• Research
• Open Access

# Degree of approximation of the conjugate of signals (functions) belonging to $\mathrm{Lip}\left(\alpha ,r\right)$-class by $\left(C,1\right)\left(E,q\right)$ means of conjugate trigonometric Fourier series

Journal of Inequalities and Applications20122012:278

https://doi.org/10.1186/1029-242X-2012-278

• Received: 2 April 2012
• Accepted: 15 October 2012
• Published:

## Abstract

In this paper, we determine the degree of approximation of the conjugate of 2π-periodic signals (functions) belonging to $Lip\left(\alpha ,r\right)$ ($0<\alpha \le 1$, $r\ge 1$)-class by using Cesàro-Euler $\left(C,1\right)\left(E,q\right)$ means of their conjugate trigonometric Fourier series. Our result generalizes the result of Lal and Singh (Tamkang J. Math. 33(3):269-274, 2002).

MSC:41A10.

## Keywords

• conjugate Fourier series
• $Lip\left(\alpha ,r\right)$-class
• $\left(C,1\right)\left(E,q\right)$ means

## 1 Introduction

Let ${\sum }_{n=0}^{\mathrm{\infty }}{u}_{n}$ be a given infinite series with $\left\{{s}_{n}\right\}$, the sequence of its n th partial sum. The sequence-to-sequence transform
${C}_{n}^{1}=\frac{1}{n+1}\sum _{k=0}^{n}{s}_{k},\phantom{\rule{1em}{0ex}}n=0,1,2,\dots ,$
(1.1)
defines the Cesàro means of order one of $\left\{{s}_{n}\right\}$. The series ${\sum }_{n=0}^{\mathrm{\infty }}{u}_{n}$ is said to be $\left(C,1\right)$ summable to s, if ${lim}_{n\to \mathrm{\infty }}{C}_{n}^{1}=s$. The sequence-to-sequence transform
${E}_{n}^{q}=\frac{1}{{\left(1+q\right)}^{n}}\sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){q}^{n-k}{s}_{k},\phantom{\rule{1em}{0ex}}q>0,n=0,1,2,\dots ,$
(1.2)
defines the Euler means of order $q>0$ of $\left\{{s}_{n}\right\}$. By super imposing the $\left(C,1\right)$ means on $\left(E,q\right)$ means of $\left\{{s}_{n}\right\}$, we get $\left(C,1\right)\left(E,q\right)$ means of $\left\{{s}_{n}\right\}$ denoted by ${C}_{n}^{1}{E}_{n}^{q}$ and defined by
${C}_{n}^{1}{E}_{n}^{q}=\frac{1}{n+1}\sum _{k=0}^{n}{E}_{k}^{q}=\frac{1}{n+1}\sum _{k=0}^{n}{\left(q+1\right)}^{-k}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}{s}_{v}.$
(1.3)

The series ${\sum }_{n=0}^{\mathrm{\infty }}{u}_{n}$ is said to be $\left(C,1\right)\left(E,q\right)$ summable to s, if ${lim}_{n\to \mathrm{\infty }}{C}_{n}^{1}{E}_{n}^{q}=s$.

For a given 2π-periodic Lebesgue integrable signal (function), let
${s}_{n}\left(f;x\right)=\frac{{a}_{0}}{2}+\sum _{k=1}^{n}\left({a}_{k}coskx+{b}_{k}sinkx\right)$
(1.4)

denote the $\left(n+1\right)$th partial sum, called trigonometric polynomial of degree n (or order n), of the Fourier series of $f\in {L}_{1}\left[-\pi ,\pi \right]$.

The conjugate of Fourier series of f is defined by
$\sum _{k=1}^{\mathrm{\infty }}\left({b}_{k}coskx-{a}_{k}sinkx\right),$
(1.5)
and its n th partial sum is defined as
$\stackrel{˜}{{s}_{n}}\left(f;x\right)=\sum _{k=1}^{n}\left({b}_{k}coskx-{a}_{k}sinkx\right)$
(1.6)
The conjugate of f denoted by $\stackrel{˜}{f}$ is defined by
$2\pi \stackrel{˜}{f}\left(x\right)=-\underset{ϵ\to 0}{lim}{\int }_{ϵ}^{\pi }\psi \left(t\right)cot\left(t/2\right)\phantom{\rule{0.2em}{0ex}}dt,$

where $\psi \left(t\right)=f\left(x+t\right)-f\left(x-t\right)$ [, p.131].

