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Strong converse inequality for left Bernstein-Durrmeyer quasi-interpolants

Journal of Inequalities and Applications20152015:367

https://doi.org/10.1186/s13660-015-0890-2

  • Received: 13 August 2015
  • Accepted: 9 November 2015
  • Published:

Abstract

For the left Bernstein-Durrmeyer quasi-interpolants \(M_{n}^{(2r-1)}f\), we prove that, for some l,
$$\omega_{\varphi}^{2r}\biggl(f,\frac{1}{\sqrt{n}} \biggr)_{p}\leqslant C \bigl(\bigl\Vert M_{n}^{(2r-1)}f-f \bigr\Vert _{p}+\bigl\Vert M_{ln}^{(2r-1)}f-f\bigr\Vert _{p} \bigr) \quad (1< p\leq\infty). $$
This is a strong converse inequality of type B.

Keywords

  • strong converse inequality
  • modulus of smoothness
  • K-functional
  • Bernstein-Durrmeyer left quasi-interpolants

MSC

  • 41A27
  • 41A40

1 Introduction

For \(f\in L_{p}[0,1]\) the Bernstein-Durrmeyer operators are given by
$$M_{n}(f,x)=(n+1)\sum^{n}_{k=0}p_{n,k}(x) \int_{0}^{1}p_{n,k}(t)f(t)\, dt, $$
where \(p_{n,k}(x)= {n\choose k}x^{k}(1-x)^{n-k}\) (cf. [1, 2] and [3, 4] for more integral type operators). The rate of convergence and the inverse theorem for \(M_{n}(f,x)\) and their combination have been investigated in [5]. Recently Sablonnière (cf. [6, 7]) introduced a family of operators, so-called quasi-interpolants. Many quasi-interpolants of different operators were studied (e.g. [812]).
In the following \(\Pi_{j}\) denotes the space of algebraic polynomials of degree at most j. Because \(M_{n}\) is an automorphism of \(\Pi_{n}\), \(M_{n}\) and its inverse \(M_{n}^{-1}\) can be expressed as linear differential operators with polynomial coefficients in the forms \(M_{n}=\sum^{n}_{j=0}\beta^{n}_{j}(x)D^{j}\) and \(M_{n}^{-1}=\sum^{n}_{j=0}\alpha^{n}_{j}(x)D^{j}\), where \(D^{0}=id\), \(D=\frac{d}{dx}\). The polynomials \(\alpha^{n}_{j}(x)\in \Pi_{j}\) are expressed explicitly in terms of shifted Jacobi polynomials (cf. [6, 9, 12]) as
$$\alpha^{n}_{j}(x)=\sum^{[j/2]}_{s=0}(-1)^{s}X^{s}J^{(s,s)}_{j-2s}(x)/s!(n)_{j-s}, $$
where \(X=x(1-x)\), \((n)_{j}=n(n-1)\cdots(n+j-1)\), and
$$J^{(s,s)}_{j-2s}(x)=\sum^{j-2s}_{i=0}{j-s \choose i} {j-s\choose j-2s-i}(x-1)^{j-2s-i}x^{i}. $$
Now we give the definition of left Bernstein-Durrmeyer quasi-interpolants (cf. [6, 9, 12]):
$$ M_{n}^{(r)}(f,x)=\sum^{r}_{j=0} \alpha^{n}_{j}(x)D^{j}M_{n}(f,x)=:\sum ^{r}_{j=0}\alpha^{n}_{j}(x)M_{n,j}(f,x), $$
(1.1)
where \(M_{n,j}=D^{j}M_{n}\). It is well known that \(\alpha^{n}_{0}(x)=1\), \(M_{n}^{(r)}\) is exact on \(\Pi_{r}\), i.e. \(M_{n}^{(r)}p=p\) for all \(p\in\Pi_{r}\), \(0\leq r\leq n\).

For \(M_{n}^{(2r-1)}(f,x)\) the global approximation equivalent theorem has been obtained in [9] as follows.

Theorem

[9]

