Equivalent theorem of approximation by linear combination of weighted Baskakov–Kantorovich operators in Orlicz spaces

Abstract

In this paper, we introduce the Orlicz space corresponding to the Young function and, by virtue of the equivalent theorem between the modified K-functional and modulus of smoothness, establish the direct, inverse, and equivalent theorems for linear combination of the Jacobi weighted Baskakov–Kantorovich operators in the Orlicz spaces.

1 Motivations and main results

In recent years, since the Orlicz spaces are more general than the classical $L_{p}$ spaces, which are composed of measurable functions $f(x)$ such that $|f(x)|^{p}$ are integrable, there is growing interest in problems of approximation in Orlicz spaces.

For proceeding smoothly, we recall from [22] some definitions and related results.

A continuous convex function $\varPhi(t)$ on $[0,\infty)$ is called a Young function if

$$ \lim_{t\to0^{+}}\frac{\varPhi(t)}{t}=0\quad\text{and}\quad\lim _{t\to\infty}\frac{\varPhi(t)}{t}=\infty. $$

For a Young function $\varPhi(t)$, its complementary Young function is denoted by $\varPsi(t)$.

It is clear that the convexity of $\varPhi(t)$ leads to $\varPhi (\alpha t) \le\alpha\varPhi(t)$ for $\alpha\in[0,1]$. In particular, we have $\varPhi(\alpha t)<\alpha\varPhi(t)$ for $\alpha\in(0,1)$.

A Young function $\varPhi(t)$ is said to satisfy the $\Delta_{2}$-condition, denoted by $\varPhi\in\Delta_{2}$, if there exist $t_{0}>0$ and $C\ge1$ such that $\varPhi(2t)\le C\varPhi(t)$ for $t\ge t_{0}$.

Let $\varPhi(t)$ be a Young function. We define the Orlicz class $L_{\varPhi}[0,\infty)$ as the collection of all Lebesgue-measurable functions $u(x)$ on $[0,\infty)$. Since the integral

$$ \rho(u,\varPhi)= \int_{0}^{\infty}\varPhi\bigl( \bigl\vert u(x) \bigr\vert \bigr)\,\mathrm{d}x $$

is finite, we define the Orlicz space $L_{\varPhi}^{*}[0,\infty)$ as the linear hull of $L_{\varPhi}[0,\infty)$ under the Luxembourg norm

$$ \Vert u \Vert _{(\varPhi)}=\inf_{\lambda>0} \biggl\{ \lambda: \rho\biggl(\frac{u}{ \lambda},\varPhi\biggr)\le1 \biggr\} . $$

The Orlicz norm, which is equivalent to the Luxembourg norm on $L_{\varPhi}^{*}[0,\infty)$, is given by

$$ \Vert u \Vert _{\varPhi}= \sup_{\rho(v,\varPsi)\le1} \biggl\vert \int_{0}^{\infty}u(x)v(x) \,\mathrm{d}x \biggr\vert $$

and satisfies

$$ \Vert u \Vert _{(\varPhi)}\le \Vert u \Vert _{\varPhi}\le2 \Vert u \Vert _{(\varPhi)}. $$

(1.1)

Throughout this paper, we use C to denote a constant independent of n and x, which may be not necessarily the same in different cases.

Let $f\in L_{\varPhi}^{*}[0,\infty)$ and $r\in\mathbb{N}$. Then the weighted K-functional $K_{r,\varphi}(f,t^{r})_{w,\varPhi}$, the weighted modified K-functional $\bar{K}_{r,\varphi}(f,t^{r})_{w, \varPhi}$, and the weighted modulus of smoothness $\omega_{{r,\varphi }}(f,t)_{w,\varPhi}$ are given respectively by

$$\begin{aligned} &K_{r,\varphi}\bigl(f,t^{r}\bigr)_{w,\varPhi} =\inf _{g} \bigl\{ \bigl\Vert w(f-g) \bigr\Vert _{\varPhi}+t ^{r} \bigl\Vert w\varphi^{r}g^{(r)} \bigr\Vert _{\varPhi}:g^{(r-1)}\in AC_{ \mathrm{loc}} \bigr\} , \\ &\bar{K}_{r,\varphi}\bigl(f,t^{r}\bigr)_{w,\varPhi} =\inf _{g} \bigl\{ \bigl\Vert w(f-g) \bigr\Vert _{\varPhi} +t^{r} \bigl\Vert w\varphi^{r} g^{(r)} \bigr\Vert _{\varPhi}+t^{2r} \bigl\Vert wg^{(r)} \bigr\Vert _{\varPhi}: g^{(r-1)}\in AC_{\mathrm{loc}} \bigr\} , \end{aligned}$$

and

$$ \omega_{r,\varphi}(f,t)_{w,\varPhi}= \textstyle\begin{cases} \sup_{0< h\le t} \Vert w\Delta_{h\varphi}^{r}(f) \Vert _{\varPhi}, & a=0, \\ \sup_{0< h\le t} \Vert w\Delta_{h\varphi}^{r}(f) \Vert _{\varPhi[t^{*}, \infty)}+\sup_{0< h\le t^{*}} \Vert w\vec{\Delta}_{h}^{r}(f) \Vert _{\varPhi[0,12t^{*}]}, & a>0, \end{cases} $$

where

$$\begin{aligned}& \Delta_{h\varphi}^{r}(f,x)=\sum^{r}_{k=0}(-1)^{k} \binom{r}{k}f \biggl(x+r \biggl[\frac{h\varphi(x)}{2} \biggr]-kh\varphi (x) \biggr), \\& t^{*}=r^{2}t^{2}, \qquad\varphi(x)= \sqrt{x} ,\qquad\varphi(x)= \sqrt{x(1+x)} , \\& \varphi(x)=x, \qquad w(x)=x^{a}(1+x)^{b} \end{aligned}$$

for $a,b\in\mathbb{R}$ is the Jacobi weight function, and $g^{(r-1)} \in AC_{\mathrm{loc}}$ means that g is $r-1$ times differentiable and $g^{(r-1)}$ is absolutely continuous in every closed finite interval $[c,d]\subset[0,\infty)$.

Between the weighted modulus of smoothness and the weighted modified K̄-functional, there are the following equivalent theorems.

Theorem A

([13])

Let $f\in L_{\varPhi}^{*}[0,\infty)$ and $r\in\mathbb{N}$. Then there exist constants C and $t_{0}$ such that

$$ \frac{\omega_{{r,\varphi}}(f,t)_{w,\varPhi}}{C}\le \bar{K}_{r,\varphi } \bigl(f,t^{r}\bigr)_{w,\varPhi}\le C\omega_{{r,\varphi }}(f,t)_{w,\varPhi} $$

(1.2)

for $0< t\le t_{0}$.

Theorem B

([12])

Let $f\in L_{\varPhi}^{*}[0,\infty)$ and $r\in\mathbb{N}$. Then there exist constants C and $t_{0}$ such that

$$ \frac{\omega_{{r,\varphi}}(f,t)_{w,\varPhi }}{C}\le K_{r,\varphi}\bigl(f,t ^{r}\bigr)_{w,\varPhi}\le C\omega_{{r,\varphi}}(f,t)_{w,\varPhi} $$

(1.3)

for $0< t\le t_{0}$.

