dots, j_{s})} ‖}_{L^{p} (ℝ^{s})} \leq C \prod_{i = 1}^{s} {(\frac{n T_{i}^{1 / 2} (a_{n}^{(i)})}{a_{n}^{(i)}})}^{j_{i}} {‖ W v_{n}^{[s]} (f) ‖}_{L^{p} (ℝ^{s})} \\ \leq C \prod_{i = 1}^{s} {(\frac{n T_{i}^{1 / 2} (a_{n}^{(i)})}{a_{n}^{(i)}})}^{j_{i}} {‖ W v_{n}^{[s]} (f) ‖}_{L^{p} (| x_{i} | \leq a_{n}^{(i)}, i = 1, 2, \dots, s)} \\ \leq C \prod_{i = 1}^{s} {(\frac{n}{a_{n}^{(i)}})}^{j_{i}} \prod_{i = 1}^{s} T_{i}^{(2 j_{i} + 1) / 4} (a_{n}^{(i)}) {‖ \frac{W}{\prod_{i = 1}^{s} T_{i}^{1 / 4}} v_{n}^{[s]} (f) ‖}_{L^{p} (ℝ^{s})} \\ \leq C \prod_{i = 1}^{s} {(\frac{n}{a_{n}^{(i)}})}^{j_{i}} \prod_{i = 1}^{s} T_{i}^{(2 j_{i} + 1) / 4} (a_{n}^{(i)}) {‖ W f ‖}_{L^{p} (ℝ^{s})} . \end{array}$ #

In the rest of only this section, we suppose

$w = \exp (- Q) = w_{i} = \exp (- Q_{i}), i = 1, 2, \dots, s,$

$a_{n} = a_{n}^{(i)}, T = T_{i}, i = 1, 2, \dots, s .$

Let

$W_{i} : = W_{i} (x_{1}, \dots, x_{i - 1}, x_{i + 1}, \dots, x_{s}) : = \prod_{j \neq i, 1 \leq j \leq s} w_{j} (x_{j}), i = 1, 2, \dots, s .$

In ( [7] , Corollary 8) we give the Favard-type inequalities:

Proposition 5.4 [7] . Let $w \in F (C^{2} +)$ , and let $r \geq 0$ be an integer. Let $1 \leq p \leq \infty$ , and let $w f^{(r)} \in L_{p} (ℝ)$ . Then we have

$E_{p, n} (f, w) \leq C {(\frac{a_{n}}{n})}^{k} {‖ w f^{(k)} ‖}_{L_{p} (ℝ)}, k = 1, 2, \dots, r,$

and equivalently,

$E_{p, n} (f, w) \leq C {(\frac{a_{n}}{n})}^{k} E_{p, n - k} (f^{(k)}, w) .$

The following theorem is a generalization of Proposition 5.4.

Theorem 5.5. We suppose

$w_{j} = \exp (- Q_{j}) \in F_{λ} (C^{3} +) (0 < λ < 3 / 2), j = 1, 2, \dots, s$ , and let (3.3) satisfy, that is,

$T_{j} (a_{n}) \leq c {(\frac{n}{a_{n}})}^{2 / 3}, j = 1, 2, \dots, s .$

Let $W f^{(r, r, \dots, r)} \in L_{p} (ℝ^{s})$ for some positive integer r. Then we have

$E_{n, p; s} (W; f) \leq C {(\frac{a_{n}}{n})}^{r} {‖ T^{〈 s 〉} W f^{(r, r, \dots, r)} ‖}_{L^{p} (ℝ^{s})} .$

Equivalently,

$E_{n, p; s} (W; f) \leq C {(\frac{a_{n}}{n})}^{r} E_{n - r, p; s} (T^{〈 s 〉} W; f^{(r, r, \dots, r)}) .$

Proof. Using

$f - v_{n}^{[s]} = (f - v_{n}^{[1]} (f)) + (v_{n}^{[1]} (f) - v_{n}^{[2]} (f)) + \dots + (v_{n}^{[s - 1]} (f) - v_{n}^{[s]} (f))$ , we get from Lemma 3.9 (2) and (3.2),

$\begin{matrix} E_{n, p; s} \leq {‖ W (f - v_{n}^{[s]} (f)) ‖}_{L^{p} (ℝ^{s})} \\ \leq {‖ W (f - v_{n}^{[1]} (f)) ‖}_{L^{p} (ℝ^{s})} + \sum_{j = 2}^{s} {‖ W (v_{n}^{[j - 1]} (f) - v_{n}^{[j]} (f)) ‖}_{L^{p} (ℝ^{s})} \\ \leq C [{‖ W (f - v_{n, 1} (f)) ‖}_{L^{p} (ℝ^{s})} + \sum_{j = 2}^{s} {‖ W v_{n}^{[j - 1]} (f - v_{n, j} (f)) ‖}_{L^{p} (ℝ^{s})}] \\ \leq C [{‖ W (f - v_{n, 1} (f)) ‖}_{L^{p} (ℝ^{s})} + \sum_{j = 2}^{s} {‖ W (\prod_{k = 1}^{j - 1} T_{i}^{1 / 4}) (f - v_{n, j} (f)) ‖}_{L^{p} (ℝ^{s})}] \end{matrix}$