Let \(f\in L_{p}[0,1] \), \(1< p\leq\infty\), \(\varphi(x)=\sqrt{x(1-x)}\), \(n\geq4r\), \(r\in N\), \(0<\alpha<r\), then
$$\bigl\Vert M_{n}^{(2r-1)}f-f\bigr\Vert _{p}=O \bigl(n^{-\alpha}\bigr)\quad \Leftrightarrow\quad \omega_{\varphi }^{2r}(f,t)_{p}=O \bigl(t^{2\alpha}\bigr). $$
Here
$$\begin{aligned}& \omega_{\varphi}^{s}(f,t)_{p}=\sup _{0< h\leq t}\bigl\Vert \bigtriangleup^{s}_{h\varphi}f(x) \bigr\Vert _{p}, \\& \bigtriangleup^{s}_{h\varphi}f(x)=\sum ^{s}_{k=0}(-1)^{k}{s \choose k}f(x+sh/2-kh). \end{aligned}$$
This is Ditzian-Totik modulus of smoothness, it is equivalent to K-functional
$$ K^{s}_{\varphi}\bigl(f,t^{s} \bigr)_{p}=\inf_{g\in W^{s}(\varphi)} \bigl\{ \Vert f-g\Vert _{p}+t^{s}\bigl\Vert \varphi^{s} g^{(s)}\bigr\Vert _{p} \bigr\} , $$
(1.2)
where \(W^{s}(\varphi)=:\{g\in L_{p}[0,1], g^{(s-1)}\in \mathit{A.C.}_{[0,1]},\|\varphi^{s} g^{(s)}\|_{p}<\infty\}\). It was proved that \(\omega_{\varphi}^{s}(f,t)_{p}\sim K^{s}_{\varphi}(f,t^{s})_{p}\), i.e. there exists \(A>0\) such that (cf. [13])
$$ A^{-1}\omega_{\varphi}^{s}(f,t)_{p} \leq K^{s}_{\varphi}\bigl(f,t^{s}\bigr)_{p} \leq A\omega_{\varphi}^{s}(f,t)_{p}. $$
(1.3)

The strong converse inequality is an important problem of operator approximation theory. The strong converse inequalities for various operators have been investigated in subsequent papers (e.g. [14, 15]). In most of these results the second order moduli of smoothness \(\omega_{\varphi}^{2}(f,t)_{p}\) were used. The intention of this paper is to prove a strong converse inequality of type B for the quasi-interpolants \(M_{n}^{(2r-1)}f\) by using high order modulus. To this end we have to prove several key lemmas presented in Section 2. Application of these lemmas enables us to prove our main result in Section 3.

Throughout this paper C denotes a positive constant independent of n and x not necessarily the same at each occurrence.

2 Lemmas

In this section we give some lemmas.

Lemma 2.1

(cf. [9, 10])

For \(j\geq1\), \(r\in N\), we have
$$ \bigl\vert \alpha^{n}_{j}(x)\bigr\vert \leq Cn^{-\frac{j}{2}}\delta_{n}^{j}(x),\qquad \bigl\vert D^{r}\alpha^{n}_{j}(x)\bigr\vert \leq Cn^{\frac{-j+r}{2}}\delta_{n}^{j-r}(x), $$
(2.1)
where \(\delta_{n}(x)=\varphi(x)+\frac{1}{\sqrt{n}}\sim\max \{\varphi(x),\frac{1}{\sqrt{n}}\}\).

Lemma 2.2

Let \(E_{n}=[\frac{1}{n},1-\frac{1}{n}]\), \(\varphi(x)=\sqrt{x(1-x)}\), \(f\in W^{2r+1}(\varphi)\) and \(R_{2r+1}(f,t,x)=\frac{1}{(2r)!}\int^{t}_{x}(t-u)^{2r}f^{(2r+1)}(u)\, du\), then we have, for \(1< p\leq\infty\),
$$ \bigl\Vert M_{n}^{(2r-1)}\bigl(R_{2r+1}(f, \cdot,x),x\bigr)\bigr\Vert _{p}^{E_{n}}\leq Cn^{-r-\frac{1}{2}} \bigl\Vert \varphi^{2r+1}f^{(2r+1)}\bigr\Vert _{p}. $$
(2.2)