For $f\in C([0,\infty))$, the classical Baskakov operators are defined in [3] as

$$ V_{n}(f;x)=\sum_{k=0}^{\infty}f \biggl(\frac{k}{n} \biggr)v_{n,k}(x), $$

where $v_{n,k}(x)=\binom{n+k-1 }{k}\frac{x^{k}}{(1+x)^{n+k}}$, $k, n\in\mathbb{N}$. To approximate functions in the $L_{p}$-norm, Ditzian and Totik [5] defined the Kantorovich modifications

$$ \tilde{V}_{n}(f;x)=(n-1)\sum _{k=0}^{\infty}v_{n,k}(x) \int_{k/(n-1)} ^{(k+1)/(n-1)}f(u)\,\mathrm{d}u $$

(1.4)

and obtained the direct inequality

$$ \Vert \tilde{V}_{n}f-f \Vert _{p}\le C \biggl[ \omega_{2,\varphi} \biggl(f,\frac{1}{n^{1/2}} \biggr)_{p}+ \frac{1}{n} \Vert f \Vert _{p} \biggr] $$

and the week converse inequality

$$ \Vert \tilde{V}_{n}f-f \Vert _{p}=O \biggl( \frac{1}{n^{\alpha/2}} \biggr)\quad \Longleftrightarrow\quad \omega_{2,\varphi }(f,h)_{p}=O \bigl(h^{\alpha}\bigr) $$

for $\alpha<2$, where $f\in L_{p}[0,\infty)$, $1\le p<\infty$, and $\varphi(x)=\sqrt{x(1+x)}$.

There are many approximation results on operators of the Baskakov type in the space $C[0,\infty)$ or $L_{p}[0,\infty)$. See [1,2,3,4,5,6,7,8,9,10,11,12, 14,15,16,17,18,19,20,21, 23, 24, 28, 29] and closely related references therein. Gupta and Acu [10] discussed a uniform estimate and established a quantitative result for the modified Baskakov–Szász–Mirakyan operators. Kumar and Acar [14] introduced a modification of generalized Baskakov–Durrmeyer operators of the Stancu type and studied their approximation properties. Goyal and Agrawal [8] introduced the Bézier variant of the generalized Baskakov–Kantorovich operators, established a direct approximation theorem with the aid of the Ditzian–Totik modulus of smoothness, and studied the rate of convergence for the functions having the derivatives of bounded variation for these operators. Zhang and Zhu [29] studied preservation properties, such as monotonicity, convexity, and smoothness, as well as those under the average, of the Baskakov–Kantorovich operators. Gadjev [7] studied the approximation of functions by the Baskakov–Kantorovich operator in the space $L_{p}[0,\infty)$ and obtained the double inequality

$$ \frac{1}{C} \Vert \tilde{V}_{n}f-f \Vert _{p}\le\tilde{K} \biggl(f, \frac{1}{n} \biggr)_{p}\le C\frac{\ell}{n} \bigl( \Vert \tilde{V} _{n}f-f \Vert _{p}+ \Vert \tilde{V}_{\ell}f-f \Vert _{p} \bigr) $$

for $\ell\in\mathbb{N}$ with $\ell\ge C_{1}n$, where $C_{1}$ is a positive constant, and

$$\begin{aligned} \begin{aligned} \tilde{K}(f,t)_{p} ={}&\inf\biggl\{ \Vert f-g \Vert _{p}+t \Vert \tilde{D}g \Vert _{p}: f-g\in L_{p}[0,\infty), \\ &{}g\in\tilde{W}_{p}[0,\infty), \tilde{D}= \frac{\mathrm{d}}{\mathrm{d}x} \biggl[\varphi^{2}(x)\frac{ \mathrm{d}}{\mathrm{d}x} \biggr] \biggr\} \end{aligned} \end{aligned}$$

is a K-functional.

For $n, r\in\mathbb{N}$ such that $n\geq2r$, the linear combinations of the Baskakov–Kantorovich operator are defined as

$$ L_{n,r}(f;x)=\sum_{i=0}^{2r-1}c_{i}(n,r) \tilde{V}_{n_{i}}(f;x), $$

(1.5)

where the coefficients $c_{i}(n,r)$ only dependent of n, r and satisfy the following conditions:

$$\begin{aligned}& n\le n_{0}\le n_{1}< \cdots< n_{2r-1}\le C_{n}, \quad\sum_{i=0}^{2r-2}c _{i}(n,r)=1, \\& \sum_{i=0}^{2r-2} \bigl\vert c_{i}(n,r) \bigr\vert \leq C, \end{aligned}$$

(1.6)

$$\begin{aligned}& \sum_{i=0}^{2r-1}c_{i}(n,r) \tilde{V}_{n_{i}} \bigl((t-x)^{k}; x \bigr)=0,\quad k=1,2, \ldots, 2r-1. \end{aligned}$$

(1.7)

There are few results on the linear combinations of the Baskakov–Kantorovich operators. In [11], we obtained approximation properties for linear combinations of modified summation operators of integral type in the Orlicz space. Basing on these conclusions, we discover in this paper approximation properties for linear combinations of the Baskakov–Kantorovich operators.

Our main results in this paper can be stated as the following three theorems.

Theorem 1.1

(Direct theorem)

Let $f\in L_{\varPhi}^{*}[0,\infty)$, $\varPsi\in\Delta_{2}$, $\varphi(x)=\sqrt{x(1+x)}$, $n,a,b\in\mathbb{N}$, and $0\le a,b< n-1$. Then

$$ \bigl\Vert w\bigl(L_{n,r}(f)-f\bigr) \bigr\Vert _{\varPhi} \le C\omega_{2r,\varphi} \biggl(f,\frac{1}{n ^{1/2}} \biggr)_{w,\varPhi}. $$

Theorem 1.2

(Inverse theorem)

Let $f\in L_{\varPhi}^{*}[0,\infty)$, $n\ge2r$, $\varphi(x)= \sqrt{x(1+x)}$, $a,b\in\mathbb{N}$, and $0\le a,b< n-1$. Then

$$ \omega_{2r,\varphi} \biggl(f,\frac{1}{n^{r/2}} \biggr)_{w,\varPhi} \le\frac{C}{n^{r}}\sum_{k=1}^{n}k^{r-1} \bigl\Vert w\bigl(L_{k,r}(f)-f\bigr) \bigr\Vert _{\varPhi}. $$

Theorem 1.3

(Equivalence theorem)

Let $f\in L_{\varPhi}^{*}[0,\infty)$, $n\ge2r$, $\varphi(x)= \sqrt{x(1+x)}$, $\varPsi\in\Delta_{2}$, $a,b\in\mathbb{N}$, and $0\le a,b< n-1$. Then

$$\begin{aligned}& \begin{gathered} \bigl\Vert w\bigl(L_{n,r}(f)-f\bigr) \bigr\Vert _{\varPhi}=O \biggl(\psi\biggl(\frac{1}{n^{1/2}} \biggr) \biggr), \quad n\to\infty \\ \quad \Longleftrightarrow\quad\omega_{2r,\varphi}(f,t)_{w,\varPhi }=O\bigl( \psi(t)\bigr), \quad t\to0^{+}. \end{gathered} \end{aligned}$$

These main results are stronger than the results mentioned before.

2 Proof of the direct theorem

To prove the direct theorem, we need several lemmas.

Lemma 2.1

The operators $\tilde{V}_{n}(f;x)$ defined in (1.4) satisfy

$$ \tilde{V}_{n}(1;x)=1 \quad\textit{and}\quad\tilde{V}_{n} \bigl((t-x)^{2r};x \bigr)\le \frac{C\delta_{n}^{2r}(x)}{n^{r}}, $$

where $\delta_{n}^{2r}(x)=\max \{\varphi^{2r}(x), \frac{1}{n^{r}} \}$, $\varphi(x)=\sqrt{x(1+x)}$, $r\in \mathbb{N}$, and C is a positive constant.

