Higher order derivatives of approximation polynomials on $\mathbb{R}$

Abstract

Leviatan has investigated the behavior of higher order derivatives of approximation polynomials of a differentiable function f on $[-1,1]$. Especially, when $P_{n}$ is the best approximation of f, he estimates the differences $\|f^{(k)}-P_{n}^{(k)}\|_{L_{\infty}([-1,1])}$, $k=0,1,2,\ldots$ . In this paper, we give the analogies for them with respect to the differentiable functions on $\mathbb{R}$.

1 Introduction

Let $\mathbb{R}=(-\infty,\infty)$ and ${\mathbb{R}}^{+}=[0,\infty)$. We say that $f: (0,\infty) \rightarrow{\mathbb{R}^{+}}$ is quasi-increasing in $(0,\infty)$ if there exists $C>0$ such that $f(x)\leqslant Cf(y)$ for $0< x< y$. The notation $f(x)\sim g(x)$ means that there are positive constants $C_{1}$, $C_{2}$ such that for the relevant range of x, $C_{1}\leqslant f(x)/g(x)\leqslant C_{2}$. A similar notation is used for sequences and sequences of functions. Throughout $C,C_{1},C_{2},\ldots$ denote positive constants independent of n, x, t. The same symbol does not necessarily denote the same constant in different occurrences. We denote the class of polynomials with degree n by $\mathcal{P}_{n}$.

First, we introduce some classes of weights. Levin and Lubinsky [1] introduced the class of weights on ${\mathbb{R}}$ as follows.

Definition 1.1

Let $Q: \mathbb{R}\rightarrow[0,\infty)$ be a continuous even function, and satisfy the following properties:

• (a) $Q'(x) >0$ for $x>0$ and is continuous in $\mathbb {R}$, with $Q(0)=0$.

• (b) $Q''(x)$ exists and is positive in $\mathbb {R}\backslash\{0\}$.

• (c) $\lim_{x\rightarrow\infty}Q(x)=\infty$.

• (d) The even function

  $$ T_{w}(x):=\frac{xQ'(x)}{Q(x)}, \quad x\neq0 $$

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

  $$ T_{w}(x)\ge\Lambda>1, \quad x\in\mathbb{R}\backslash\{0\}. $$

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

  $$ \frac{Q''(x)}{|Q'(x)|}\le C_{1}\frac{|Q'(x)|}{Q(x)}, \quad \mbox{a.e. } x\in \mathbb{R}. $$

  Furthermore, if there also exist a compact subinterval J (∋0) of $\mathbb{R}$ and $C_{2}>0$ such that

  $$ \frac{Q''(x)}{|Q'(x)|}\ge C_{2}\frac{|Q'(x)|}{Q(x)}, \quad \mbox{a.e. } x\in \mathbb{R}\backslash J, $$

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

For convenience, we denote T instead of $T_{w}$, if there is no confusion. Next, we give some typical examples of $\mathcal{F}(C^{2}+)$.

Example 1.2

[2]

• (1) If $T(x)$ is bounded, then we call the weight $w=\exp(-Q(x))$ the Freud-type weight and we write $w\in\mathcal{F}^{*}\subset\mathcal{F}(C^{2}+)$.

• (2) When $T(x)$ is unbounded, then we call the weight $w=\exp(-Q(x))$ the Erdös-type weight: For $\alpha>1$, $l\ge1$ we define

  $$ Q(x):=Q_{l,\alpha}(x)=\exp_{l}\bigl(|x|^{\alpha}\bigr)- \exp_{l}(0), $$

  where $\exp_{l}(x)=\exp(\exp(\exp\cdots\exp x)\cdots)$ (l times). More generally, we define

  $$ Q_{l,\alpha,m}(x)=|x|^{m}\bigl\{ \exp_{l} \bigl(|x|^{\alpha}\bigr) -\tilde{\alpha}\exp _{l}(0)\bigr\} ,\quad \alpha+m>1, m\ge0, \alpha\ge0, $$

  where $\tilde{\alpha}=0$ if $\alpha=0$, and otherwise $\tilde {\alpha}=1$. We note that $Q_{l,0,m}$ gives a Freud-type weight, and $Q_{l,\alpha,m}$ ($\alpha>0$) gives an Erdös-type weight.

• (3) For $\alpha>1$, $Q_{\alpha}(x)=(1+|x|)^{|x|^{\alpha}} -1$ gives also an Erdös-type weight.

For a continuous function $f : [-1,1] \to\mathbb{R}$, let

$$ E_{n}(f)=\inf_{P\in\mathcal{P}_{n}}\|f-P\|_{L_{\infty}([-1,1])}=\inf _{P\in\mathcal{P}_{n}}\max_{x\in[-1,1]}\bigl\vert f(x)-P(x)\bigr\vert . $$

Leviatan [3] has investigated the behavior of the higher order derivatives of approximation polynomials for the differentiable function f on $[-1,1]$, as follows.

Theorem

(Leviatan [3])

For $r\ge0$ we let $f\in C^{(r)}[-1,1]$, and let $P_{n}\in\mathcal{P}_{n}$ denote the polynomial of best approximation of f on $[-1,1]$. Then for each $0\le k\le r$ and every $-1\le x\le1$,

$$ \bigl\vert f^{(k)}(x)-P_{n}^{(k)}(x)\bigr\vert \le\frac{C_{r}}{n^{k}}\Delta _{n}^{-k}(x)E_{n-k} \bigl(f^{(k)} \bigr),\quad n\ge k, $$

where $\Delta_{n}(x):=\sqrt{1-x^{2}}/n+1/n^{2}$ and $C_{r}$ is an absolute constant which depends only on r.

In this paper, we will give an analogy of Leviatan’s theorem for some exponential-type weight. In Section 2, we give the theorems in the space $L_{\infty}(\mathbb{R})$, and we also make a certain assumption and some notations which are needed in order to state the theorems. In Section 3, we give some lemmas and the proofs of the theorems.