Proof

Let \(\psi(u)=\varphi^{2r+1}(u)f^{(2r+1)}(u)\), \(G(x)=M(\psi,x)=\sup_{t}\vert \frac{1}{t-x}\int^{t}_{x}|\psi (u)|\, du\vert \), i.e. \(G(x)\) is the maximal function of ψ. Noting that (cf. [7])
$$\bigl\vert D^{j}p_{n,k}(x)\bigr\vert \leq C\sum ^{j}_{i=0} \biggl(\frac{\sqrt{n}}{\varphi(x)} \biggr)^{j+i} \biggl\vert \frac{k}{n}-x\biggr\vert ^{i}p_{n,k}(x),\quad x\in E_{n}, $$
we have, for \(x\in E_{n}\),
$$\bigl\vert D^{j}M_{n}(f,x)\bigr\vert =\bigl\vert M_{n,j}(f,x)\bigr\vert \leq C\sum^{j}_{i=0} \biggl(\frac{\sqrt {n}}{\varphi(x)} \biggr)^{j+i}\sum^{n}_{k=0}p_{n,k}(x) \biggl\vert \frac {k}{n}-x\biggr\vert ^{i}\bigl\vert a_{k}(n)\bigr\vert , $$
where \(a_{k}(n)=(n+1)\int_{0}^{1}p_{n,k}(t)f(t)\, dt\). So, for \(x\in E_{n}\),
$$\begin{aligned}& \bigl\vert M_{n}^{(2r-1)}\bigl(R_{2r+1}(f,\cdot,x),x \bigr)\bigr\vert \\& \quad \leq \sum_{j=0}^{2r-1}\bigl\vert \alpha^{n}_{j}(x)\bigr\vert \bigl\vert M_{n,j} \bigl(R_{2r+1}(f,\cdot ,x),x\bigr)\bigr\vert \\& \quad \leq C\sum_{j=0}^{2r-1}\bigl\vert \alpha^{n}_{j}(x)\bigr\vert \sum ^{j}_{i=0} \biggl(\frac {\sqrt{n}}{\varphi(x)} \biggr)^{j+i}\sum^{n}_{k=0}p_{n,k}(x) \biggl\vert \frac{k}{n}-x\biggr\vert ^{i}\bigl\vert \overline{a}_{k}(n)\bigr\vert \\& \quad =:C\sum_{j=0}^{2r-1}I_{j}, \end{aligned}$$
where \(\overline{a}_{k}(n)=\frac{n+1}{(2r)!}\int_{0}^{1}p_{n,k}(t)\int^{t}_{x}(t-u)^{2r}f^{(2r+1)}(u)\, du\, dt\). Using (9.6.1) in [13], we have
$$\begin{aligned} \bigl\vert \overline{a}_{k}(n)\bigr\vert &=\frac{n+1}{(2r)!}\int _{0}^{1}p_{n,k}(t)\biggl\vert \int ^{t}_{x}\frac{(t-u)^{2r}}{\varphi^{2r+1}(u)}\varphi ^{2r+1}(u)f^{(2r+1)}(u)\, du\biggr\vert \, dt \\ &\leq\frac{n+1}{(2r)!}\varphi^{-(2r+1)}(x)G(x)\int_{0}^{1}p_{n,k}(t)|t-x|^{2r+1} \, dt. \end{aligned}$$
Hence by Hölder’s inequality, we have, for \(x\in E_{n}\),
$$\begin{aligned} \begin{aligned} \|I_{j}\|_{p}^{E_{n}}\leq{}&\bigl\Vert G(x)\bigr\Vert _{p}\Biggl\Vert \alpha^{n}_{j}(x)\varphi ^{-(2r+1)}(x)\sum^{j}_{i=0} \biggl( \frac{\sqrt{n}}{\varphi(x)} \biggr)^{j+i}\sum^{n}_{k=0}p_{n,k}(x) \biggl\vert \frac{k}{n}-x\biggr\vert ^{i} \\ &{}\times(n+1)\int_{0}^{1}p_{n,k}(t)|t-x|^{2r+1} \, dt\Biggr\Vert _{p}^{E_{n}} \\ \leq{}&\bigl\Vert G(x)\bigr\Vert _{p}\Biggl\Vert \alpha^{n}_{j}(x)\varphi^{-(2r+1)}(x)\sum ^{j}_{i=0} \biggl(\frac{\sqrt{n}}{\varphi(x)} \biggr)^{j+i} \Biggl(\sum^{n}_{k=0}p_{n,k}(x) \biggl(\frac{k}{n}-x \biggr)^{2i} \Biggr)^{1/2} \\ &{}\times \Biggl(\sum^{n}_{k=0}p_{n,k}(x) (n+1)\int_{0}^{1}p_{n,k}(t) (t-x)^{4r+2}\, dt \Biggr)^{1/2}\Biggr\Vert _{p}^{E_{n}}. \end{aligned} \end{aligned}$$
From (9.4.14) in [13] and (6.4) in [5], we have, for \(x\in E_{n}\),
$$\begin{aligned}& \Biggl(\sum^{n}_{k=0}p_{n,k}(x) \biggl(\frac{k}{n}-x \biggr)^{2i} \Biggr)^{1/2}\leq C \frac{\varphi^{i}(x)}{n^{\frac{i}{2}}}, \\& \Biggl(\sum^{n}_{k=0}p_{n,k}(x) (n+1)\int_{0}^{1}p_{n,k}(t) (t-x)^{4r+2}\, dt \Biggr)^{1/2} \leq C\frac{\varphi^{2r+1}(x)}{n^{r+{\frac{1}{2}}}}. \end{aligned}$$
Together with (2.1) and the fact that
$$\bigl\Vert G(x)\bigr\Vert _{p}\leq C_{p}\bigl\Vert \varphi^{2r+1}f^{(2r+1)}\bigr\Vert _{p}, $$
we obtain (2.2). □

Lemma 2.3

For \(n\geq2r\), we have
$$\begin{aligned}& M_{n}^{(2r-1)}\bigl((t-x)^{2r},x \bigr) \\& \quad = (-1)^{r+1}n^{-r}\varphi^{2r}(x) \frac {(2r)!}{2^{r}(r!)}+\varphi^{2r}(x)o \biggl(\frac{1}{n^{r}} \biggr) \\& \qquad {}+(2r)! \biggl(b^{n}_{2r}\frac{1}{n^{2r}}+b^{n}_{2r-1} \frac{\varphi ^{2}(x)}{n^{2r-1}}+\cdots+b^{n}_{r+1}\frac{\varphi^{2r-2}(x)}{n^{r+1}} \biggr) \biggl(1+O \biggl(\frac{1}{n} \biggr) \biggr), \end{aligned}$$
(2.3)
where \(b_{j}^{n}\) are uniformly bounded in n and independent of x.