Proof

This follows from simple calculation. □

Lemma 2.2

([5])

If t locates between x and u, then

$$ \frac{(x-t)^{2r-1}}{\varphi^{2r}(t)}\le\frac{ \vert x-u \vert ^{2r-1}}{\varphi ^{2r-2}(x)}\frac{1}{x} \biggl( \frac{1}{1+x}+\frac{1}{1+u} \biggr). $$

Lemma 2.3

([5])

For $w(x)=x^{a}(1+x)^{b}$ and $a,b\in\mathbb{R}$, we have

$$ \frac{w(x)}{w(u)}\le2^{|b|} \biggl[ \biggl(\frac{x}{u} \biggr)^{a}+ \biggl(\frac{x}{u} \biggr)^{b} \biggr]. $$

Lemma 2.4

Let $f\in L_{\varPhi}^{*}[0,\infty)$, $w(x)=x^{a}(1+x)^{b}$, $a,b\in\mathbb{N}$, and $0\le a,b< n-1$. Then

$$ \bigl\Vert wL_{n,r}(f) \bigr\Vert _{\varPhi}\le C \Vert wf \Vert _{\varPhi}. $$

Proof

By Lemma 2.3 we can write

$$\begin{aligned} \bigl\vert w(x)\tilde{V}_{n}(f;x) \bigr\vert ={}& \Biggl\vert \sum_{k=0}^{\infty}v_{n,k}(x)w(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)}f(u)\,\mathrm{d}u \Biggr\vert \\ \le{}&\sum_{k=0}^{\infty}v_{n,k}(x) (n-1)2^{ \vert b \vert+1} \biggl[ \biggl( \frac{x(n-1)}{k+1} \biggr)^{a} \\ &{}+ \biggl(\frac{x(n-1)}{k+1} \biggr)^{b} \biggr] \int_{k/(n-1)} ^{(k+1)/(n-1)}w(u) \bigl\vert f(u) \bigr\vert \,\mathrm{d}u \\ \triangleq {}&I_{1}+I_{2}. \end{aligned}$$

Using (1.1) and Jensen’s inequality gives

$$\begin{aligned} \Vert I_{1} \Vert _{\varPhi}\le{}&2 \Vert I_{1} \Vert _{(\varPhi)} \\ ={}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty} \varPhi\Biggl( \Biggl\vert \sum _{k=0}^{\infty}v_{n,k}(x) (n-1)2^{ \vert b \vert+1} \biggl[ \frac{x(n-1)}{k+1} \biggr]^{a} \\ &{}\times \int_{k/(n-1)}^{(k+1)/(n-1)}\frac{w(u)f(u)}{\lambda} \,\mathrm{d}u \Biggr\vert \Biggr)\,\mathrm{d}x\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty}\varPhi\Biggl(\sum _{k=0}^{\infty}v_{n-a,k+a}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)}\frac{Cw(u) \vert f(u) \vert}{ \lambda}\,\mathrm{d}u \Biggr) \,\mathrm{d}x\le1 \Biggr\} \\ ={}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty}\varPhi\Biggl( \sum _{k=a}^{\infty}v_{n-a,k}(x) (n-1) \int_{(k-a)/(n-1)}^{(k+1-a)/(n-1)}\frac{Cw(u) \vert f(u) \vert}{ \lambda}\,\mathrm{d}u \Biggr) \,\mathrm{d}x\le1 \Biggr\} \\ ={}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty}\varPhi\Biggl( \sum _{k=a}^{\infty}v_{n-a,k}(x) (n-1) \int_{(k-a)/(n-1)}^{(k+1-a)/(n-1)}\frac{Cw(u) \vert f(u) \vert}{ \lambda}\,\mathrm{d}u \\ &{}+\sum_{k=0}^{a-1}v_{n-a,k}(x) (n-1) \int_{\max \{0,\frac{k-a}{n-1} \}}^{\max \{0, \frac{k+1-a}{n-1} \}} \frac{Cw(u) \vert f(u) \vert}{\lambda}\, \mathrm{d}u \Biggr) \,\mathrm{d}x\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty}\sum_{k=0} ^{\infty}v_{n-a,k}(x)\varPhi\biggl((n-1) \int_{\max \{0,\frac{k-a}{n-1} \}}^{\max \{0, \frac{k+1-a}{n-1} \}}\frac{Cw(u) \vert f(u) \vert}{\lambda}\, \mathrm{d}u \biggr) \,\mathrm{d}x\le1 \Biggr\} \\ ={}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty}\sum_{k=a} ^{\infty}v_{n-a,k}(x)\varPhi\biggl((n-1) \int_{(k-a)/(n-1)}^{(k+1-a)/(n-1)}\frac{Cw(u) \vert f(u) \vert}{ \lambda}\,\mathrm{d}u \biggr) \,\mathrm{d}x\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty}\sum_{k=a} ^{\infty}v_{n-a,k}(x) (n-1) \int_{(k-a)/(n-1)}^{(k+1-a)/(n-1)}\varPhi\biggl( \frac{Cw(u) \vert f(u) \vert}{\lambda} \biggr)\,\mathrm{d}u\, \mathrm{d}x\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda:\sum _{k=0}^{\infty} \frac{n-1}{n-a-1} \int_{k/(n-1)}^{(k+1)/(n-1)}\varPhi\biggl( \frac{Cw(u) \vert f(u) \vert}{ \lambda} \biggr)\,\mathrm{d}u\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \biggl\{ \lambda: \int_{0}^{\infty}\varPhi\biggl( \frac{Cw(u) \vert f(u) \vert}{\lambda} \biggr)\,\mathrm{d}u\le1 \biggr\} \\ ={}&C \Vert wf \Vert _{(\varPhi)}\le C \Vert wf \Vert _{\varPhi}. \end{aligned}$$

Similarly, we have

$$ \Vert I_{2} \Vert _{\varPhi}\le C \Vert wf \Vert _{\varPhi}. $$

Consequently, we arrive at

$$\begin{aligned} \bigl\Vert w(x)\tilde{V}_{n}(f;x) \bigr\Vert _{\varPhi}\le C \Vert wf \Vert _{\varPhi}. \end{aligned}$$

(2.1)

Combining (1.5), (1.6), and (2.1), it follows that

$$\begin{aligned} \begin{aligned} \bigl\Vert wL_{n,r}(f) \bigr\Vert _{\varPhi}&= \Biggl\Vert \sum_{i=0}^{2r-1}c_{i}(n,r)w \tilde{V}_{n_{i}}(f) \Biggr\Vert _{\varPhi}\le\sum _{i=0}^{2r-1} \bigl\Vert c_{i}(n,r)w \tilde{V}_{n_{i}}(f) \bigr\Vert _{\varPhi} \\ &\le\sum_{i=0}^{2r-1} \bigl\vert c_{i}(n,r) \bigr\vert \bigl\Vert w\tilde{V}_{n_{i}}(f) \bigr\Vert _{\varPhi}\le C\sum_{i=0}^{2r-1} \bigl\vert c_{i}(n,r) \bigr\vert \Vert wf \Vert _{\varPhi} \le C \Vert wf \Vert _{\varPhi}. \end{aligned} \end{aligned}$$

The proof of Lemma 2.4 is complete. □

Lemma 2.5

([13])

For $f\in L_{\varPhi}^{*}[0,\infty)$ and $\varPsi\in\Delta_{2}$, we have

$$ \bigl\Vert \theta(f) \bigr\Vert _{\varPhi}\le C \Vert f \Vert _{\varPhi}, $$

where

$$ \theta(f;x)=\sup_{\substack{0\le t< \infty \\ t\ne x}}\frac{1}{t-x} \int_{x}^{t}f(u)\,\mathrm{d}u $$

is the Hardy–Littlewood function of $f(x)$.

We now in a position to prove Theorem 1.1.