2 Theorems and preliminaries

First, we introduce some well-known notations. If f is a continuous function on $\mathbb{R}$, then we define

$$ \Vert fw\Vert _{L_{\infty}(\mathbb{R})}:=\sup_{t\in\mathbb {R}}\bigl\vert f(t)w(t)\bigr\vert , $$

and for $1\le p<\infty$ we denote

$$ \|fw\|_{L_{p}(\mathbb{R})}:= \biggl(\int_{\mathbb{R}} \bigl\vert f(t)w(t)\bigr\vert ^{p}\, dt \biggr)^{1/p}. $$

Let $1\le p\le\infty$. If $\|wf\|_{L_{p}(\mathbb{R})}<\infty$, then we write $wf\in L_{p}(\mathbb{R})$, and here if $p=\infty$, we suppose that $f\in C(\mathbb{R})$ and $\lim_{|x|\rightarrow\infty}|w(x)f(x)|=0$. We denote the rate of approximation of f by

$$ E_{p,n}(w,f):=\inf_{P\in\mathcal{P}_{n}}\bigl\Vert (f-P)w\bigr\Vert _{L_{p}(\mathbb{R})}. $$

The Mhaskar-Rakhmanov-Saff numbers $a_{x}$ is defined as follows:

$$ x=\frac{2}{\pi}\int_{0}^{1}\frac{a_{x}uQ'(a_{x}u)}{\sqrt{1-u^{2}}}\, du, \quad x>0. $$

To write our theorems we need some preliminaries. We need further assumptions.

Definition 2.1

Let $w=\exp(-Q)\in\mathcal{F}(C^{2}+)$ and let $r \ge1$ be an integer. Then for $0< \lambda<(r+2)/(r+1)$ we write $w\in\mathcal{F}_{\lambda}(C^{r+2}+)$ if $Q\in C^{(r+2)}(\mathbb{R}\backslash\{0\})$ and there exist two constants $C>1$ and $K\ge1$ such that for all $|x|\ge K$,

$$ \frac{|Q'(x)|}{Q^{\lambda}(x)} \leq C \quad \mbox{and}\quad \biggl\vert \frac{Q''(x)}{Q'(x)} \biggr\vert \sim\biggl\vert \frac {Q^{(k+1)}(x)}{Q^{(k)}(x)} \biggr\vert $$

for every $k=2,\ldots,r$ and also

$$ \biggl\vert \frac{Q^{(r+2)}(x)}{Q^{(r+1)}(x)} \biggr\vert \leq C \biggl\vert \frac {Q^{(r+1)}(x)}{Q^{(r)}(x)} \biggr\vert . $$

In particular, $w\in\mathcal{F}_{\lambda}(C^{3}+)$ means that $Q\in C^{(3)}(\mathbb{R}\backslash\{0\})$ and

$$ \frac{|Q'(x)|}{Q^{\lambda}(x)} \leq C \quad \mbox{and}\quad \biggl\vert \frac{Q'''(x)}{Q''(x)} \biggr\vert \le C \biggl\vert \frac{Q''(x)}{Q'(x)} \biggr\vert $$

hold for $|x| \geq K$. In addition, let $\mathcal{F}_{\lambda}(C^{2}+):=\mathcal{F}(C^{2}+)$.

From [2], we know that Example 1.2(2), (3) satisfy all conditions of Definition 2.1. Under the same condition as of Definition 2.1 we obtain an interesting theorem as follows.

Theorem 2.2

([4], Theorems 4.1, 4.2 and (4.11))

Let r be a positive integer, $0< \lambda<(r+2)/(r+1)$ and let $w=\exp(-Q)\in\mathcal{F}_{\lambda}(C^{r+2}+)$. Then, for any $\mu, \nu, \alpha, \beta\in\mathbb{R}$, we can construct a new weight $w_{\mu,\nu,\alpha,\beta}\in\mathcal {F}_{\lambda}(C^{r+1}+)$ such that

$$ T_{w}^{\mu}(x) \bigl(1+x^{2}\bigr)^{\nu}\bigl(1+Q(x)\bigr)^{\alpha}\bigl(1+\bigl\vert Q'(x)\bigr\vert \bigr)^{\beta}w(x)\sim w_{\mu,\nu,\alpha,\beta}(x) $$

on $\mathbb{R}$, and for some $c \ge1$,

$$\begin{aligned}& a_{n/c}(w)\le a_{n}(w_{\mu.\nu,\alpha,\beta})\le a_{cn}(w), \\& T_{w_{\mu,\nu,\alpha,\beta}}(x)\sim T_{w}(x) \end{aligned}$$

hold on $\mathbb{R} \backslash\{0\}$.

For a given $\mu\in\mathbb{R}$ and $w\in\mathcal{F}_{\lambda }(C^{3}+)$ ($0< \lambda< 3/2$), we let $w_{\mu}\in\mathcal{F}(C^{2}+)$ satisfy $w_{\mu}(x) \sim T_{w}^{\mu}(x)w(x)$ (see Theorem 4.1 in [4]). Let $P_{n;f,w_{\mu}}\in\mathcal{P}_{n}$ be the best approximation of f with respect to the weight $w_{\mu}$, that is,

$$ \bigl\Vert (f-P_{n;f,w_{\mu}})w_{\mu}\bigr\Vert _{L_{\infty}(\mathbb {R})}=E_{n}(w_{\mu},f) :=\inf_{P\in\mathcal{P}_{n}} \bigl\Vert (f-P)w_{\mu}\bigr\Vert _{L_{\infty}(\mathbb{R})}. $$

Then we have the main result as follows.

Theorem 2.3

Let $r \ge0$ be an integer. Let $w=\exp(-Q)\in\mathcal{F}_{\lambda}(C^{r+3}+)$, where $0< \lambda<(r+3)/(r+2)$. Suppose that $f\in C^{(r)}(\mathbb{R})$ with

$$ \lim_{|x|\rightarrow\infty}T^{1/4}(x)f^{(r)}(x)w(x)=0. $$

Then there exists an absolute constant $C_{r}>0$ which depends only on r such that, for $0\le k\le r$ and $x\in\mathbb{R}$,

$$\begin{aligned} \bigl\vert \bigl(f^{(k)}(x)-P_{n;f,w}^{(k)}(x) \bigr)w(x)\bigr\vert \le& C_{r} T^{k/2}(x)E_{n-k} \bigl(w_{1/4},f^{(k)} \bigr) \\ \le& C_{r} T^{k/2}(x) \biggl(\frac{a_{n}}{n} \biggr)^{r-k}E_{n-r} \bigl(w_{1/4},f^{(r)} \bigr). \end{aligned}$$

When $w\in\mathcal{F}^{*}$, we can replace $w_{1/4}$ with cw (c is a constant) in the above.

Applying Theorem 2.3 with w or $w_{-1/4}$, we have the following corollaries.