A function $f\in Lip\alpha$, if
$|f\left(x+t\right)-f\left(x\right)|=O\left({t}^{\alpha }\right),$
and $f\in Lip\left(\alpha ,r\right)$ if
${\left({\int }_{0}^{2\pi }{|f\left(x+t\right)-f\left(x\right)|}^{r}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r}=O\left({t}^{\alpha }\right),\phantom{\rule{1em}{0ex}}0<\alpha \le 1,r\ge 1.$
The ${L}_{r}$-norm for $f\in {L}_{r}\left[-\pi ,\pi \right]$ is defined by
${\parallel f\parallel }_{r}={\left({\int }_{0}^{2\pi }{|f\left(x\right)|}^{r}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r},\phantom{\rule{1em}{0ex}}r\ge 1.$
The ${L}_{\mathrm{\infty }}$-norm is defined by
${\parallel f\parallel }_{\mathrm{\infty }}=sup\left\{|f\left(x\right)|:x\in R\right\}.$
The degree of approximation ${E}_{n}\left(f\right)$ of a function $f\in Lip\left(\alpha ,r\right)$ by trigonometric polynomials ${T}_{n}\left(x\right)$ of degree n is given by
${E}_{n}\left(f\right)=\underset{{T}_{n}}{min}{\parallel f-{T}_{n}\parallel }_{r}.$
This method of approximation is called trigonometric Fourier approximation (tfa). We also write
${K}_{n}\left(t\right)=\frac{1}{n+1}\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}\frac{cos\left(v+1/2\right)t}{sin\left(t/2\right)}$

and $\tau =\left[1/t\right]$, the integral part of $1/t$.

## 2 Known result

Various investigators such as Dhakal , Lal and Singh , Mittal et al. [4, 5], Nigam , Qureshi [7, 8] have studied the degree of approximation in various function spaces such as Lipα, $Lip\left(\alpha ,r\right)$, $Lip\left(\xi \left(t\right),r\right)$ and weighted $\left({L}_{r},\xi \left(t\right)\right)$ by using triangular matrix summability and product summability $\left(C,1\right)\left(E,1\right)$, $\left(N,{p}_{n}\right)\left(E,1\right)$. Recently, Lal and Singh  have determined the degree of approximation of the conjugate of $f\in Lip\left(\alpha ,r\right)$ by $\left(C,1\right)\left(E,1\right)$ means of conjugate Fourier series. Lal and Singh  have proved the following.

Theorem 1 

If $f:R\to R$ is a 2π-periodic and $Lip\left(\alpha ,r\right)$ function, then the degree of approximation of its conjugate function $\stackrel{˜}{f}\left(x\right)$ by the $\left(C,1\right)\left(E,1\right)$ product means of conjugate series of Fourier series of f satisfies, for $n=0,1,2,\dots$ ,
${M}_{n}\left(\stackrel{˜}{f}\right)=Min{\parallel {\left(CE\right)}_{n}^{1}-\stackrel{˜}{f}\parallel }_{r}=O\left({n}^{-\alpha +1/r}\right),$
(2.1)
where
${\left(CE\right)}_{n}^{1}=\frac{1}{n+1}\sum _{k=0}^{n}\left(\frac{1}{{2}^{k}}\sum _{i=0}^{k}\left(\begin{array}{c}k\\ i\end{array}\right){S}_{i}\right),$

is $\left(C,1\right)\left(E,1\right)$ means of series (1.5).