Proof of Theorem 1.1

Let

$$ U=W_{\varPhi}^{2r} \bigl\{ g: g^{(2r-1)}\in AC_{\mathrm{loc}}, \varphi^{2r}g^{(2r)} \in L_{\varPhi}^{*}[0,\infty) \bigr\} . $$

Taylor’s formula with integral remainder of $g\in U$ gives

$$ g(u)=\sum_{i=0}^{2r-1} \frac{1}{i!}(u-x)^{i}g^{(i)}(x)+R_{2r}(g;u,x), $$

where

$$ R_{2r}(g;u,x)=\frac{1}{(2r-1)!} \int_{x}^{u}(u-t)^{2r-1}g^{(2r)}(t) \,\mathrm{d}t,\quad x\in[0,\infty). $$

From (1.7) it follows that

$$ w(x)\bigl[L_{n,r}(g;x)-g(x)\bigr]=w(x)L_{n,r} \bigl(R_{2r}(g;u,x);x\bigr) $$

and

$$ \bigl\Vert w\bigl(L_{n,r}(g)-g\bigr) \bigr\Vert _{\varPhi}= \bigl\Vert wL_{n,r}\bigl(R_{2r}(g; \cdot,x),x\bigr) \bigr\Vert _{ \varPhi}. $$

(2.2)

We now estimate $|w(x)\tilde{V}_{n}(R_{2r}(g;u,x);x) |$. As $x\in[\frac{1}{n},\infty)$, we have $\delta_{n}^{2r}(x)=\varphi^{2r}(x)$. Applying Lemma 2.2 leads to

$$\begin{aligned}& \begin{gathered} \bigl\vert w(x)\tilde{V}_{n} \bigl(R_{2r}(g;u,x);x\bigr) \bigr\vert \\ \quad = \Biggl\vert w(x)\sum _{k=0} ^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{1}{(2r-1)!} \\ \qquad{}\times \int_{x}^{u}(u-t)^{2r-1}g^{(2r)}(t) \,\mathrm{d}t\,\mathrm{d}u \Biggr\vert \\ \quad \le w(x)\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{1}{(2r-1)!} \frac{(u-x)^{2r}}{\varphi^{2r-2}(x)} \\ \qquad{}\times\biggl[\frac{1}{x(1+x)}+\frac{1}{x(1+u)} \biggr] \biggl[ \frac{1}{w(u)}+ \frac{1}{w(x)} \biggr]\,\mathrm{d}u \bigl\vert \theta \bigl(w\delta_{n}^{2r}g^{(2r)};x\bigr) \bigr\vert \\ \quad \triangleq(J_{1}+J_{2}+J_{3}+J_{4}) \bigl\vert \theta\bigl(w\delta_{n}^{2r}g^{(2r)};x \bigr) \bigr\vert . \end{gathered} \end{aligned}$$

From Cauchy’s integral inequality [25, 26] and Lemma 2.1 it follows that

$$\begin{aligned} J_{1} ={}&\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{1}{(2r-1)!} \frac{(u-x)^{2r}}{\varphi^{2r}(x)}\frac{w(x)}{w(u)} \,\mathrm{d}u \\ \le{}&\frac{1}{(2r-1)!}\frac{1}{\varphi^{2r}(x)} \Biggl[\sum _{k=0}^{ \infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{w^{2}(x)}{w ^{2}(u)}\,\mathrm{d}u \Biggr]^{1/2} \\ &{}\times\Biggl[\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)} ^{(k+1)/(n-1)}(u-x)^{4r}\,\mathrm{d}u \Biggr]^{1/2} \\ \le{}&\frac{1}{(2r-1)!}\frac{1}{\varphi^{2r}(x)}\frac{C\delta_{n}^{2r}(x)}{n ^{r}} = \frac{C}{n^{r}}, \end{aligned}$$

where

$$ \sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)}\frac{w ^{2}(x)}{w^{2}(u)}\,\mathrm{d}u\le C. $$

Similarly, we can also obtain

$$\begin{aligned} J_{2}&=\frac{1}{(2r-1)!}\frac{w(x)}{\varphi^{2r}(x)} \sum _{k=0}^{ \infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{(u-x)^{2r}}{w(x)}\,\mathrm{d}u \\ &=\frac{1}{(2r-1)!}\frac{1}{\varphi^{2r}(x)}\tilde{V}_{n} \bigl((u-x)^{2r};x \bigr)\le\frac{C}{n^{r}}. \end{aligned}$$

Now we estimate $J_{3}$. By Cauchy’s integral inequality [25, 26] and Lemma 2.1 we derive that

$$\begin{aligned} \begin{aligned} J_{3} ={}&\frac{w(x)}{(2r-1)!\varphi^{2r-2}(x)x}\sum_{k=0}^{\infty}v _{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{(u-x)^{2r}}{(1+u)w(u)}\,\mathrm{d}u \\ ={}&\frac{w(x)(1+x)}{(2r-1)!\varphi^{2r}(x)}\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{(u-x)^{2r}}{(1+u)w(u)}\,\mathrm{d}u \\ ={}&\frac{w_{1}(x)}{(2r-1)!\varphi^{2r}(x)}\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{(u-x)^{2r}}{w_{1}(u)}\,\mathrm{d}u \\ \le{}&\frac{1}{\varphi^{2r}(x)} \Biggl[\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{w_{1}^{2}(x)}{w_{1}^{2}(u)}\, \mathrm{d}u \Biggr]^{1/2} \\ &{}\times\Biggl[\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)} ^{(k+1)/(n-1)}(u-x)^{4r}\,\mathrm{d}u \Biggr]^{1/2} \\ \le{}&\frac{C}{n^{r}}, \end{aligned} \end{aligned}$$

where $w_{1}(x)=x^{a}(1+x)^{b+1}$.

Finally, we estimate $J_{4}$. Applying Cauchy’s integral inequality [25, 26] and Lemma 2.1 yields

$$\begin{aligned} J_{4} ={}&\frac{1}{(2r-1)!}\frac{w(x)}{\varphi^{2r-2}(x)}\sum _{k=0}^{ \infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{(u-x)^{2r}}{w(x)} \frac{1}{x(1+u)}\,\mathrm{d}u \\ \le{}&\frac{1}{x\varphi^{2r-2}(x)} \Biggl[\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)}(u-x)^{4r}\,\mathrm{d}u \Biggr]^{1/2} \\ &{}\times\Biggl[\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)} ^{(k+1)/(n-1)}\frac{\mathrm{d}u}{(1+u)^{2}} \Biggr]^{1/2} \\ \le{}&\frac{1}{x\varphi^{2r-2}(x)}\frac{C\delta _{n}^{2r}(x)}{n^{r}}\frac{ \sqrt{2} }{1+x} \le \frac{C}{n^{r}}, \end{aligned}$$

where

$$ \sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)}\frac{ \,\mathrm{d}u}{(1+u)^{2}}\le \frac{2}{(1+x)^{2}}. $$

From the previous inequalities and Lemma 2.5 it follows that

$$ \begin{aligned}[b] \bigl\Vert w\tilde{V}_{n}\bigl(R_{2r}(g;\cdot,x);x\bigr) \bigr\Vert _{\varPhi[ \frac{1}{n},\infty)} &\le\frac{C}{n^{r}} \bigl\Vert \theta\bigl(w\delta _{n} ^{2r}g^{(2r)}\bigr) \bigr\Vert _{\varPhi[\frac{1}{n},\infty)} \\ &\le\frac{C}{n ^{r}} \bigl\Vert w\delta_{n}^{2r}g^{(2r)} \bigr\Vert _{\varPhi[\frac{1}{n}, \infty)}. \end{aligned} $$

(2.3)