Corollary 2.4

• (1) Let $w=\exp(-Q)\in\mathcal{F}_{\lambda}(C^{r+3}+)$ and $0< \lambda<(r+3)/(r+2)$, $r \ge0$. We suppose that $f\in C^{(r)}(\mathbb{R})$ with

  $$ \lim_{|x|\rightarrow\infty}T^{1/4}(x)f^{(r)}(x)w(x)=0, $$

  then for $0\le k\le r$ we have

  $$\begin{aligned} \bigl\Vert \bigl(f^{(k)}-P_{n;f,w}^{(k)} \bigr)w_{-k/2}\bigr\Vert _{L_{\infty}(\mathbb{R})} \le& C_{r} E_{n-k} \bigl(w_{1/4},f^{(k)} \bigr) \\ \le& C_{r} \biggl(\frac{a_{n}}{n} \biggr)^{r-k}E_{n-r} \bigl(w_{1/4},f^{(r)} \bigr). \end{aligned}$$

• (2) Let $w=\exp(-Q)\in\mathcal{F}_{\lambda}(C^{r+4}+)$, $0< \lambda<(r+4)/(r+3)$, $r\ge0$. We suppose that $f\in C^{(r)}(\mathbb{R})$ with

  $$ \lim_{|x|\rightarrow\infty}f^{(r)}(x)w(x)=0, $$

  then for $0\le k\le r$ we have

  $$\begin{aligned} \bigl\Vert \bigl(f^{(k)}-P_{n;f,w_{-1/4}}^{(k)} \bigr)w_{-(2k+1)/4}\bigr\Vert _{L_{\infty}(\mathbb{R})} \le& C_{r} E_{n-k} \bigl(w,f^{(k)} \bigr) \\ \le& C_{r} \biggl(\frac{a_{n}}{n} \biggr)^{r-k}E_{n-r} \bigl(w,f^{(r)} \bigr). \end{aligned}$$

When $w\in\mathcal{F}^{*}$, we can replace $w_{\mu}$ ($\mu=-k/2$, $\mu=-(2k+1)/4$, $0\le k\le r$, and $\mu=1/4$) with cw (c is a constant) in the above.

Corollary 2.5

Let $r \ge0$ be an integer. Let $w=\exp(-Q)\in\mathcal{F}_{\lambda}(C^{r+4}+)$, $0< \lambda<(r+4)/(r+3)$, and let $w_{(2r+1)/4}f^{(r)}\in L_{\infty}(\mathbb{R})$. Then, for each k ($0\le k\le r$) and the best approximation polynomial $P_{n;f,w_{k/2}}$;

$$ \bigl\Vert (f-P_{n;f,w_{k/2}} )w_{k/2}\bigr\Vert _{L_{\infty}(\mathbb{R})}=E_{n} (w_{k/2},f ), $$

we have

$$\begin{aligned} \bigl\Vert \bigl(f^{(k)}-P_{n;f,w_{k/2}}^{(k)} \bigr)w\bigr\Vert _{L_{\infty}(\mathbb{R})} \le& C_{r} E_{n-k} \bigl(w_{(2k+1)/4},f^{(k)} \bigr) \\ \le& C_{r} \biggl(\frac{a_{n}}{n} \biggr)^{r-k}E_{n-r} \bigl(w_{(2k+1)/4},f^{(r)} \bigr). \end{aligned}$$

When $w\in\mathcal{F}^{*}$, we can replace $w_{\mu}$ ($\mu=k/2$, $\mu =(2k+1)/4$, $0\le k\le r$) with cw (c is a constant) in the above.

3 Proofs of theorems

We give the proofs of the theorems. First, we give some lemmas to prove the theorems. We construct the orthonormal polynomials $p_{n}(x)=p_{n}(w^{2},x)$ of degree n for $w^{2}(x)$, that is,

$$ \int_{-\infty}^{\infty} p_{n} \bigl(w^{2},x\bigr)p_{m}\bigl(w^{2},x \bigr)w^{2}(x)\, dx=\delta_{mn}\quad (\text{Kronecker delta}). $$

Let $fw\in L_{2}(\mathbb{R})$. The Fourier-type series of f is defined by

$$ \tilde{f}(x):=\sum_{k=0}^{\infty} a_{k}\bigl(w^{2},f\bigr)p_{k}\bigl(w^{2},x \bigr),\quad a_{k}\bigl(w^{2},f\bigr):=\int _{-\infty}^{\infty} f(t)p_{k}\bigl(w^{2},t \bigr)w^{2}(t)\, dt. $$

We denote the partial sum of $\tilde{f}(x)$ by

$$ s_{n}(f,x):=s_{n}\bigl(w^{2},f,x\bigr):=\sum _{k=0}^{n-1} a_{k} \bigl(w^{2},f\bigr)p_{k}\bigl(w^{2},x\bigr). $$

Moreover, we define the de la Vallée Poussin means by

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

Theorem 3.1

(Theorem 1.1, (1.5), Corollary 6.2, (6.5) in [5])

Let $w\in\mathcal{F}_{\lambda}(C^{3}+)$, $0<\lambda<3/2$, and let $1\le p\le\infty$. When $T^{1/4}wf\in L_{p}(\mathbb{R})$, we have, for $n\ge1$,

$$ \bigl\Vert v_{n}(f)w\bigr\Vert _{L_{p}(\mathbb{R})}\le C \bigl\Vert T^{1/4}wf \bigr\Vert _{L_{p}(\mathbb{R})}, $$

and so

$$ \bigl\Vert \bigl(f-v_{n}(f)\bigr)w \bigr\Vert _{L_{p}(\mathbb{R})}\le C E_{p,n} \bigl(T^{1/4}w,f \bigr). $$

So, equivalently,

$$ \bigl\Vert v_{n}(f)w\bigr\Vert _{L_{p}(\mathbb{R})}\le C \Vert w_{1/4}f \Vert _{L_{p}(\mathbb{R})}, $$

and so

$$ \bigl\Vert \bigl(f-v_{n}(f)\bigr)w\bigr\Vert _{L_{p}(\mathbb{R})}\le C E_{p,n} (w_{1/4},f ). $$

(3.1)

When $w\in\mathcal{F}^{*}$, we can replace $w_{1/4}$ with cw.

Lemma 3.2

Let $w\in\mathcal{F}(C^{2}+)$.