Proof

First we note \(M_{n}^{(2r)}p=p\) for all \(p\in\Pi_{2r}\), so we have
$$M_{n}^{(2r)}\bigl((t-x)^{2r},x\bigr)=0, $$
then
$$M_{n}^{(2r)}\bigl((t-x)^{2r},x\bigr)-M_{n}^{(2r-1)} \bigl((t-x)^{2r},x\bigr)=\alpha ^{n}_{2r}(x)M_{n,2r} \bigl((t-x)^{2r},x\bigr). $$
Therefore we have
$$ M_{n}^{(2r-1)}\bigl((t-x)^{2r},x\bigr)=- \alpha^{n}_{2r}(x)M_{n,2r}\bigl((t-x)^{2r},x \bigr). $$
(2.4)
Using (cf. [5])
$$M_{n,2r}(f,x)=\frac{(n+1)!n!}{(n-2r)!(n+2r)!}\sum_{k=0}^{n-2r}p_{n-2r,k}(x) \int^{1}_{0}p_{n+2r,k+2r}(t)f^{(2r)}(t)\, dt, $$
we have
$$M_{n,2r}\bigl((t-x)^{2r},x\bigr)=\frac{(n+1)!n!(2r)!}{(n-2r)!(n+2r)!}=(2r)! \biggl(1+O \biggl(\frac{1}{n} \biggr) \biggr). $$
Therefore
$$ M_{n}^{(2r-1)}\bigl((t-x)^{2r},x \bigr)=-(2r)!\alpha^{n}_{2r}(x) \biggl(1+O \biggl( \frac {1}{n} \biggr) \biggr). $$
(2.5)
Also we have (cf. (3.11) in [10])
$$ \alpha^{n}_{2r}(x)=b^{n}_{2r} \frac{1}{n^{2r}}+b^{n}_{2r-1}\frac{\varphi ^{2}(x)}{n^{2r-1}}+ \cdots+b^{n}_{r}\frac{\varphi^{2r}(x)}{n^{r}}, $$
(2.6)
where \(b_{j}^{n}\) are uniformly bounded in n and independent of x.
By Theorem 4.2 and Table 2 in [7] we know that \(\lim_{n}n^{r}\alpha^{n}_{2r}(x)\) exists and
$$ \lim_{n}n^{r}\alpha^{n}_{2r}(x)= \frac{(-1)^{r}\varphi^{2r}(x)}{2^{r}(r!)}. $$
(2.7)
With this relation and (2.6), we get the representation of the coefficient \(b^{n}_{r}\) in (2.6), i.e.
$$ \lim_{n}b_{r}^{n}= \frac{(-1)^{r}}{2^{r}(r!)}. $$
(2.8)
From (2.5)-(2.8) we get (2.3). □

Lemma 2.4

For \(f\in W^{2r}(\varphi)\), \(1< p\leq\infty\), we have
$$ \bigl\Vert \varphi^{2r+1}D^{2r+1} \bigl(M_{n}^{(2r-1)}f\bigr)\bigr\Vert _{p}\leq C \sqrt{n}\bigl\Vert \varphi^{2r}f^{(2r)}\bigr\Vert _{p}. $$
(2.9)

Proof

By (2.6) in [5] one has, for \(x\in[0,1]\), \(1\leq p\leq\infty\), \(r,s\in N_{0}=N\cup\{0\}\),
$$\bigl\Vert \delta_{n}^{s}(x)\varphi^{2r}(x)D^{2r+s}M_{n}(f,x) \bigr\Vert _{p}\leq Cn^{\frac{s}{2}}\bigl\Vert \varphi^{2r}f^{(2r)}\bigr\Vert _{p}. $$
So we have
$$ \bigl\Vert \varphi^{2r+m}(x)D^{2r+m}M_{n}(f,x) \bigr\Vert _{p}^{E_{n}}\leq Cn^{\frac{m}{2}}\bigl\Vert \varphi^{2r}f^{(2r)}\bigr\Vert _{p}, \quad m\geq0. $$
(2.10)
Using (2.1) and (2.10), we have
$$\begin{aligned}& \Biggl\Vert \varphi^{2r+1}(x)D^{2r+1}\sum _{j=0}^{2r-1}\alpha ^{n}_{j}(x)M_{n,j}(f,x) \Biggr\Vert _{p}^{E_{n}} \\& \quad \leq \sum_{j=0}^{2r-1}\sum _{i=0}^{j}\biggl\Vert \varphi^{2r+1}(x){2r+1 \choose i}\bigl(D^{i}\alpha^{n}_{j}(x) \bigr)M_{n,2r+1+j-i}(f,x)\biggr\Vert _{p}^{E_{n}} \\& \quad \leq C\sum_{j=0}^{2r-1}\sum _{i=0}^{j}\bigl\Vert \varphi ^{2r+1}(x)n^{\frac{-j+i}{2}} \varphi^{j-i}(x)M_{n,2r+1+j-i}(f,x)\bigr\Vert _{p}^{E_{n}} \\& \quad \leq C\sqrt{n}\bigl\Vert \varphi^{2r}f^{(2r)}\bigr\Vert _{p}. \end{aligned}$$
(2.11)
Since \((\varphi^{2r+1}(x)D^{2r+1}M_{n}^{(2r-1)}(f,x) )^{2}\) are polynomials, we can use a result of the weight polynomial approximation [13], Theorem 8.4.8, translating the interval \([-1,1]\) to \([0,1]\) to obtain the estimate
$$ \bigl\Vert \bigl(\varphi^{2r+1}D^{2r+1} \bigl(M_{n}^{(2r-1)}f\bigr) \bigr)^{2}\bigr\Vert _{p}^{[0,1]}\leq M\bigl\Vert \bigl(\varphi^{2r+1}D^{2r+1} \bigl(M_{n}^{(2r-1)}f\bigr) \bigr)^{2}\bigr\Vert _{p}^{E_{n}}, $$
(2.12)
where M does not depend on n. From (2.11) and (2.12) we obtain (2.9). □