For $x\in [0,\frac{1}{n} )$, we have $\delta_{n}^{2r}(x)=\frac{1}{n ^{r}}$. Accordingly,

$$\begin{aligned} \begin{aligned} &\bigl\vert w(x)\tilde{V}_{n}\bigl(R_{2r}(g;u,x);x \bigr) \bigr\vert \\ &\quad \le\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{w(x)n ^{r}}{(2r-1)!}(u-x)^{2r} \biggl[\frac{1}{w(u)} + \frac{1}{w(x)} \biggr]\,\mathrm{d}u \bigl\vert \theta \bigl(w\delta _{n}^{2r}g^{(2r)},x\bigr) \bigr\vert \\ &\quad \triangleq(E_{1}+E_{2}) \bigl\vert \theta\bigl(w \delta_{n}^{2r}g^{(2r)};x\bigr) \bigr\vert . \end{aligned} \end{aligned}$$

Using Cauchy’s integral inequality [25, 26] and Lemma 2.1 yields

$$\begin{aligned} E_{1} ={}&\frac{w(x)n^{r}}{(2r-1)!}\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{(u-x)^{2r}}{w(u)}\,\mathrm{d}u \\ \le{}&\frac{n^{r}}{(2r-1)!} \Biggl[\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)}(u-x)^{4r}\,\mathrm{d}u \Biggr]^{1/2} \\ &{}\times\Biggl[\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)} ^{(k+1)/(n-1)}\frac{w^{2}(x)}{w^{2}(u)}\,\mathrm{d}u \Biggr]^{1/2} \\ \le{}&\frac{n^{r}}{(2r-1)!}\frac{C}{n^{r}}\delta_{n}^{2r}(x) =\frac{C}{n ^{r}} \end{aligned}$$

and

$$\begin{aligned} E_{2}&=\frac{w(x)n^{r}}{(2r-1)!}\sum_{k=0}^{\infty}v_{n,k}(x) (n-1) \int_{k/(n-1)}^{(k+1)/(n-1)} \frac{(u-x)^{2r}}{w(x)}\,\mathrm{d}u \\ &=\frac{n^{r}}{(2r-1)!}\tilde{V}_{n} \bigl((u-x)^{2r};x \bigr)\le\frac{C}{n ^{r}}. \end{aligned}$$

Therefore we have

$$ \bigl\vert w(x)\tilde{V}_{n}\bigl(R_{2r}(g;u,x);x \bigr) \bigr\vert \le\frac{C}{n^{r}} \bigl\vert \theta\bigl(w\delta _{n}^{2r}g^{(2r)};x\bigr) \bigr\vert $$

and

$$ \begin{aligned}[b] \bigl\Vert w\tilde{V}_{n}\bigl(R_{2r}(g;\cdot,x);x\bigr) \bigr\Vert _{\varPhi[0, \frac{1}{n})} &\le\frac{C}{n^{r}} \bigl\Vert \theta\bigl(w\delta _{n} ^{2r}g^{(2r)} \bigr) \bigr\Vert _{\varPhi[0,\frac{1}{n})} \\ &\le\frac{C}{n ^{r}} \bigl\Vert w\delta_{n}^{2r}g^{(2r)} \bigr\Vert _{\varPhi[0,\frac{1}{n})}. \end{aligned} $$

(2.4)

Hence, by virtue of (2.3) and (2.4), we derive

$$ \bigl\Vert w\tilde{V}_{n}\bigl(R_{2r}(g;\cdot,x);x \bigr) \bigr\Vert _{\varPhi[0,\infty )} \le\frac{C}{n^{r}} \bigl\Vert w \delta_{n}^{2r}g^{(2r)} \bigr\Vert _{ \varPhi[0,\infty)}, $$

and, consequently,

$$\begin{aligned}& \begin{aligned} \bigl\Vert w L_{n,r}\bigl(R_{2r}(g;\cdot,x);x\bigr) \bigr\Vert _{\varPhi}&\le\sum_{i=0} ^{2r-1} \bigl\Vert c_{i}(n,r)w\tilde{V}_{n_{i}}\bigl(R_{2r}(g;\cdot,x);x\bigr) \bigr\Vert _{\varPhi} \\ &\le\frac{C}{n^{r}}\sum_{i=0}^{2r-1} \bigl\vert c_{i}(n,r) \bigr\vert \bigl\Vert w\delta _{n} ^{2r}g^{(2r)} \bigr\Vert _{\varPhi}\le\frac{C}{n^{r}} \bigl\Vert w\delta_{n} ^{2r}g^{(2r)} \bigr\Vert _{\varPhi}. \end{aligned} \end{aligned}$$

Combining this inequality with (1.2), (2.2), and Lemma 2.4 results in

$$\begin{aligned} \bigl\Vert w\bigl(L_{n,r}(f)-f\bigr) \bigr\Vert _{\varPhi} &\le\bigl\Vert w \bigl(L_{n,r}(f-g)-(f-g)\bigr) \bigr\Vert _{\varPhi}+ \bigl\Vert w\bigl(L _{n,r}(g)-g\bigr) \bigr\Vert _{\varPhi} \\ &\le C \bigl\Vert w(f-g) \bigr\Vert _{\varPhi}+\frac{C}{n^{r}} \bigl\Vert w\delta_{n}^{2r}g^{(2r)} \bigr\Vert _{\varPhi}\le C\omega_{2r,\varphi} \biggl(f, \frac{1}{\sqrt{n} } \biggr)_{w,\varPhi}. \end{aligned}$$

The proof of the direct theorem is complete. □

3 Proofs of the inverse and equivalence theorems

We first prepare several lemmas for proving Theorems 1.2 and 1.3.

Lemma 3.1

If $f\in L_{\varPhi}^{*}[0,\infty)$, $n\ge2r$, $a,b\in\mathbb{N}$, and $0\le a,b< n-1$, then

$$ \bigl\Vert w\varphi^{2r}L_{n,r}^{(2r)}(f) \bigr\Vert _{\varPhi}\le Cn^{r} \Vert wf \Vert _{\varPhi}. $$

Proof

From [7, Eq. (16)] it follows that

$$ \tilde{V}_{n}^{(2r)}(f;x)=\frac{(n+2r-1)!}{(n-1)!}\sum _{k=0}^{\infty }\Delta^{2r}a_{k}(n-1) v_{n+2r, k}(x), $$

where

$$ a_{k}(n-1)=(n-1) \int_{k/(n-1)}^{(k+1)/(n-1)}f(u)\,\mathrm{d}u, \qquad\Delta a _{k}=a_{k+1}-a_{k},\qquad\Delta ^{m} a_{k}=\Delta\bigl(\Delta^{m-1}a _{k} \bigr). $$

By Lemma 2.3 it follows that

$$\begin{aligned} \begin{aligned}[b] &w(x)\varphi^{2r}(x) \tilde{V}_{n}^{(2r)}(f;x) \\ &\quad = \frac{(n+2r-1)!}{(n-1)!}\sum _{k=0}^{\infty}\Delta^{2r}a_{k}(n-1)v _{n+2r,k}(x)x^{r}(1+x)^{r}w(x) \\ &\quad \le Cn^{r}\sum_{k=0}^{\infty} \Biggl\vert \sum_{i=0}^{2r} \binom{2r}{i}(-1)^{i}a_{k+(2r-i)}(n-1) \Biggr\vert v _{n,k+r}(x)w(x) \\ &\quad \le Cn^{r}\sum _{i=0}^{2r}\binom{2r}{i}\sum _{k=0}^{\infty}v_{n,k+r}(x)w(x) (n-1) \int_{(k+2r-i)/(n-1)}^{(k+1+2r-i)/(n-1)} \frac{w(u) \vert f(u) \vert }{w(u)} \,\mathrm{d}u \\ &\quad \le Cn^{r}\sum_{i=0}^{2r} \binom{2r}{i}\sum_{k=0}^{\infty}v _{n,k+r}(x)2^{ \vert b \vert+1} \biggl[ \biggl(\frac{(n-1)x}{k+2r-i+1} \biggr)^{a} \\ &\qquad{}+ \biggl(\frac{(n-1)x}{k+2r-i+1} \biggr)^{b} \biggr](n-1) \int_{(k+2r-i)/(n-1)}^{(k+1+2r-i)/(n-1)}w(u) \bigl\vert f(u) \bigr\vert \,\mathrm{d}u \\ &\quad \triangleq\sum_{i=0}^{2r} \binom{2r}{i}(F_{1}+F_{2}), \end{aligned} \end{aligned}$$