• (1) (Lemma 3.5(a) in [1]) Let $L>0$ be fixed. Then, uniformly for $t>0$,

  $$ a_{Lt}\sim a_{t}. $$

• (2) (Lemma 3.4, (3.17) in [1]) For $x >1$, we have

  $$ \bigl\vert Q'(a_{x})\bigr\vert \sim \frac{x \sqrt{T(a_{x})}}{a_{x}} \quad \textit{and}\quad \bigl\vert Q(a_{x})\bigr\vert \sim\frac{x}{ \sqrt{T(a_{x})}}. $$

• (3) (Proposition 3 in [6]) If $T(x)$ is unbounded, then for any $\eta>0$ there exists $C(\eta )>0$ such that for $t\ge1$,

  $$ a_{t}\le C(\eta)t^{\eta}. $$

To prove the results, we need the following notations. We set

$$ \sigma(t):=\inf \biggl\{ a_{u}: \frac{a_{u}}{u}\le t \biggr\} ,\quad t>0 $$

and

$$ \Phi_{t}(x):=\sqrt{\biggl\vert 1-\frac{|x|}{\sigma(t)} \biggr\vert }+T^{-1/2}\bigl(\sigma(t)\bigr), \quad x\in\mathbb{R}. $$

Define for $fw\in L_{p}(\mathbb{R})$, $0< p\le\infty$,

$$\begin{aligned} \omega_{p}(f,w,t) :=&\sup_{0< h\le t} \biggl\Vert w(x) \biggl\{ f \biggl(x+\frac{h}{2}\Phi_{t}(x) \biggr)- f \biggl(x- \frac{h}{2}\Phi_{t}(x) \biggr) \biggr\} \biggr\Vert _{L_{p}(|x|\le \sigma(2t))} \\ &{} +\inf_{c\in\mathbb{R}}\bigl\Vert w(x) (f-c) (x)\bigr\Vert _{L_{p}(|x|\ge\sigma(4t))} \end{aligned}$$

(see [7, 8]).

Proposition 3.3

(cf. Theorem 1.2 in [8], Corollary 1.4 in [7])

Let $w\in\mathcal{F}(C^{2}+)$. Let $0< p\le\infty$. Then for $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $fw\in L_{p}(\mathbb{R})$ (where for $p=\infty$, we require f to be continuous, and fw to vanish at ±∞), we have, for $n\ge C_{3}$,

$$ E_{p,n} (w,f )\le C_{1}\omega_{p} \biggl(f,w,C_{2}\frac {a_{n}}{n} \biggr), $$

where $C_{j}$, $j=1,2,3$, do not depend on f and n.

Proof

Damelin and Lubinsky [8] or Damelin [7] have treated a certain class $\mathcal{E}_{1}$ of weights containing the ones satisfying conditions (a)-(d) in Definition 1.1 and

$$ \frac{yQ'(y)}{xQ'(x)}\le \biggl(\frac{Q(y)}{Q(x)} \biggr)^{C},\quad y\ge x > 0, $$

(3.2)

where $C >0$ is a constant, and they obtain this Proposition for $w\in\mathcal{E}_{1}$. Therefore, we may show $\mathcal {F}(C^{2}+)\subset\mathcal{E}_{1}$. In fact, from Definition 1.1(d) and (e), we have, for $y\ge x>0$,

$$ \frac{Q'(y)}{Q'(x)}=\exp \biggl(\int_{x}^{y} \frac{Q''(t)}{Q'(t)}\, dt \biggr) \le\exp \biggl(C_{1}\int _{x}^{y}\frac{Q'(t)}{Q(t)}\, dt \biggr)= \biggl( \frac {Q(y)}{Q(x)} \biggr)^{C_{1}} $$

and

$$ \frac{y}{x}=\exp \biggl(\int_{x}^{y} \frac{1}{t}\, dt \biggr) \le\exp \biggl(\frac{1}{\Lambda}\int _{x}^{y}\frac{Q'(t)}{Q(t)}\, dt \biggr) = \biggl( \frac{Q(y)}{Q(x)} \biggr)^{\frac{1}{\Lambda}}. $$

Therefore, we obtain (3.2) with $C=C_{1}+\frac{1}{\Lambda}$, that is, we see $\mathcal{F}(C^{2}+)\subset\mathcal{E}_{1}$. □

Theorem 3.4

Let $w\in\mathcal{F}(C^{2}+)$.

• (1) If f is a function having bounded variation on any compact interval and if

  $$ \int_{-\infty}^{\infty} w(x)\bigl\vert df(x)\bigr\vert < \infty, $$

  then there exists a constant $C>0$ such that, for every $t>0$,

  $$ \omega_{1}(f,w,t)\le C t\int_{-\infty}^{\infty} w(x)\bigl\vert df(x)\bigr\vert , $$

  and so

  $$ E_{1,n}(w,f)\le C\frac{a_{n}}{n}\int_{-\infty}^{\infty} w(x)\bigl\vert df(x)\bigr\vert . $$

• (2) If f is continuous and $\lim_{|x|\rightarrow \infty}|(\sqrt{T}wf)(x)|=0$, then we have

  $$ \lim_{t\rightarrow0}\omega_{\infty}(f,w,t)=0. $$

To prove this theorem we need the following lemma.

Lemma 3.5

(Lemma 2.5(b) in [7] and Lemma 7 in [6])

Let $w\in\mathcal{F}(C^{2}+)$. Uniformly for $u>0$ large enough and $|x|, |y| \le a_{u}$ such that

$$ |x-y|\le t\Phi_{t}(x), \quad t=a_{u}/u, $$

then

$$ w(x) \sim w(y). $$

Proof of Theorem 3.4

(1) Let $g(x):=f(x)-f(0)$. For $t>0$ small enough let $0< h\le t$ and $|x|\le\sigma(2t)<\sigma(t)$. Hence we have $\Phi_{t}(x)\le2$ for $|x| \le\sigma(2t)$. Then by Lemma 3.5,

$$\begin{aligned}& \int_{|x| \le\sigma(2t)} w(x)\biggl\vert g \biggl(x+\frac{h}{2} \Phi _{t}(x) \biggr)-g \biggl(x-\frac{h}{2}\Phi_{t}(x) \biggr)\biggr\vert \, dx \\& \quad =\int_{|x| \le\sigma(2t)} w(x)\biggl\vert \int _{x-\frac{h}{2}\Phi _{t}(x)}^{x+\frac{h}{2}\Phi_{t}(x)}\, df(v)\biggr\vert \, dx \le C\int _{|x| \le\sigma(2t)} \biggl\vert \int_{x-\frac{h}{2}\Phi _{t}(x)}^{x+\frac{h}{2}\Phi_{t}(x)}w(v) \, df(v)\biggr\vert \, dx \\& \quad \le\int_{-\infty}^{\infty} \int_{x-h}^{x+h} w(v)\bigl\vert df(v)\bigr\vert \, dx \le\int_{-\infty}^{\infty} w(v)\int_{v-h\le x\le v+h} \, dx\bigl\vert df(v)\bigr\vert \\& \quad \le2h\int_{-\infty}^{\infty} w(v)\bigl\vert df(v) \bigr\vert . \end{aligned}$$