Lemma 2.5

((4.2) in [9])

For \(f\in L_{p}[0,1]\), \(1\leq p\leq\infty\), we have
$$ \bigl\Vert \varphi^{2r}D^{2r} \bigl(M_{n}^{(2r-1)}f\bigr)\bigr\Vert _{p}\leq Cn^{r}\|f\|_{p}. $$
(2.13)

Lemma 2.6

For \(f\in W^{2r+1}(\varphi)\), we have
$$\begin{aligned}& M_{n}^{(2r-1)}(f,x)-f(x)-\frac{(-1)^{r+1}\varphi ^{2r}(x)}{2^{r}n^{r}(r!)}f^{(2r)}(x) \\& \quad =o \biggl(\frac{1}{n^{r}} \biggr)\varphi ^{2r}(x)f^{(2r)}(x) + \biggl(b^{n}_{2r}\frac{1}{n^{2r}}+b^{n}_{2r-1} \frac{\varphi ^{2}(x)}{n^{2r-1}}+\cdots+b^{n}_{r+1}\frac{\varphi ^{2r-2}(x)}{n^{r+1}} \biggr) \\& \qquad {}\times f^{(2r)}(x) \biggl(1+O \biggl(\frac{1}{n} \biggr) \biggr) +M_{n}^{(2r-1)}\bigl(R_{2r+1}(f,\cdot,x),x \bigr), \end{aligned}$$
(2.14)
where \(\{b^{n}_{2r-1},\ldots, b^{n}_{r+1}\}\) are uniformly bounded in n and independent of x.

Proof

By Taylor’s formula we expand f as follows:
$$f(t)=f(x)+(t-x)f'(x)+\cdots+\frac{(t-x)^{2r}}{(2r)!}f^{(2r)}(x)+R_{2r+1}(f,t,x), $$
where \(R_{2r+1}(f,t,x)=\frac{1}{(2r)!}\int^{t}_{x}(t-u)^{2r}f^{(2r+1)}(u)\, du\).
Noting \(M_{n}^{(2r-1)}p=p\) for all \(p\in\Pi_{2r-1}\) (cf. [7]), we obtain
$$M_{n}^{(2r-1)}(f,x)-f(x)=M_{n}^{(2r-1)} \biggl( \frac {(t-x)^{2r}}{(2r)!},x \biggr)f^{(2r)}(x)+M_{n}^{(2r-1)} \bigl(R_{2r+1}(f,\cdot,x),x\bigr). $$
Using Lemma 2.3 we obtain (2.14). □

3 Main result

Using the lemmas in Section 2 we are able to prove the following main result, which is the strong converse inequality for left Bernstein-Durrmeyer quasi-interpolants of type B.

Theorem 3.1

Let \(f\in L_{p}[0,1]\), \(1< p\leq\infty\), \(\varphi(x)=\sqrt{x(1-x)}\), \(n\geq4r\), \(r\in N\), then there exists a constant k such that, for \(l\geq kn\),
$$\omega_{\varphi}^{2r}\biggl(f,\frac{1}{\sqrt{n}} \biggr)_{p}\leq C \biggl(\frac{l}{n} \biggr)^{r}\bigl( \bigl\Vert M_{n}^{(2r-1)}f-f\bigr\Vert _{p}+\bigl\Vert M_{l}^{(2r-1)}f-f\bigr\Vert _{p}\bigr). $$

