# 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:

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

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

□

## References

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3. Leviatan, D: The behavior of the derivatives of the algebraic polynomials of best approximation. J. Approx. Theory 35, 169-176 (1982)

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## Acknowledgements

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

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### Corresponding author

Correspondence to Hee Sun Jung.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript and participated in the sequence alignment. All authors read and approved the final manuscript.

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Jung, H.S., Sakai, R. Higher order derivatives of approximation polynomials on $$\mathbb{R}$$ . J Inequal Appl 2015, 268 (2015). https://doi.org/10.1186/s13660-015-0789-y