Hence we have

$$ \int_{|x| \le\sigma(2t)} w(x)\biggl\vert g \biggl(x+ \frac{h}{2}\Phi _{t}(x) \biggr) -g \biggl(x-\frac{h}{2} \Phi_{t}(x) \biggr)\biggr\vert \, dx \le2t \int_{-\infty}^{\infty} w(x)\bigl\vert df(x)\bigr\vert . $$

(3.3)

Moreover, we see

$$ \inf_{c\in\mathbb{R}}\bigl\Vert w(x) (f-c) (x)\bigr\Vert _{L_{1}(|x|\ge \sigma(4t))} \le\frac{1}{Q'(\sigma(4t))}\bigl\Vert Q'(x)w(x)g(x) \bigr\Vert _{L_{1}(|x|\ge\sigma(4t))}. $$

(3.4)

From Lemma 3.2(2), for $4t=:\frac{a_{u}}{u}$,

$$ Q'\bigl(\sigma(4t)\bigr)= Q'(a_{u})\sim \frac{u\sqrt{T(a_{u})}}{a_{u}}\sim\frac {\sqrt{T(\sigma(4t))}}{t}. $$

On the other hand, we have

$$\begin{aligned} \int_{0}^{\infty} Q'(x)w(x)\bigl\vert g(x)\bigr\vert \, dx =& \int_{0}^{\infty} Q'(x)w(x)\biggl\vert \int_{0}^{x}dg(u) \biggr\vert \, dx \\ \le& \int_{0}^{\infty} Q'(x)w(x)\int _{0}^{x}\bigl\vert df(u)\bigr\vert \, dx \\ =& {\bigr.{-}w(x)\int_{0}^{x}\bigl\vert df(u)\bigr\vert \bigl|_{0}^{\infty} }+\int_{0}^{\infty} w(u)\bigl\vert df(u)\bigr\vert . \end{aligned}$$

Here we see

$$ \biggl\vert -w(x)\int_{0}^{x}\bigl\vert df(u) \bigr\vert \biggr\vert \le\int_{0}^{x} w(u) \bigl\vert df(u)\bigr\vert . $$

Therefore, we have

$$ \int_{0}^{\infty} Q'(x)w(x)\bigl\vert g(x)\bigr\vert \, dx \le2\int_{0}^{\infty} w(u)\bigl\vert df(u)\bigr\vert . $$

Similarly, for $x < 0$ we see

$$ \int_{-\infty}^{0} \bigl\vert Q'(x)w(x)g(x) \bigr\vert \, dx \le2\int_{-\infty}^{0} w(x)\bigl\vert df(x)\bigr\vert . $$

Consequently, we have

$$ \int_{-\infty}^{\infty} \bigl\vert Q'(x)w(x)g(x) \bigr\vert \, dx \le2\int_{-\infty}^{\infty} w(x)\bigl\vert df(x)\bigr\vert . $$

Hence we have

$$ \bigl\Vert Q'wg\bigr\Vert _{L_{1}(\mathbb{R})}\le2 \int_{-\infty}^{\infty} w(u)\bigl\vert df(u)\bigr\vert . $$

(3.5)

Therefore, using (3.4) and (3.5), we have

$$ \inf_{c\in\mathbb{R}}\bigl\Vert w(x) (f-c) (x)\bigr\Vert _{L_{1}(|x|\ge \sigma(4t))} = O(t) \int_{-\infty}^{\infty} w(x) \bigl\vert df(x)\bigr\vert . $$

(3.6)

Consequently, by (3.3) and (3.6) we have

$$ \omega_{1}(f,w,t)\le C t\int_{-\infty}^{\infty} w(x)\bigl\vert df(x)\bigr\vert . $$

Hence, setting $t=C_{2}\frac{a_{n}}{n}$, if we use Proposition 3.3, then

$$ E_{1,n}(w,f)\le C\frac{a_{n}}{n}\int_{-\infty}^{\infty} w(x)\bigl\vert df(x)\bigr\vert . $$

(2) Given $\varepsilon>0$, and let us take $L=L(\varepsilon)>0$ such that

$$ \sup_{|x|\ge L}\bigl\vert w(x)f(x)\bigr\vert \le\sup _{|x|\ge L}\bigl\vert \sqrt {T(x)}w(x)f(x)\bigr\vert < \varepsilon, $$

since $T(x)>1$. Hence, if $|x| \ge2L$ and $0 < t < t_{0}$, then

$$\begin{aligned}& \biggl\vert w(x) \biggl\{ f \biggl(x+\frac{h}{2}\Phi_{t}(x) \biggr)-f \biggl(x-\frac{h}{2}\Phi_{t}(x) \biggr) \biggr\} \biggr\vert \\& \quad \le C \biggl[ \biggl\vert \sqrt{T \biggl(x+\frac{h}{2} \Phi_{t}(x) \biggr)}w \biggl(x+\frac {h}{2}\Phi_{t}(x) \biggr)f \biggl(x+\frac{h}{2}\Phi_{t}(x) \biggr) \biggr\vert \\& \qquad {}+\biggl\vert \sqrt{T \biggl(x-\frac{h}{2}\Phi_{t}(x) \biggr)}w \biggl(x-\frac {h}{2}\Phi_{t}(x) \biggr)f \biggl(x- \frac{h}{2}\Phi_{t}(x) \biggr) \biggr\vert \biggr] \\& \quad \le 2C\varepsilon, \end{aligned}$$

where for the first inequality we used Lemma 3.5(2), and for the second inequality we used the fact that $|x \pm\frac {h}{2}\Phi_{t}(x)| \ge L$. On the other hand,

$$ \lim_{t\rightarrow0}\sup_{0< h\le t} \biggl\Vert w(x) \biggl\{ f \biggl(x+\frac{h}{2}\Phi_{t}(x) \biggr) -f \biggl(x- \frac{h}{2}\Phi_{t}(x) \biggr) \biggr\} \biggr\Vert _{L_{\infty}(|x|\le2L)}=0. $$

Finally, we will show

$$ \inf_{c\in\mathbb{R}}\bigl\| w(f-c)\bigr\| _{L_{\infty}(|x|\geq\sigma(4t))}\to0,\quad t\to0. $$