## 3 Main result

Recently, Nigam and Sharma  have studied the degree of approximation of functions belonging to $Lip\left(\xi \left(t\right),r\right)$-class through $\left(C,1\right)\left(E,q\right)$ means of their Fourier series. In this paper, we use the $\left(C,1\right)\left(E,q\right)$ means of conjugate Fourier series of $f\in Lip\left(\alpha ,r\right)$ to determine the degree of approximation of the conjugate of f, which in turn generalizes the result of Lal and Singh . More precisely we prove

Theorem 2 Let $f\left(x\right)$ be a 2π-periodic, Lebesgue integrable function and belong to the $Lip\left(\alpha ,r\right)$-class with $r\ge 1$ and $\alpha r\ge 1$. Then the degree of approximation of $\stackrel{˜}{f}\left(x\right)$, the conjugate of $f\left(x\right)$ by $\left(C,1\right)\left(E,q\right)$ means of its conjugate Fourier series is given by
${\parallel {C}_{n}^{1}{E}_{n}^{q}-\stackrel{˜}{f}\parallel }_{r}=O\left({n}^{-\alpha +1/r}\right),\phantom{\rule{1em}{0ex}}n=0,1,2,\dots ,$
(3.1)

where δ is an arbitrary number such that $\left(\alpha +\delta \right)s+1<0$ and $1/r+1/s=1$ for $r>1$.

Remark 1 The authors have used conditions ${\left({\int }_{0}^{\pi /\left(n+1\right)}{|\frac{t\psi \left(t\right)}{{t}^{\alpha }}|}^{r}\phantom{\rule{0.2em}{0ex}}dt\right)}^{1/r}=O\left(1\right)$ implied by (3.2) and (3.3), but not mentioned in the statement of Theorem 1 [, pp.271-272].

## 4 Lemmas

We need the following lemmas for the proof of our theorem.

Lemma 1 $|{K}_{n}\left(t\right)|=O\left(1/t\right)+O\left(\left(n+1\right)t\right)$ for $0\le t\le \pi /\left(n+1\right)\le \pi /\left(v+1\right)$.

Proof
$\begin{array}{rcl}|{K}_{n}\left(t\right)|& \le & \frac{1}{2\pi \left(n+1\right)}|\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}\frac{cos\left(v+1/2\right)t}{sin\left(t/2\right)}|\\ =& \frac{1}{2\pi \left(n+1\right)}|\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}\frac{cos\left(v+1-1/2\right)t}{sin\left(t/2\right)}|\\ \le & \frac{1}{\left(n+1\right)}\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}|\frac{cos\left(v+1\right)tcos\left(t/2\right)+sin\left(v+1\right)tsin\left(t/2\right)}{sin\left(t/2\right)}|\\ =& \frac{1}{\left(n+1\right)}\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}\left[O\left(1/t\right)+O\left(sin\left(v+1\right)t\right)\right]\\ =& O\left[\frac{1}{\left(n+1\right)t}\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}\right]\\ +O\left[\frac{1}{\left(n+1\right)}\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}\left(v+1\right)t\right]\\ =& O\left[\frac{1}{\left(n+1\right)t}\left(n+1\right)\right]+O\left[\frac{1}{\left(n+1\right)}\left(n+1\right)\left(n+1\right)t\right]\\ =& O\left(1/t\right)+O\left(\left(n+1\right)t\right),\end{array}$

in view of $sin\left(v+1\right)t\le \left(v+1\right)t$ for $0\le t<\pi /\left(v+1\right)$ and ${\left(sin\left(t/2\right)\right)}^{-1}<\pi /t$ for $0 [, p.247]. □

Lemma 2 $|{K}_{n}\left(t\right)|=O\left(1/t\right)+O\left(1\right)$ for $\pi /\left(v+1\right)\le t\le \pi$.

Proof
$\begin{array}{rcl}|{K}_{n}\left(t\right)|& \le & \frac{1}{2\pi \left(n+1\right)}|\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}\frac{cos\left(v+1/2\right)t}{sin\left(t/2\right)}|\\ =& \frac{1}{2\pi \left(n+1\right)}|\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}\frac{cos\left(v+1-1/2\right)t}{sin\left(t/2\right)}|\\ \le & \frac{1}{\left(n+1\right)}\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}|\frac{cos\left(v+1\right)tcos\left(t/2\right)+sin\left(v+1\right)tsin\left(t/2\right)}{sin\left(t/2\right)}|\\ =& \frac{1}{\left(n+1\right)}\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}\left[O\left(1/t\right)+O\left(1\right)\right]\\ =& O\left[\frac{1}{\left(n+1\right)t}\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}\right]\\ +O\left[\frac{1}{\left(n+1\right)}\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}\right]\\ =& O\left[\frac{1}{\left(n+1\right)t}\left(n+1\right)\right]+O\left[\frac{1}{\left(n+1\right)}\left(n+1\right)\right]\\ =& O\left(1/t\right)+O\left(1\right),\end{array}$