(3.1)

where

$$\begin{aligned}& \begin{aligned} F_{1} &=\sum_{k=0}^{\infty}v_{n,k+r}(x) \biggl[\frac{(n-1)x}{k+2r-i+1} \biggr]^{a} (n-1) \int_{(k+2r-i)/(n-1)}^{(k+1+2r-i)/(n-1)}Cn^{r}w(u) \bigl\vert f(u) \bigr\vert \,\mathrm{d}u \\ &\le\sum_{k=0}^{\infty}v_{n-a,k+a+r}(x) (n-1) \int_{(k+2r-i)/(n-1)} ^{(k+1+2r-i)/(n-1)}Cn^{r}w(u) \bigl\vert f(u) \bigr\vert \,\mathrm{d}u, \end{aligned} \\& \begin{aligned} \Vert F_{1} \Vert _{\varPhi}\le{}&2 \Vert F_{1} \Vert _{(\varPhi)} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty} \varPhi\Biggl(\sum _{k=0}^{\infty}v_{n-a,k+a+r}(x) (n-1)\\ &{}\times \int_{(k+2r-i)/(n-1)} ^{(k+1+2r-i)/(n-1)} Cn^{r}w(u) \frac{ \vert f(u) \vert}{\lambda}\,\mathrm{d}u \Biggr) \,\mathrm {d}x\le1 \Biggr\} \\ ={}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty}\varPhi\Biggl( \sum _{k=a+r}^{\infty} v_{n-a,k}(x) (n-1)\\ &{}\times \int_{(k+r-a-i)/(n-1)}^{(k+1+r-a-i)/(n-1)} Cn^{r}w(u) \frac{ \vert f(u) \vert}{ \lambda}\,\mathrm{d}u \Biggr)\,\mathrm{d}x\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty}\varPhi\Biggl(\sum _{k=a+r}^{\infty} v_{n-a,k}(x) (n-1) \int_{(k+r-a-i)/(n-1)} ^{(k+1+r-a-i)/(n-1)} Cn^{r}w(u) \frac{ \vert f(u) \vert}{\lambda}\,\mathrm{d}u \\ &{}+\sum_{k=0}^{a+r-1}v_{n-a,k}(x) (n-1) \int_{\max\{0,\frac{k+r-a-i}{n-1}\}}^{\max\{0,\frac{k+1+r-a-i}{n-1} \}} Cn^{r}w(u) \frac{ \vert f(u) \vert}{\lambda}\,\mathrm{d}u \Biggr)\,\mathrm {d}x\le1 \Biggr\} \\ ={}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty}\varPhi\Biggl( \sum _{k=0}^{\infty} v_{n-a,k}(x) (n-1)\\ &{}\times \int_{\max\{0,\frac{k+r-a-i}{n-1}\}}^{\max\{0,\frac{k+1+r-a-i}{n-1} \}} Cn^{r}w(u) \frac{ \vert f(u) \vert}{\lambda}\,\mathrm{d}u \Biggr)\,\mathrm {d}x\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty}\sum_{k=0} ^{\infty} v_{n-a,k}(x)\\ &{}\times\varPhi\biggl((n-1) \int_{\max\{0,\frac{k+r-a-i}{n-1}\}}^{\max\{0,\frac{k+1+r-a-i}{n-1} \}} Cn^{r}w(u) \frac{ \vert f(u) \vert}{\lambda}\,\mathrm{d}u \biggr)\,\mathrm {d}x\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty}\sum_{k=a+i-r} ^{\infty} v_{n-a,k}(x) (n-1)\\ &{}\times \int_{(k+r-a-i)/(n-1)}^{(k+1+r-a-i)/(n-1)} \varPhi\biggl(Cn^{r}w(u) \frac{ \vert f(u) \vert}{\lambda} \biggr)\,\mathrm{d}u\,\mathrm {d}x\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda:\sum _{k=a+i-r}^{\infty} \frac{n-1}{n-a-1} \int_{(k+r-a-i)/(n-1)}^{(k+1+r-a-i)/(n-1)} \varPhi\biggl(Cn^{r}w(u) \frac{ \vert f(u) \vert}{\lambda} \biggr)\,\mathrm{d}u\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda:\sum _{k=0}^{\infty} \int_{k/(n-1)}^{(k+1)/(n-1)} \varPhi\biggl(Cn^{r}w(u) \frac{ \vert f(u) \vert}{ \lambda} \biggr)\,\mathrm{d}u\le1 \Biggr\} \\ \le{}& Cn^{r} \Vert wf \Vert _{(\varPhi)} \le Cn^{r} \Vert wf \Vert _{\varPhi}, \end{aligned} \end{aligned}$$

and, similarly,

$$ \Vert F_{2} \Vert _{\varPhi}\le Cn^{r} \Vert wf \Vert _{\varPhi}. $$

(3.2)

Employing inequalities between (3.1) and (3.2) yields

$$\begin{aligned} \bigl\Vert w\varphi^{2r}(x)\tilde{V}_{n}^{(2r)}(f) \bigr\Vert _{\varPhi} &\le\Biggl\Vert \sum _{i=0}^{2r}\binom{2r}{i}(F_{1}+F_{2}) \Biggr\Vert _{ \varPhi} \\ &\le\Biggl\Vert \sum_{i=0}^{2r} \binom{2r}{i}F_{1} \Biggr\Vert _{\varPhi}+ \Biggl\Vert \sum_{i=0}^{2r} \binom{2r}{i}F_{2} \Biggr\Vert _{\varPhi} \\ &\le\sum_{i=0}^{2r}\binom{2r}{i} \bigl( \Vert F_{1} \Vert _{\varPhi}+ \Vert F_{2} \Vert _{\varPhi}\bigr) \le Cn^{r} \Vert wf \Vert _{\varPhi}. \end{aligned}$$

Combining this inequality with (1.5) and (1.6) results in

$$ \bigl\Vert w\varphi^{2r}L_{n,r}^{(2r)}(f) \bigr\Vert _{\varPhi}= \Biggl\Vert \sum_{i=0}^{2r-1}c_{i}(n,r)w \varphi^{2r}\tilde{V}_{n_{i}}^{(2r)}(f,x) \Biggr\Vert _{\varPhi}\le Cn^{r} \Vert wf \Vert _{\varPhi}. $$

The proof of Lemma 3.1 is complete. □

Lemma 3.2

Let $f\in L_{\varPhi}^{*}[0,\infty)$, $n\ge2r$, $a,b\in\mathbb {N}$, and $0\le a,b< n-1$. Then

$$ \bigl\Vert w\varphi^{2r}L_{n,r}^{(2r)}(f) \bigr\Vert _{\varPhi}\le C \bigl\Vert w \varphi^{2r}f^{(2r)} \bigr\Vert _{\varPhi}. $$