We estimate each term. From Proposition 5.4 with the weight $T_{j}^{1 / 4} w_{j}$ ,

$\begin{array}{l} {‖ W (\prod_{k = 1}^{j - 1} T_{k}^{1 / 4}) (f - v_{n, j} (f)) ‖}_{L^{p} (ℝ^{s})} \\ = {‖ {‖ W_{i} (\prod_{k = 1}^{j - 1} T_{k}^{1 / 4}) w_{j} (f_{{\hat{X}}_{j}} - v_{n, j} (f_{{\hat{X}}_{j}})) ‖}_{L^{p} (ℝ_{(j)})} ‖}_{L^{p} (ℝ_{j}^{s - 1})} \\ \leq C {‖ W_{i} (\prod_{k = 1}^{j - 1} T_{k}^{1 / 4}) E_{n, p} (T_{j}^{1 / 4} w_{j}, f_{{\hat{X}}_{j}}) ‖}_{L^{p} (ℝ_{j}^{s - 1})} \\ \leq C {(\frac{a_{n}}{n})}^{r} {‖ \prod_{i \neq j} w_{i} (\prod_{k = 1}^{j - 1} T_{k}^{1 / 4}) {‖ T_{j}^{1 / 4} w_{j} f_{{\hat{X}}_{j}}^{(r, 0, \dots, 0)} ‖}_{L^{p} (ℝ_{(j)})} ‖}_{L^{p} (ℝ_{j}^{s - 1})} . \end{array}$

Now, we use Theorem 4.1 and the fact

$T_{i} (x_{i}) \leq {Q^{'}}_{i} {(x_{i})}^{r}, i = 1,2, \dots, s,$

then we have

$\begin{array}{l} {‖ W (\prod_{k = 1}^{j - 1} T_{k}^{1 / 4}) (f - v_{n, j} (f)) ‖}_{L^{p} (ℝ^{s})} \\ \leq C {(\frac{a_{n}}{n})}^{r} {‖ \prod_{i \neq j} {Q^{'}}_{i} {(x_{i})}^{r} w_{i} {‖ T_{j}^{1 / 4} w_{j} f_{{\hat{X}}_{j}}^{(r, 0, \dots, 0)} ‖}_{L^{p} (ℝ_{(j)})} ‖}_{L^{p} (ℝ_{j}^{s - 1})} \\ \leq C {(\frac{a_{n}}{n})}^{r} {‖ T^{〈 s 〉} W f^{(r, r, \dots, r)} ‖}_{L^{p} (ℝ^{s})} . \end{array}$

Consequently, we have

${‖ W (f - v_{n}^{[s]} (f)) ‖}_{L^{p} (ℝ^{s})} \leq C {(\frac{a_{n}}{n})}^{r} {‖ T^{〈 s 〉} W f^{(r, r, \dots, r)} ‖}_{L^{p} (ℝ^{s})} .$

Corollary 5.6. Under the conditions of Theorem 5.5, if w is a Freud-type weight, then

$E_{n, p; s} (W; f) \leq C {(\frac{a_{n}}{n})}^{r} {‖ W f^{(r, r, \dots, r)} ‖}_{L^{p} (ℝ^{s})} .$

Equivalently,

$E_{n, p; s} (W; f) \leq C {(\frac{a_{n}}{n})}^{r} E_{n - r, p; s} (W; f^{(r, r, \dots, r)}) .$

Let $1 \leq p \leq \infty$ . For ${‖ W f ‖}_{L_{p} (ℝ^{s})} < \infty$ we define the K-functional $K_{r, p} (W; f, δ)$ by

$K_{r, p; s} (W; f, δ) : = \underset{g}{i n f} {{‖ W (f - g) ‖}_{L_{p} (ℝ^{s})} + δ^{r} {‖ W g^{(r, r, \dots, r)} ‖}_{L_{p} (ℝ^{s})}},$

where the infimum is over all functions $g^{(r - 1, r - 1, \dots, r - 1)}$ which are absolutely

continuous and ${‖ W g^{(r, r, \dots, r)} ‖}_{L_{p} (ℝ^{s})} < \infty$ . We have the following.