(3.7)

If we let $4t := \frac{a_{n}}{n}$, then we see $n \to \infty$ and $\sigma(4t) = a_{n} \to \infty$ as $t \to 0$. Hence using $\lim_{|x|\to \infty}|(\sqrt{T}wf)(x)|=0$, we have for $|x|\geq \sigma(4t)$,

$$a_{n} < x \to \infty \quad \Rightarrow\quad \bigl|f(x)w(x)\bigr|\leq \bigl|T^{1/2}(x)f(x)w(x)\bigr|\to 0 $$

and $|cw(x)|\leq cw(a_{n}) \to 0$ as $t \to 0$. Therefore, (3.7) is proved. Consequently, we have the result. □

Lemma 3.6

(cf. Lemma 4.4 in [9])

Let g be a real valued function on $\mathbb{R}$ satisfying $\|gw\| _{L_{\infty}(\mathbb{R})}<\infty$ and, for some $n \ge1$,

$$ \int_{-\infty}^{\infty} gPw^{2}\, dt=0, \quad P\in\mathcal{P}_{n}. $$

(3.8)

Then we have

$$ \biggl\Vert w(x)\int_{0}^{x} g(t) \, dt \biggr\Vert _{L_{\infty}(\mathbb{R})} \le C \frac{a_{n}}{n}\|gw\|_{L_{\infty}(\mathbb{R})}. $$

(3.9)

Especially, if $w\in\mathcal{F}_{\lambda}(C^{3}+)$, $0<\lambda<3/2$ and $T^{1/4}wf'\in L_{\infty}(\mathbb{R})$, then we have

$$ \biggl\Vert w(x)\int_{0}^{x} \bigl(f'(t)-v_{n}\bigl(f'\bigr) (t) \bigr) \, dt \biggr\Vert _{L_{\infty}(\mathbb{R})} \le C \frac{a_{n}}{n}E_{n} \bigl(w_{1/4},f' \bigr). $$

(3.10)

When $w\in\mathcal{F}^{*}$, we also have (3.10) replacing $w_{1/4}$ with cw.

Proof

We let

$$ \phi_{x}(t)= \left \{ \textstyle\begin{array}{l@{\quad}l} w^{-2}(t),& 0\le t\le x; \\ 0,& \mbox{otherwise}, \end{array}\displaystyle \right . $$

(3.11)

then we have, for arbitrary $P_{n}\in\mathcal{P}_{n}$,

$$\begin{aligned} \biggl\vert \int_{0}^{x} g(t)\, dt \biggr\vert =&\biggl\vert \int_{-\infty}^{\infty} g(t) \phi_{x}(t)w^{2}(t)\, dt\biggr\vert \\ =&\biggl\vert \int _{-\infty}^{\infty} g(t) \bigl(\phi _{x}(t)-P_{n}(t) \bigr)w^{2}(t)\, dt\biggr\vert . \end{aligned}$$

(3.12)

Therefore, we have

$$\begin{aligned} \biggl\vert \int_{0}^{x} g(t)\, dt\biggr\vert \le& \|gw\|_{L_{\infty}(\mathbb{R})} \inf_{P_{n}\in\mathcal{P}_{n}}\int_{-\infty}^{\infty} \bigl\vert \phi _{x}(t)-P_{n}(t)\bigr\vert w(t)\, dt \\ =&\|gw\|_{L_{\infty}(\mathbb{R})}E_{1,n}(w,\phi_{x}). \end{aligned}$$

Here, from Theorem 3.4 we see that

$$\begin{aligned} E_{1,n}(w,\phi_{x}) \le& C\frac{a_{n}}{n}\int _{-\infty}^{\infty} w(t)\bigl\vert d\phi_{x}(t) \bigr\vert \\ \le& C\frac{a_{n}}{n}\int_{0}^{x} w(t)\bigl\vert Q'(t)\bigr\vert w^{-2}(t)\, dt \\ =& C\frac{a_{n}}{n}\int_{0}^{x} Q'(t)w^{-1}(t)\, dt \\ \le& C\frac{a_{n}}{n}w^{-1}(x). \end{aligned}$$

So, we have

$$\begin{aligned} \biggl\vert w(x)\int_{0}^{x} g(t)\, dt\biggr\vert \le& \|gw\|_{L_{\infty}(\mathbb{R})}w(x)E_{1,n} (w,\phi_{x} ) \\ \le& C\frac{a_{n}}{n}\|gw\|_{L_{\infty}(\mathbb{R})}. \end{aligned}$$

Therefore, we have (3.9). Next we show (3.10). Since

$$ v_{n}\bigl(f'\bigr) (t)=\frac{1}{n}\sum _{j=n+1}^{2n}s_{j}\bigl(f',t \bigr), $$

and, for any $P\in\mathcal{P}_{n}$, $j\ge n+1$,

$$ \int_{-\infty}^{\infty} \bigl(f'(t)-s_{j} \bigl(f';t\bigr) \bigr)P(t)w^{2}(t)\, dt=0, $$

we have

$$ \int_{-\infty}^{\infty} \bigl(f'(t)-v_{n} \bigl(f'\bigr) (t) \bigr)P(t)w^{2}(t)\, dt=0. $$

(3.13)

Using (3.9) and (3.1), we have (3.10). □

Lemma 3.7

Let $w=\exp(-Q)\in\mathcal{F}_{\lambda}(C^{3}+)$, $0<\lambda<3/2$. Let $\Vert w_{1/4}f'\Vert _{L_{\infty}(\mathbb{R})}<\infty$, and let $q_{n-1}\in\mathcal{P}_{n-1}$ ($n \ge1$) be the best approximation of $f'$ with respect to the weight w, that is,

$$ \bigl\Vert \bigl(f'-q_{n-1}\bigr)w \bigr\Vert _{L_{\infty}(\mathbb{R})}=E_{n-1}\bigl(w,f'\bigr). $$

Now we set

$$ F(x):=f(x)-\int_{0}^{x}q_{n-1}(t)\, dt, $$

then there exists $S_{2n}\in\mathcal{P}_{2n}$ such that

$$ \bigl\Vert w (F-S_{2n} ) \bigr\Vert _{L_{\infty}(\mathbb{R})} \le C \frac{a_{n}}{n}E_{n} \bigl(w_{1/4},f' \bigr) $$

and

$$ \bigl\Vert wS_{2n}' \bigr\Vert _{L_{\infty}(\mathbb{R})}\le C E_{n-1} \bigl(w_{1/4},f' \bigr). $$

When $w\in\mathcal{F}^{*}$, we have the same results replacing $w_{1/4}$ with cw.