Proof

Since

$$\begin{aligned} &\varphi^{2r}(x)\tilde{V}_{n}^{(2r)}(f;x)\\ &\quad = \frac{(n+2r-1)!}{(n-1)!} \sum_{k=0}^{\infty} \Delta^{2r}a_{k}(n-1)v_{n+2r,k}(x)x^{r}(1+x)^{r} \\ &\quad =\frac{(n+2r-1)!}{(n-1)!}\sum_{k=0}^{\infty} \Delta^{2r}a_{k}(n-1)v _{n+2r,k}(x)\sum _{i=0}^{r}\binom{r}{i}x^{2r-i} \\ &\quad =\sum_{i=0}^{r}\binom{r}{i} \sum_{k=0}^{\infty}\Delta^{2r}a_{k}(n-1)v _{n+i,k+2r-i}(x)\frac{(k+2r-i)!(n+i-1)!}{k!(n-1)!} \\ &\quad =\sum_{i=0}^{r}\binom{r}{i} \sum_{k=0}^{\infty} \frac{(k+2r-i)!(n+i-1)!}{k!(n-1)!} (n-1) \int_{0}^{1/(n-1)} \int_{0} ^{1/(n-1)}\cdots \\ &\qquad{} \int_{0}^{1/(n-1)}f^{(2r)} \biggl( \frac{k}{n-1}+t_{1}+t _{2}+\cdots +t_{2r} \biggr)\,\mathrm{d}t_{1}\,\mathrm{d}t_{2}\cdots\,\mathrm{d} t_{2r}v_{n+i,k+2r-i}(x) \\ &\quad \le C\sum_{k=0}^{\infty} (n-1)^{2r} \int_{0}^{1/(n-1)} \int_{0}^{1/(n-1)} \cdots \\ &\qquad{} \int_{0}^{1/(n-1)}\sum_{i=0}^{r} \binom{r}{i} \biggl( \frac{k}{n-1}+t_{1}+t_{2}+ \cdots+t_{2r} \biggr)^{2r-i} \\ &\qquad{}\times\biggl\vert f^{(2r)} \biggl(\frac{k}{n-1}+t_{1}+t_{2}+ \cdots+t_{2r} \biggr) \biggr\vert \,\mathrm{d}t_{1}\,\mathrm{d} t_{2}\cdots\,\mathrm{d}t_{2r} v_{n+i,k+2r-i}(x) \\ &\quad \le C\sum_{k=0}^{\infty}(n-1)^{2r} \int_{0}^{1/(n-1)} \int_{0}^{1/(n-1)} \cdots \\ &\qquad{} \int_{0}^{1/(n-1)} \varphi^{2r} \biggl( \frac{k}{n-1}+t _{1}+t_{2}+\cdots +t_{2r} \biggr) \\ &\qquad{}\times\biggl\vert f^{(2r)} \biggl(\frac{k}{n-1}+t_{1}+t_{2}+ \cdots+t_{2r} \biggr) \biggr\vert \,\mathrm{d}t_{1}\,\mathrm{d} t_{2}\cdots\,\mathrm{d}t_{2r}\sum_{i=0} ^{r}v_{n+i,k+2r-i}(x), \end{aligned}$$

we obtain

$$\begin{aligned}& w(x)\varphi^{2r}(x)\tilde{V}_{n}^{(2r)}(f;x)\\& \quad \le C\sum_{i=0}^{r} \sum _{k=0}^{\infty}v_{n+i,k+2r-i}(x) (n-1)^{2r} \int_{0}^{1/(n-1)} \cdots \\& \qquad \int_{0}^{1/(n-1)}w \biggl(\frac{k}{n-1}+t_{1}+ \cdots+t_{2r} \biggr) \varphi^{2r} \biggl( \frac{k}{n-1}+t_{1}+\cdots+t_{2r} \biggr) \\& \qquad {}\times\biggl\vert f^{(2r)} \biggl(\frac{k}{n-1}+t_{1}+ \cdots+t_{2r} \biggr) \biggr\vert \frac{w(x)}{w (\frac {k}{n-1}+t_{1}+\cdots+t_{2r} )} \,\mathrm{d} t_{1}\cdots\,\mathrm{d}t_{2r} \\& \quad \le C\sum_{i=0}^{r}\sum _{k=0}^{\infty}v_{n+i,k+2r-i}(x) (n-1)^{2r}2^{b+1} \biggl[ \biggl(\frac{n-1}{k+1} \biggr)^{a}x^{a}+ \biggl(\frac{n-1}{k+1} \biggr)^{b}x^{b} \biggr] \\& \qquad {}\times \int_{0}^{1/(n-1)}\cdots \int_{0}^{1/(n-1)}w \biggl( \frac{k}{n-1}+t_{1}+ \cdots+t_{2r} \biggr) \varphi^{2r} \biggl( \frac{k}{n-1}+t_{1}+\cdots+t_{2r} \biggr) \\& \qquad {}\times\biggl\vert f^{(2r)} \biggl(\frac{k}{n-1}+t_{1}+ \cdots+t_{2r} \biggr) \biggr\vert \,\mathrm{d}t_{1}\cdots\,\mathrm{d} t_{2r} \triangleq G_{1}+G_{2}, \end{aligned}$$

where

$$\begin{aligned} G_{1} ={}&C\sum_{i=0}^{r} \sum_{k=0}^{\infty}(n-1)^{2r}v_{n+i,k+2r-i}(x)2^{b+1} \biggl(\frac{n-1}{k+1} \biggr)^{a}x^{a} \int_{0}^{1/(n-1)}\cdots \int_{0}^{1/(n-1)} \\ &{}\times w \biggl(\frac{k}{n-1}+t_{1}+\cdots +t_{2r} \biggr) \varphi^{2r} \biggl(\frac{k}{n-1}+t_{1}+ \cdots+t_{2r} \biggr) \\ &{}\times\biggl\vert f^{(2r)} \biggl(\frac{k}{n-1}+t_{1}+ \cdots+t _{2r} \biggr) \biggr\vert \,\mathrm{d}t_{1}\cdots\, \mathrm{d} t_{2r} \\ \le{}& C\sum_{i=0}^{r}\sum _{k=0}^{\infty}v_{n+i-a,k+2r+a-i}(x) (n-1)^{2r} \int_{0}^{1/(n-1)}\cdots \\ &{}\cdots \int_{0}^{1/(n-1)}w \biggl(\frac{k}{n-1}+t_{1}+ \cdots+t _{2r} \biggr) \\ &{}\times\varphi^{2r} \biggl(\frac{k}{n-1}+t_{1}+ \cdots+t_{2r} \biggr) \biggl\vert f^{(2r)} \biggl( \frac{k}{n-1}+t_{1}+\cdots+t_{2r} \biggr) \biggr\vert \,\mathrm{d}t_{1}\cdots\,\mathrm{d}t_{2r}, \end{aligned}$$

