Open Access

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

Journal of Inequalities and Applications20152015:268

https://doi.org/10.1186/s13660-015-0789-y

Received: 20 March 2015

Accepted: 18 August 2015

Published: 4 September 2015

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 \(\mathbb{R}\).

Keywords

the polynomial of best approximation the exponential-type weight

MSC

41A10 41A50

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:
  1. (a)

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

     
  2. (b)

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

     
  3. (c)

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

     
  4. (d)
    The even function
    $$ T_{w}(x):=\frac{xQ'(x)}{Q(x)}, \quad x\neq0 $$
    is quasi-increasing in \((0,\infty)\), with
    $$ T_{w}(x)\ge\Lambda>1, \quad x\in\mathbb{R}\backslash\{0\}. $$
     
  5. (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. (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. (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. (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. (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. (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. (1)
    (Lemma 3.5(a) in [1]) Let \(L>0\) be fixed. Then, uniformly for \(t>0\),
    $$ a_{Lt}\sim a_{t}. $$
     
  2. (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. (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. (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. (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
$$ \bigl\Vert \bigl(f^{(k)}-P_{n;f,w_{k/2}}^{(k)} \bigr)w\bigr\Vert _{L_{\infty}(\mathbb{R})} \le CE_{n-k} \bigl(w_{(2k+1)/4},f^{(k)} \bigr). $$
 □

Declarations

Acknowledgements

The authors thank to Prof. Dany Leviatan for many kind suggestions and comments.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics Education, Sungkyunkwan University
(2)
Department of Mathematics, Meijo University

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© Jung and Sakai 2015