Proof

To prove our result at first we estimate K-functional \(K^{2r}_{\varphi}(f,n^{-r})_{p}\). We choose the function
$$g=K_{n}^{(2r-1)}\bigl(K_{n}^{(2r-1)}f \bigr)=:K_{n}^{2(2r-1)}f. $$
By the definition of the K-functional and the boundedness of \(K_{n}^{(2r-1)}\) (cf. [7], p.243, (3.2) in [9]), we have
$$\begin{aligned} K_{\varphi}^{2r}\bigl(f,n^{-r}\bigr)_{p} \leq& \|f-g\|_{p}+n^{-r}\bigl\Vert \varphi^{2r}g^{(2r)} \bigr\Vert _{p} \\ =&\bigl\Vert f-M_{n}^{2(2r-1)}f\bigr\Vert _{p}+n^{-r}\bigl\Vert \varphi^{2r}D^{2r} \bigl(M_{n}^{2(2r-1)}f \bigr)\bigr\Vert _{p} \\ \leq&\bigl\Vert f-M_{n}^{(2r-1)}f\bigr\Vert _{p}+\bigl\Vert M_{n}^{(2r-1)}f-M_{n}^{2(2r-1)}f \bigr\Vert _{p} \\ &{}+n^{-r}\bigl\Vert \varphi^{2r}D^{2r} \bigl(M_{n}^{2(2r-1)}f \bigr)\bigr\Vert _{p} \\ \leq& C\bigl\Vert f-M_{n}^{(2r-1)}f\bigr\Vert _{p}+n^{-r}\bigl\Vert \varphi^{2r}D^{2r} \bigl(M_{n}^{2(2r-1)}f \bigr)\bigr\Vert _{p}. \end{aligned}$$
Therefore we only need to estimate \(\varphi^{2r}g^{(2r)}=\varphi^{2r}D^{2r}(M_{n}^{2(2r-1)}f)\). We recall Lemma 2.6 with \(g=M_{n}^{2(2r-1)}f\) in place of f and l in place of n to obtain
$$\begin{aligned}& M_{l}^{(2r-1)}(g,x)-g(x)-\frac{(-1)^{r+1}\varphi ^{2r}(x)}{2^{r}l^{r}(r!)}g^{(2r)}(x) \\& \quad =o \biggl(\frac{1}{l^{r}} \biggr)\varphi ^{2r}(x)g^{(2r)}(x)+ \biggl(b^{l}_{2r}\frac{1}{l^{2r}}+b^{l}_{2r-1} \frac{\varphi ^{2}(x)}{l^{2r-1}}+\cdots+b^{l}_{r+1}\frac{\varphi ^{2r-2}(x)}{l^{r+1}} \biggr) \\& \qquad {}\times g^{(2r)}(x) \biggl(1+O \biggl(\frac{1}{l} \biggr) \biggr) +M_{l}^{(2r-1)}\bigl(R_{2r+1}(g,\cdot,x),x \bigr). \end{aligned}$$
(3.1)
For \(x\in E_{n}\), \(n\varphi^{2}(x)\geq1\). So we have
$$ \begin{aligned} &\biggl\Vert \frac{1}{l^{2r}}g^{(2r)}(x) \biggr\Vert _{p}^{E^{n}}=\biggl\Vert \frac {n^{r}\varphi^{2r}(x)}{l^{2r}n^{r}\varphi^{2r}(x)}g^{(2r)}(x) \biggr\Vert _{p}^{E^{n}}\leq \frac{1}{l^{r}} \biggl( \frac{n}{l} \biggr)^{r}\bigl\Vert \varphi^{2r}g^{(2r)} \bigr\Vert _{p}, \\ &\biggl\Vert \frac{\varphi^{2}(x)}{l^{2r-1}}g^{(2r)}(x)\biggr\Vert _{p}^{E^{n}}= \biggl\Vert \frac{n^{r-1}\varphi^{2r}(x)}{l^{2r-1}n^{r-1}\varphi ^{2r-2}(x)}g^{(2r)}(x) \biggr\Vert _{p}^{E^{n}}\leq \frac{1}{l^{r}} \biggl( \frac{n}{l} \biggr)^{r-1}\bigl\Vert \varphi^{2r}g^{(2r)} \bigr\Vert _{p}, \\ &\ldots, \\ &\biggl\Vert \frac{\varphi^{2r-2}(x)}{l^{r+1}}g^{(2r)}(x)\biggr\Vert _{p}^{E^{n}}=\biggl\Vert \frac{n\varphi^{2r}(x)}{l^{r+1}n\varphi ^{2}(x)}g^{(2r)}(x) \biggr\Vert _{p}^{E^{n}}\leq \frac{1}{l^{r}} \biggl( \frac{n}{l} \biggr)\bigl\Vert \varphi^{2r}g^{(2r)}\bigr\Vert _{p}. \end{aligned} $$
(3.2)
By Lemma 2.2 we have
$$ \bigl\Vert M_{l}^{(2r-1)}\bigl(R_{2r+1}(g, \cdot,x),x\bigr)\bigr\Vert _{p}^{E_{n}}\leq Cl^{-r-\frac{1}{2}} \bigl\Vert \varphi^{2r+1}g^{(2r+1)}\bigr\Vert _{p}. $$