and, by Jensen’s inequality,

$$\begin{aligned} \Vert G_{1} \Vert _{\varPhi}\le{}&2 \Vert G_{1} \Vert _{(\varPhi)} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty} \varPhi\Biggl(C\sum _{i=0}^{r}\sum_{k=0}^{\infty}v_{n+i-a,k+a+2r-i}(x) (n-1)^{2r} \int_{0}^{1/(n-1)}\,\mathrm{d}t_{1}\cdots \\ & \int_{0}^{1/(n-1)}\,\mathrm{d}t_{2r-1} \int_{k/(n-1)}^{(k+1)/(n-1)}w(t _{1}+\cdots +t_{2r-1}+t_{2r}) \\ &{}\times\varphi^{2r}(t_{1}+\cdots +t_{2r-1}+t_{2r}) \frac{ \vert f^{(2r)}(t_{1}+\cdots+t_{2r-1}+t_{2r}) \vert}{\lambda}\, \mathrm{d}t _{2r} \Biggr)\,\mathrm{d}x\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty}\varPhi\Biggl(\sum _{i=0}^{r} \Biggl[\sum _{k=2r+a-i}^{\infty}v_{n+i-a,k}(x) (n-1)^{2r} \int_{0}^{1/(n-1)}\,\mathrm{d}t_{1}\cdots \\ & \int_{0}^{1/(n-1)}\,\mathrm{d}t_{2r-1} \int_{(k-2r-a+i)/(n-1)}^{(k-2r-a+1+i)/(n-1)}w(t _{1}+\cdots +t_{2r-1}+t_{2r}) \\ &{}\times\varphi^{2r}(t_{1}+\cdots +t_{2r-1}+t_{2r})\frac{C \vert f^{(2r)}(t_{1}+\cdots+t_{2r-1}+t_{2r}) \vert}{\lambda}\, \mathrm{d}t _{2r} \\ &{}+\sum_{k=0}^{2r+a-i-1}v_{n+i-a,k}(x) (n-1)^{2r} \int_{0}^{1/(n-1)} \,\mathrm{d}t_{1}\cdots \int_{0}^{1/(n-1)}\,\mathrm{d}t_{2r-1} \\ &{}\times \int_{\max\{0,\frac{k-2r-a+i}{n-1}\}}^{\max\{0, \frac{k-2r-a+1+i}{n-1}\}} w(t_{1}+\cdots +t_{2r-1}+t_{2r})\varphi^{2r}(t _{1}+\cdots+t_{2r-1}+t_{2r}) \\ &{}\times\frac{C \vert f^{(2r)}(t_{1}+\cdots+t_{2r-1}+t_{2r}) \vert}{\lambda}\,\mathrm{d}t_{2r} \Biggr] \Biggr)\,\mathrm{d}x\le 1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty} \frac{1}{2^{r}}\sum _{i=0}^{r}\binom{r}{i} \sum _{k=0}^{\infty}v_{n+i-a,k}(x) (n-1)^{2r-1} \int_{0}^{1/(n-1)}\,\mathrm{d}t_{1}\cdots \\ & \int_{0}^{1/(n-1)}\,\mathrm{d}t_{2r-1}\varPhi \biggl((n-1) \int_{\max\{0,\frac{k-2r-a+i}{n-1}\}}^{\max\{0, \frac{k-2r-a+1+i}{n-1}\}} C2^{r}w(t_{1}+ \cdots+t_{2r-1}+t_{2r}) \\ &{}\times\varphi^{2r}(t_{1}+\cdots +t_{2r-1}+t_{2r})\frac{ \vert f^{(2r)}(t_{1}+\cdots+t_{2r-1}+t_{2r}) \vert}{\lambda} \, \mathrm{d}t _{2r} \biggr)\,\mathrm{d}x\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda: \int_{0}^{\infty} \frac{1}{2^{r}}\sum _{i=0}^{r}\binom{r}{i} \sum _{k=0}^{\infty}v_{n+i-a,k+2r+a-i}(x) (n-1)^{2r} \int_{0}^{1/(n-1)}\,\mathrm{d}t_{1}\cdots \\ & \int_{0}^{1/(n-1)}\,\mathrm{d}t_{2r-1} \int_{k/(n-1)}^{(k+1)/(n-1)} \varPhi\biggl(Cw(t_{1}+ \cdots+t_{2r-1}+t_{2r}) \\ &{}\times\varphi^{2r}(t_{1}+\cdots +t_{2r-1}+t_{2r}) \frac{ \vert f^{(2r)}(t_{1}+\cdots+t_{2r-1}+t_{2r}) \vert}{\lambda} \biggr) \,\mathrm{d} t_{2r}\,\mathrm{d}x\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda:\frac{1}{2^{r}}\sum _{i=0} ^{r}\binom{r}{i}C_{1} \sum_{k=0}^{\infty} (n-1)^{2r-1} \int_{0}^{1/(n-1)} \,\mathrm{d}t_{1}\cdots \int_{0}^{1/(n-1)}\,\mathrm{d}t_{2r-1} \\ &{}\times \int_{k/(n-1)}^{(k+1)/(n-1)}\varPhi\biggl(Cw(t_{1}+ \cdots+t_{2r-1}+t_{2r})\varphi^{2r}(t_{1}+ \cdots+t_{2r-1}+t_{2r}) \\ &{}\times\frac{ \vert f^{(2r)}(t_{1}+\cdots+t_{2r-1}+t_{2r}) \vert}{\lambda} \biggr)\,\mathrm{d}t_{2r}\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \Biggl\{ \lambda:\sum _{k=0}^{\infty}(n-1)^{2r-1} \int_{0}^{1/(n-1)}\,\mathrm{d}t_{1}\cdots \int_{0}^{1/(n-1)}\,\mathrm{d}t_{2r-1} \\ &{}\times \int_{k/(n-1)}^{(k+1)/(n-1)}\varPhi\biggl(w(t_{1}+ \cdots+t_{2r-1}+t_{2r})\varphi^{2r}(t_{1}+ \cdots+t_{2r-1}+t_{2r})C \\ &{}\times C_{1}\frac{ \vert f^{(2r)}(t_{1}+\cdots+t_{2r-1}+t_{2r}) \vert}{\lambda} \biggr)\,\mathrm{d} t_{2r}\le1 \Biggr\} \\ \le{}&2\inf_{\lambda>0} \biggl\{ \lambda: \int_{0}^{\infty}\varPhi\biggl(Cw(t)\varphi ^{2r}(t) \frac{ \vert f^{(2r)}(t) \vert}{\lambda} \biggr) \,\mathrm {d}t\le1 \biggr\} \\ \le{}& C \bigl\Vert w\varphi^{2r}f^{(2r)} \bigr\Vert _{\varPhi}, \end{aligned}$$

where $C_{1}\ge1$. Similarly, we have

$$ \Vert G_{2} \Vert _{\varPhi}\le C \bigl\Vert w\varphi ^{2r}f^{(2r)} \bigr\Vert _{\varPhi}. $$

Consequently, it follows that

$$ \bigl\Vert w\varphi^{2r}\tilde{V}_{n}^{(2r)}(f) \bigr\Vert _{\varPhi}\le C \bigl\Vert w\varphi^{2r}f^{(2r)} \bigr\Vert _{\varPhi}. $$

Combining this inequality with (1.5) and (1.6) yields

$$ \bigl\Vert w\varphi^{2r}L_{n,r}^{(2r)}(f) \bigr\Vert _{\varPhi}= \Biggl\Vert \sum_{i=0}^{2r-1}c_{i}(n,r)w \varphi^{2r}\tilde{V}_{n_{i}}^{(2r)}(f) \Biggr\Vert _{\varPhi}\le C \bigl\Vert w\varphi^{2r}f^{(2r)} \bigr\Vert _{\varPhi}. $$

The proof of Lemma 3.2 is complete. □

We now in a position to prove Theorems 1.2 and 1.3.

Proof of Theorem 1.2

From Lemmas 3.1 and 3.2 and [27, Theorem 2.2] we obtain

$$ K_{2r,\varphi} \biggl(f,\frac{1}{n^{r/2}} \biggr)_{w,\varPhi}\le \frac{C}{n ^{r}} \sum_{k=1}^{n}k^{r-1} \bigl\Vert w\bigl(L_{k,r}(f)-f\bigr) \bigr\Vert _{\varPhi} . $$

Application of inequality (1.3) concludes the inverse theorem. □

Proof of Theorem 1.3

Using the so-called order function $\psi(t)=t^{\alpha}|\ln t|^{ \beta}e^{|\ln t|^{\gamma}}$ for $0<\alpha<1$, $\beta\in \mathbb{R}$, and $\gamma<1$ and combining Theorems 1.1 and 1.2 conclude the equivalence theorem. □