Proof

Let

$$ S_{2n}(x)=f(0)+\int_{0}^{x} v_{n} \bigl(f'-q_{n-1} \bigr) (t)\, dt, $$

(3.14)

then, by Lemma 3.6 and (3.10),

$$\begin{aligned} \begin{aligned} &\bigl\Vert w (F-S_{2n} )\bigr\Vert _{L_{\infty}(\mathbb{R})} \\ &\quad = \biggl\Vert w \biggl(f-\int_{0}^{x}q_{n-1}(t) \, dt -f(0)-\int_{0}^{x} v_{n} \bigl(f'-q_{n-1} \bigr) (t)\,dt \biggr)\biggr\Vert _{L_{\infty}(\mathbb{R})} \\ &\quad = \biggl\Vert w \biggl(\int_{0}^{x} \bigl[f'(t)-v_{n}\bigl(f'\bigr) (t) \bigr] \, dt \biggr)\biggr\Vert _{L_{\infty}(\mathbb{R})} \le C\frac{a_{n}}{n}E_{n} \bigl(w_{1/4},f' \bigr). \end{aligned} \end{aligned}$$

Now by Theorem 3.1, (3.1),

$$\begin{aligned} \bigl\Vert wS_{2n}'\bigr\Vert _{L_{\infty}(\mathbb{R})} =& \bigl\Vert w \bigl(v_{n}\bigl(f'-q_{n-1}\bigr) \bigr)\bigr\Vert _{L_{\infty}(\mathbb {R})} \\ \le&\bigl\Vert \bigl(f'-v_{n} \bigl(f' \bigr) \bigr)w\bigr\Vert _{L_{\infty}(\mathbb{R})} +\bigl\Vert \bigl(f'-q_{n-1} \bigr)w\bigr\Vert _{L_{\infty}(\mathbb{R})} \\ \le& E_{n} \bigl(w_{1/4},f' \bigr)+E_{n-1} \bigl(w,f' \bigr) \le2E_{n-1} \bigl(w_{1/4},f' \bigr). \end{aligned}$$

 □

To prove Theorem 2.3 we need the following theorems with $p=\infty$.

Theorem 3.8

(Corollary 3.4 in [6])

Let $w\in\mathcal{F}(C^{2}+)$, and let $r\ge0$ be an integer. Let $1\le p\le\infty$, and let $wf^{(r)}\in L_{p}(\mathbb{R})$. Then we have, for $n\ge r$,

$$ E_{p,n}(f,w)\le C \biggl(\frac{a_{n}}{n} \biggr)^{k} \bigl\Vert f^{(k)}w\bigr\Vert _{L_{p}(\mathbb{R})}, \quad k=1,2,\ldots,r, $$

and equivalently,

$$ E_{p,n}(w,f)\le C \biggl(\frac{a_{n}}{n} \biggr)^{k}E_{p,n-k} \bigl(w,f^{(k)} \bigr). $$

Theorem 3.9

(Corollary 6.2 in [4])

Let $r\ge1$ be an integer and $w\in\mathcal{F}_{\lambda}(C^{r+2}+)$, $0< \lambda<(r+2)/(r+1)$, and let $1\le p\le\infty$. Then there exists a constant $C>0$ such that, for any $1\le k\le r$, any integer $n\ge1$, and any polynomial $P\in \mathcal{P}_{n}$,

$$ \bigl\Vert P^{(k)}w\bigr\Vert _{L_{p}(\mathbb{R})} \le C \biggl( \frac{n}{a_{n}} \biggr)^{k}\bigl\Vert T^{k/2}Pw\bigr\Vert _{L_{p}(\mathbb{R})}. $$

Proof of Theorem 2.3

We show that for $k=0,1,\ldots,r$,

$$ \bigl\vert \bigl(f^{(k)}(x)-P_{n;f,w}^{(k)} \bigr)w(x)\bigr\vert \le CT^{k/2}(x)E_{n-k} \bigl(w_{1/4},f^{(k)} \bigr). $$

(3.15)

If $r=0$, then (3.15) is trivial. For some $r\ge0$ we suppose that (3.15) holds, and let $f\in C^{(r+1)}(\mathbb{R})$ be satisfying

$$\lim_{|x|\to\infty} T^{1/4}(x)f^{(r+1)}(x)w(x)=0. $$

Then $f'\in C^{(r)}(\mathbb{R})$, and

$$\lim_{|x|\to\infty} T^{1/4}(x) (f' )^{(r)}(x)w(x)=0. $$

So we may apply the induction assumption to $f'$, for $0 \le k\le r$. Let $q_{n-1}\in\mathcal{P}_{n-1}$ be the polynomial of best approximation of $f'$ with respect to the weight w. Then from our assumption we have, for $0\le k\le r$,

$$ \bigl\vert \bigl(f^{(k+1)}(x)-q_{n-1}^{(k)}(x) \bigr)w(x)\bigr\vert \le C T^{k/2}(x)E_{n-1-k} \bigl(w_{1/4},f^{(k+1)} \bigr), $$

that is, for $1\le k\le r+1$,

$$ \bigl\vert \bigl(f^{(k)}(x)-q_{n-1}^{(k-1)}(x) \bigr)w(x)\bigr\vert \le C T^{\frac{k-1}{2}}(x)E_{n-k} \bigl(w_{1/4},f^{(k)} \bigr). $$

(3.16)

Let

$$ F(x):=f(x)-\int_{0}^{x} q_{n-1}(t)\, dt=f(x)-Q_{n}(x), $$

(3.17)

then

$$ \bigl\vert F'(x)w(x)\bigr\vert \le C E_{n-1} \bigl(w,f' \bigr). $$

As (3.14) we set $S_{2n}=\int_{0}^{x}(v_{n}(f')(t)-q_{n-1}(t))\, dt+f(0)$, then from Lemma 3.7

$$ \bigl\Vert (F-S_{2n} )w\bigr\Vert _{L_{\infty}(\mathbb{R})} \le C \frac{a_{n}}{n}E_{n} \bigl(w_{1/4},f' \bigr) $$

(3.18)

and

$$ \bigl\Vert S_{2n}'w\bigr\Vert _{L_{\infty}(\mathbb{R})}\le C E_{n-1} \bigl(w_{1/4},f' \bigr). $$