Theorem 5.7. We suppose

$w_{j} = \exp (- Q_{j}) \in F_{λ} (C^{3} +) (0 < λ < 3 / 2), j = 1, 2, \dots, s$ , and let

$T_{j} (a_{n}) \leq c {(\frac{n}{a_{n}})}^{2 / 3}, j = 1, 2, \dots, s .$

Let $1 \leq p \leq \infty$ , and let ${‖ T^{〈 s 〉} W f ‖}_{L_{p} (ℝ^{s})} < \infty$ . Then we have

$E_{n, p; s} (W, f) \leq K_{r, p; s} (T^{〈 s 〉} W, f, \frac{a_{n}}{n}) .$

Proof. We take g as

${‖ T^{〈 s 〉} W (f - g) ‖}_{L_{p} (ℝ)} + δ^{r} {‖ T^{〈 s 〉} W g^{(r, r, \dots, r)} ‖}_{L_{p} (ℝ^{s})} \leq C K_{r, p; s} (T^{〈 s 〉} W; f, δ),$

and for this g we select $P \in P_{n} (ℝ^{s})$ such that

${‖ W (g - P) ‖}_{L^{p} (ℝ^{s})} \leq E_{n, p; s} (W; g) .$

Then, from Theorem 5.5 we see

$\begin{matrix} E_{n, p; s} (W, f) \leq {‖ W (f - P) ‖}_{L^{p} (ℝ^{s})} \\ \leq {‖ W (f - g) ‖}_{L^{p} (ℝ^{s})} + {‖ W (g - P) ‖}_{L^{p} (ℝ^{s})} \\ \leq {‖ W (f - g) ‖}_{L^{p} (ℝ^{s})} + C E_{n, p; s} (W, g) \\ \leq {‖ T^{〈 s 〉} W (f - g) ‖}_{L^{p} (ℝ^{s})} + C {(\frac{a_{n}}{n})}^{r} {‖ T^{〈 s 〉} W g^{(r, r, \dots, r)} ‖}_{L^{p} (ℝ^{s})} \\ \leq C K_{r, p} (T^{〈 s 〉} W, f, \frac{a_{n}}{n}) . \end{matrix}$ #

Corollary 5.8. Let $1 \leq p \leq \infty$ , and let $w \in F (C^{2} +)$ be a Freud-type weight. If

${‖ W f ‖}_{L_{p} (ℝ^{s})} < \infty$ , then we have

$E_{n, p; s} (W, f) \leq C K_{r, p; s} (W, f, \frac{a_{n}}{n}) .$

Let $0 < p \leq \infty$ . Damelin [11] gives a K-functional as follows:

${\bar{K}}_{r, p} (f, w, t^{r}) : = \underset{P \in P_{n}}{i n f} {{‖ w (f - P) ‖}_{L^{p} (ℝ)} + t^{r} {‖ P^{(r)} Φ_{t}^{r} w ‖}_{L^{p} (ℝ)}},$

where $t > 0$ and $r \geq 1$ are chosen in advance and

$n = n (t) : = \inf {k; \frac{a_{k}}{k} \leq t} .$

We recall the r-th order of the modulus of smoothness $ω_{r, p} (w; f; t)$ , which is defined as follows (cf. [6] and [11] ). Let r be a positive integer, and let $0 < p \leq \infty$ . We set

$Δ_{h}^{r} (f, x) : = \sum_{i = 0}^{r} (\begin{matrix} r \\ i \end{matrix}) {(- 1)}^{i} f (x + \frac{r h}{2} - i h), x \in ℝ .$

For the Freud-type weight,

${\bar{ω}}_{r, p} (f, w, t) : = {(\frac{1}{t} \int_{0}^{t} {‖ w Δ_{h}^{r} (f, x) ‖}_{L^{p} (| x | \leq σ (2 t))} d h)}^{1 / p} + \underset{P \in P_{r - 1}}{i n f} {‖ (f - P) w ‖}_{L^{p} (| x | \leq σ (4 t))} .$

For the Erdös-type weight,

${\bar{ω}}_{r, p} (f, w, t) : = {(\frac{1}{t} \int_{0}^{t} {‖ w Δ_{h Φ_{t} (x)}^{r} (f, x) ‖}_{L^{p} (| x | \leq σ (2 t))} d h)}^{1 / p} + \underset{P \in P_{r - 1}}{i n f} {‖ (f - P) w ‖}_{L^{p} (| x | \leq σ (4 t))},$

where

$Φ_{t} (x) : = \sqrt{| 1 - \frac{| x |}{σ (t)} |} + \frac{1}{\sqrt{T (σ (t))}} .$

We remark that if $T (x)$ is bounded then we see $Φ_{t} (x) ~ 1$ . So, we may consider for only the Erdös-type weight. Then the following proposition holds.

Proposition 5.9 ( [11] , Theorem 1.2, 1.3). Let $0 < p \leq \infty, r \geq 1$ , and let $w \in E_{1}$ (contains $F (C^{2} +)$ ). Let $f : ℝ \to ℝ$ for which