(3.3)
Combining (3.1)-(3.3), we obtain
$$\begin{aligned}& \frac{1}{2^{r}l^{r}(r!)}\bigl\Vert \varphi^{2r}g^{(2r)} \bigr\Vert _{p}^{E_{n}} \\& \quad \leq \bigl\Vert M_{l}^{(2r-1)}g-g\bigr\Vert _{p}+o \biggl( \frac{1}{l^{r}} \biggr)\bigl\Vert \varphi ^{2r}g^{(2r)} \bigr\Vert _{p} \\& \qquad {}+C\frac{1}{l^{r}} \biggl[ \biggl(\frac{n}{l} \biggr)^{r}+ \biggl(\frac {n}{l} \biggr)^{r-1}+\cdots+\frac{n}{l} \biggr] \biggl(1+O \biggl(\frac {1}{l} \biggr) \biggr)\bigl\Vert \varphi^{2r}g^{(2r)}\bigr\Vert _{p} \\& \qquad {}+Cl^{-r-\frac{1}{2}}\bigl\Vert \varphi^{2r+1}g^{(2r+1)}\bigr\Vert _{p}. \end{aligned}$$
(3.4)
Next we estimate the first term and the last term of the right side in (3.4). By the boundedness of \(M_{n}^{(2r-1)}f\) we have
$$\begin{aligned} \bigl\Vert M_{l}^{(2r-1)}g-g\bigr\Vert _{p} =&\bigl\Vert M_{l}^{(2r-1)} \bigl(M_{n}^{2(2r-1)}f\bigr)-M_{n}^{2(2r-1)}f\bigr\Vert _{p} \\ \leq&\bigl\Vert M_{l}^{(2r-1)} \bigl(M_{n}^{2(2r-1)}f-M_{n}^{(2r-1)}f \bigr) \bigr\Vert _{p}+\bigl\Vert M_{l}^{(2r-1)} \bigl(M_{n}^{(2r-1)}f-f \bigr)\bigr\Vert _{p} \\ &{} +\bigl\Vert M_{l}^{(2r-1)}f-f\bigr\Vert _{p}+\bigl\Vert f-M_{n}^{(2r-1)}f\bigr\Vert _{p}+\bigl\Vert M_{n}^{(2r-1)} \bigl(f-M_{n}^{(2r-1)}f \bigr)\bigr\Vert _{p} \\ \leq& C \bigl(\bigl\Vert M_{n}^{(2r-1)}f-f\bigr\Vert _{p}+\bigl\Vert M_{l}^{(2r-1)}f-f\bigr\Vert _{p} \bigr). \end{aligned}$$
(3.5)
Using (2.9) and (2.13), we obtain
$$\begin{aligned} \bigl\Vert \varphi^{2r+1}g^{(2r+1)}\bigr\Vert _{p} =&\bigl\Vert \varphi ^{2r+1}D^{2r+1} \bigl(M_{n}^{2(2r-1)}f\bigr)\bigr\Vert _{p} \\ \leq& C\sqrt{n}\bigl\Vert \varphi^{2r}D^{2r} \bigl(M_{n}^{(2r-1)}f\bigr)\bigr\Vert _{p} \\ \leq& C\sqrt{n} \bigl(\bigl\Vert \varphi^{2r}D^{2r} \bigl(M_{n}^{2(2r-1)}f\bigr)\bigr\Vert _{p}+ \bigl\Vert \varphi^{2r}D^{2r} \bigl(M_{n}^{(2r-1)} \bigl(M_{n}^{(2r-1)}f-f \bigr) \bigr)\bigr\Vert _{p} \bigr) \\ \leq& C\sqrt{n} \bigl(\bigl\Vert \varphi^{2r}g^{(2r)}\bigr\Vert _{p}+n^{r}\bigl\Vert M_{n}^{(2r-1)}f-f \bigr\Vert _{p} \bigr). \end{aligned}$$
(3.6)
Therefore with (3.4)-(3.6) we get
$$\begin{aligned}& \frac{1}{2^{r}l^{r}(r!)}\bigl\Vert \varphi^{2r}g^{(2r)} \bigr\Vert _{p}^{E_{n}} \\& \quad \leq C \bigl(\bigl\Vert M_{n}^{(2r-1)}f-f\bigr\Vert _{p}+\bigl\Vert M_{l}^{(2r-1)}f-f\bigr\Vert _{p} \bigr) \\& \qquad {}+C \biggl( \frac{n}{l} \biggr)^{r+\frac{1}{2}}\bigl\Vert M_{n}^{(2r-1)}f-f \bigr\Vert _{p}+Cl^{-r} \biggl(\frac{n}{l} \biggr)^{\frac{1}{2}}\bigl\Vert \varphi ^{2r}g^{(2r)}\bigr\Vert _{p} \\& \qquad {}+C \frac{1}{l^{r}} \biggl[ \biggl(\frac{n}{l} \biggr)^{r}+ \biggl(\frac{n}{l} \biggr)^{r-1}+\cdots+\frac{n}{l}+o(1) \biggr]\bigl\Vert \varphi ^{2r}g^{(2r)}\bigr\Vert _{p}. \end{aligned}$$
(3.7)