Here we apply Theorem 3.9 with the weight $w_{-(k-1)/2}$. In fact, by Theorem 2.2 we have $w_{-(k-1)/2}\in \mathcal{F}_{\lambda}(C^{r+2}+)$. Then, noting $a_{2n}\sim a_{n}$ from Lemma 3.2(1), we see

$$\begin{aligned} \bigl\vert S_{2n}^{(k)}(x) w_{-(k-1)/2}(x)\bigr\vert \le& C \biggl(\frac{n}{a_{n}} \biggr)^{k-1}\bigl\Vert S_{2n}'w\bigr\Vert _{L_{\infty}(\mathbb{R})} \\ \le& C \biggl(\frac{n}{a_{n}} \biggr)^{k-1}E_{n-1} \bigl(w_{1/4},f' \bigr), \end{aligned}$$

that is,

$$ \bigl\vert S_{2n}^{(k)}(x) w(x)\bigr\vert \le C \biggl(\frac{n\sqrt{T(x)}}{a_{n}} \biggr)^{k-1}E_{n-1} \bigl(w_{1/4},f' \bigr),\quad 1\le k\le r+1. $$

(3.19)

Let $R_{n}\in\mathcal{P}_{n}$ denote the polynomial of best approximation of F with w. By Theorem 3.9 with $w_{-\frac{k}{2}}$ again, for $0\le k\le r+1$, we have

$$\begin{aligned} \bigl\vert \bigl(R_{n}^{(k)}-S_{2n}^{(k)}(x) \bigr) w_{-\frac{k}{2}}(x)\bigr\vert \le& C \biggl(\frac{n}{a_{n}} \biggr)^{k}\bigl\Vert (R_{n}-S_{2n})w_{-\frac {k}{2}}(x)T^{k/2}(x) \bigr\Vert _{L_{\infty}(\mathbb{R})} \\ \le& C \biggl(\frac{n}{a_{n}} \biggr)^{k}\bigl\Vert (R_{n}-S_{2n})w\bigr\Vert _{L_{\infty}(\mathbb{R})} \end{aligned}$$

(3.20)

and by (3.18)

$$\begin{aligned} \bigl\Vert (R_{n}-S_{2n})w\bigr\Vert _{L_{\infty}(\mathbb{R})} \le& C \bigl[\bigl\Vert (F-R_{n})w\bigr\Vert _{L_{\infty}(\mathbb{R})}+\bigl\Vert (F-S_{2n})w\bigr\Vert _{L_{\infty}(\mathbb{R})} \bigr] \\ \le& C \biggl[E_{n}(w,F)+\frac{a_{n}}{n}E_{n} \bigl(w_{1/4},f' \bigr) \biggr] \\ \le& C \biggl[\frac{a_{n}}{n}E_{n-1}\bigl(w,f'\bigr)+ \frac {a_{n}}{n}E_{n-1}\bigl(w_{1/4},f'\bigr) \biggr] \\ \le& C \frac{a_{n}}{n}E_{n-1} \bigl(w_{1/4},f' \bigr). \end{aligned}$$

(3.21)

Hence, from (3.20) and (3.21) we have, for $0\le k\le r+1$,

$$\begin{aligned} \bigl\vert \bigl(R_{n}^{(k)}-S_{2n}^{(k)}(x) \bigr) w(x)\bigr\vert \le& C\bigl\vert T^{k/2}(x)\bigr\vert \bigl\vert \bigl(R_{n}^{(k)}-S_{2n}^{(k)}(x) \bigr) w_{-\frac{k}{2}}(x)\bigr\vert \\ \le& C \biggl(\frac{n\sqrt{T(x)}}{a_{n}} \biggr)^{k}\frac {a_{n}}{n}E_{n-1} \bigl(w_{1/4},f' \bigr). \end{aligned}$$

(3.22)

Therefore by (3.19), (3.22), and Theorem 3.8,

$$\begin{aligned} \bigl\vert R_{n}^{(k)}(x) w(x)\bigr\vert \le& C T^{k/2}(x) \biggl(\frac{n}{a_{n}} \biggr)^{k-1} E_{n-1} \bigl(w_{1/4},f' \bigr) \\ \le& C T^{k/2}(x)E_{n-k} \bigl(w_{1/4},f^{(k)} \bigr). \end{aligned}$$

(3.23)

Since $E_{n}(w,F)=E_{n}(w,f)$ and

$$ E_{n} (w,F )=\bigl\Vert w (F-R_{n} )\bigr\Vert _{L_{\infty}(\mathbb{R})} =\bigl\Vert w (f-Q_{n}-R_{n} ) \bigr\Vert _{L_{\infty}(\mathbb{R})} $$

(3.24)

(see (3.17)), we know that $P_{n;f,w}:=Q_{n}+R_{n}$ is the polynomial of best approximation of f with w. Now, from (3.16), (3.17), and (3.23) we have, for $1\le k\le r+1$,

$$\begin{aligned} \bigl\vert \bigl(f^{(k)}(x)-P_{n;f.w}^{(k)}(x) \bigr)w(x)\bigr\vert =&\bigl\vert \bigl(f^{(k)}(x)-Q_{n}^{(k)}(x)-R_{n}^{(k)}(x) \bigr)w(x)\bigr\vert \\ \le& \bigl\vert \bigl(f^{(k)}(x)-q_{n-1}^{(k-1)}(x) \bigr)w(x)\bigr\vert +\bigl\vert R_{n}^{(k)}(x)w(x)\bigr\vert \\ \le& C T^{k/2}(x)E_{n-k} \bigl(w_{1/4},f^{(k)} \bigr). \end{aligned}$$

For $k=0$ it is trivial. Consequently, we have (3.15) for all $r\ge0$. Moreover, using Theorem 3.8, we conclude Theorem 2.3. □

Proof of Corollary 2.4

It follows from Theorem 2.3. □

Proof of Corollary 2.5

Applying Theorem 2.3 with $w_{k/2}$, we have, for $0\le j\le r$,

$$ \bigl\Vert \bigl(f^{(j)}-P_{n;f,w_{k/2}}^{(j)}\bigr)w\bigr\Vert _{L_{\infty}(\mathbb{R})} \le C E_{n-k} \bigl(w_{(2k+1)/4},f^{(j)} \bigr). $$

Especially, when $j=k$, we obtain

$$ \bigl\Vert \bigl(f^{(k)}-P_{n;f,w_{k/2}}^{(k)} \bigr)w\bigr\Vert _{L_{\infty}(\mathbb{R})} \le CE_{n-k} \bigl(w_{(2k+1)/4},f^{(k)} \bigr). $$

 □