Since \(\varphi^{2r}g^{(2r)}=\varphi^{2r}D^{2r}(M_{n}^{2(2r-1)}f)\) are polynomials, for the same reason as (2.12) we have
$$ \bigl\Vert \varphi^{2r}g^{(2r)}\bigr\Vert _{p}^{[0,1]}\leq M\bigl\Vert \varphi^{2r}g^{(2r)} \bigr\Vert _{p}^{E_{n}}, $$
(3.8)
where M does not depend on n. Hence by (3.7) and (3.8) we obtain
$$\begin{aligned}& \frac{1}{2^{r}l^{r}(r!)}\bigl\Vert \varphi^{2r}g^{(2r)} \bigr\Vert _{p} \\& \quad \leq \frac {M}{2^{r}l^{r}(r!)}\bigl\Vert \varphi^{2r}g^{(2r)}\bigr\Vert _{p}^{E_{n}} \\& \quad \leq CM \bigl(\bigl\Vert M_{n}^{(2r-1)}f-f\bigr\Vert _{p}+\bigl\Vert M_{l}^{(2r-1)}f-f\bigr\Vert _{p} \bigr)+CM \biggl(\frac{n}{l} \biggr)^{r+\frac{1}{2}}\bigl\Vert M_{n}^{(2r-1)}f-f\bigr\Vert _{p} \\& \qquad {}+CM\frac{1}{l^{r}} \biggl[ \biggl(\frac{n}{l} \biggr)^{r}+ \biggl(\frac {n}{l} \biggr)^{r-1}+\cdots+ \frac{n}{l}+ \biggl(\frac{n}{l} \biggr)^{\frac {1}{2}}+o(1) \biggr] \bigl\Vert \varphi^{2r}g^{(2r)}\bigr\Vert _{p}. \end{aligned}$$
(3.9)
We now choose \(l\geq kn\) with k large enough such that
$$ CM\frac{1}{l^{r}} \biggl[ \biggl(\frac{n}{l} \biggr)^{r}+ \biggl(\frac {n}{l} \biggr)^{r-1}+\cdots+ \frac{n}{l}+ \biggl(\frac{n}{l} \biggr)^{\frac {1}{2}}+o(1) \biggr] \leq\frac{1}{2\cdot2^{r}l^{r}(r!)}. $$
(3.10)
By (3.9) and (3.10) we get
$$\frac{1}{2\cdot2^{r}l^{r}(r!)}\bigl\Vert \varphi^{2r}g^{(2r)}\bigr\Vert _{p} \leq C \bigl\{ \bigl\Vert M_{n}^{(2r-1)}f-f \bigr\Vert _{p}+\bigl\Vert M_{l}^{(2r-1)}f-f\bigr\Vert _{p} \bigr\} . $$
Therefore
$$\begin{aligned} K_{\varphi}^{2r}\bigl(f,n^{-r}\bigr)_{p}& \leq C\bigl\Vert f-M_{n}^{(2r-1)}f\bigr\Vert _{p}+n^{-r}\bigl\Vert \varphi^{2r}g^{(2r)} \bigr\Vert _{p} \\ &\leq C\bigl\Vert f-M_{n}^{(2r-1)}f\bigr\Vert _{p}+C \biggl(\frac{l}{n} \biggr)^{r} \bigl( \bigl\Vert M_{n}^{(2r-1)}f-f\bigr\Vert _{p}+\bigl\Vert M_{l}^{(2r-1)}f-f\bigr\Vert _{p} \bigr) \\ &\leq C \biggl(\frac{l}{n} \biggr)^{r} \bigl(\bigl\Vert M_{n}^{(2r-1)}f-f\bigr\Vert _{p}+ \bigl\Vert M_{l}^{(2r-1)}f-f\bigr\Vert _{p} \bigr). \end{aligned}$$
With (1.3) we obtain
$$\omega_{\varphi}^{2r}\biggl(f,\frac{1}{\sqrt{n}} \biggr)_{p}\leq C \biggl(\frac{l}{n} \biggr)^{r}\bigl( \bigl\Vert M_{n}^{(2r-1)}f-f\bigr\Vert _{p}+\bigl\Vert M_{l}^{(2r-1)}f-f\bigr\Vert _{p}\bigr). $$
The proof is complete. □

Declarations

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11371119 and 11401162) and by the Natural Science Foundation of Education Department of Hebei Province (Grant No. Z2014031).

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)
College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China
(2)
Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang, 050024, People’s Republic of China

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