in view of $|sin\left(v+1\right)t|\le 1$ and ${\left(sin\left(t/2\right)\right)}^{-1}\le \pi /t$ for $0 [, p.247]. □

## 5 Proof of Theorem 2

The integral representation of $\stackrel{˜}{{s}_{n}}\left(f;x\right)$ is given by
$\stackrel{˜}{{s}_{n}}\left(f;x\right)=-\frac{1}{\pi }{\int }_{0}^{\pi }\psi \left(t\right)\frac{cos\left(t/2\right)-cos\left(n+1/2\right)t}{2sin\left(t/2\right)}\phantom{\rule{0.2em}{0ex}}dt.$
Therefore, we have
${\stackrel{˜}{s}}_{n}\left(f;x\right)-\stackrel{˜}{f}\left(x\right)=\frac{1}{2\pi }{\int }_{0}^{\pi }\psi \left(t\right)\frac{cos\left(n+1/2\right)t}{sin\left(t/2\right)}\phantom{\rule{0.2em}{0ex}}dt.$
Now, denoting $\left(C,1\right)\left(E,q\right)$ transform of $\stackrel{˜}{{s}_{n}}\left(f;x\right)$ by ${C}_{n}^{1}{E}_{n}^{q}$, we write
$\begin{array}{rcl}{C}_{n}^{1}{E}_{n}^{q}-\stackrel{˜}{f}& =& \frac{1}{2\pi \left(n+1\right)}\left[\sum _{k=0}^{n}\frac{1}{{\left(1+q\right)}^{k}}{\int }_{0}^{\pi }\frac{\psi \left(t\right)}{sin\left(t/2\right)}\sum _{v=0}^{k}\left(\begin{array}{c}k\\ v\end{array}\right){q}^{k-v}cos\left(v+1/2\right)t\phantom{\rule{0.2em}{0ex}}dt\right]\\ =& \left[{\int }_{0}^{\pi /\left(n+1\right)}+{\int }_{\pi /\left(n+1\right)}^{\pi }\right]\psi \left(t\right){K}_{n}\left(t\right)\phantom{\rule{0.2em}{0ex}}dt={I}_{1}+{I}_{2},\phantom{\rule{1em}{0ex}}\text{say}.\end{array}$
(5.1)
Using Lemma 1, Hölder’s inequality, condition (3.2) and Minkowiski’s inequality, we have
$\begin{array}{rcl}|{I}_{1}|& =& {\int }_{0}^{\pi /\left(n+1\right)}|\psi \left(t\right)||{K}_{n}\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt\\ \le & {\left[{\int }_{0}^{\pi /\left(n+1\right)}{\left(|\psi \left(t\right)|/{t}^{\alpha }\right)}^{r}\phantom{\rule{0.2em}{0ex}}dt\right]}^{1/r}{\left[\underset{ϵ\to 0}{lim}{\int }_{ϵ}^{\pi /\left(n+1\right)}{\left({t}^{\alpha }|{K}_{n}\left(t\right)|\right)}^{s}\phantom{\rule{0.2em}{0ex}}dt\right]}^{1/s}\\ =& O\left({\left(n+1\right)}^{-1}\right){\left[\underset{ϵ\to 0}{lim}{\int }_{ϵ}^{\pi /\left(n+1\right)}{\left({t}^{\alpha -1}+\left(n+1\right){t}^{\alpha +1}\right)}^{s}\phantom{\rule{0.2em}{0ex}}dt\right]}^{1/s}\\ =& O\left({\left(n+1\right)}^{-1}\right)\left[{\left(\underset{ϵ\to 0}{lim}{\int }_{ϵ}^{\pi /\left(n+1\right)}{t}^{\left(\alpha -1\right)s}\phantom{\rule{0.2em}{0ex}}dt\right)}^{1/s}+{\left(\underset{ϵ\to 0}{lim}{\int }_{ϵ}^{\pi /\left(n+1\right)}\left(n+1\right){t}^{\left(\alpha +1\right)s}\phantom{\rule{0.2em}{0ex}}dt\right)}^{1/s}\right]\\ =& O\left({\left(n+1\right)}^{-1}\right)\left[{\left(n+1\right)}^{-\alpha +1-1/s}+\left(n+1\right){\left(n+1\right)}^{-\alpha -1-1/s}\right]\\ =& O\left({\left(n+1\right)}^{-1}\right)\left[{\left(n+1\right)}^{-\alpha +1/r}+\left(n+1\right){\left(n+1\right)}^{-\alpha -1-1+1/r}\right]\\ =& O\left[{\left(n+1\right)}^{-\alpha +1/r-1}+{\left(n+1\right)}^{-\alpha -2+1/r}\right]\\ =& O\left({\left(n+1\right)}^{-\alpha -1+1/r}\right).\end{array}$
(5.2)
Now, we consider
$|{I}_{2}|\le {\int }_{\pi /\left(n+1\right)}^{\pi }|\psi \left(t\right)||{K}_{n}\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt.$
Using Lemma 2, Hölder’s inequality, condition (3.3) and Minkowiski’s inequality, we have
$\begin{array}{rcl}|{I}_{2}|& \le & {\left[{\int }_{\pi /\left(n+1\right)}^{\pi }{\left(\frac{{t}^{-\delta }|\psi \left(t\right)|}{{t}^{\alpha }}\right)}^{r}\phantom{\rule{0.2em}{0ex}}dt\right]}^{1/r}{\left[{\int }_{\pi /\left(n+1\right)}^{\pi }{\left(\frac{{t}^{\alpha }|{K}_{n}\left(t\right)|}{{t}^{-\delta }}\right)}^{s}\phantom{\rule{0.2em}{0ex}}dt\right]}^{1/s}\\ =& O\left({\left(n+1\right)}^{\delta }\right){\left[{\int }_{\pi /\left(n+1\right)}^{\pi }{\left(\frac{{t}^{\alpha }}{{t}^{-\delta }}\left(O\left(1/t\right)+O\left(1\right)\right)\right)}^{s}\phantom{\rule{0.2em}{0ex}}dt\right]}^{1/s}\\ =& O\left({\left(n+1\right)}^{\delta }\right){\left[{\int }_{\pi /\left(n+1\right)}^{\pi }{\left({t}^{\alpha +\delta -1}+{t}^{\alpha +\delta }\right)}^{s}\phantom{\rule{0.2em}{0ex}}dt\right]}^{1/s}\\ =& O\left({\left(n+1\right)}^{\delta }\right)\left[{\left({\int }_{\pi /\left(n+1\right)}^{\pi }{t}^{\left(\alpha +\delta -1\right)s}\phantom{\rule{0.2em}{0ex}}dt\right)}^{1/s}+{\left({\int }_{\pi /\left(n+1\right)}^{\pi }{t}^{\left(\alpha +\delta \right)s}\phantom{\rule{0.2em}{0ex}}dt\right)}^{1/s}\right]\\ =& O\left({\left(n+1\right)}^{\delta }\right)\left[{\left(n+1\right)}^{\left(-\alpha -\delta +1\right)-1/s}+{\left(n+1\right)}^{\left(-\alpha -\delta \right)-1/s}\right]\phantom{\rule{1em}{0ex}}\left(\left(\alpha +\delta \right)s+1<0\right)\\ =& O\left[{\left(n+1\right)}^{-\alpha +1-1/s}+{\left(n+1\right)}^{-\alpha -1/s}\right]\\ =& O\left[{\left(n+1\right)}^{-\alpha +1/r}+{\left(n+1\right)}^{-\alpha -1+1/r}\right]=O\left[{\left(n+1\right)}^{-\alpha +1/r}\left(1+{\left(n+1\right)}^{-1}\right)\right]\\ =& O\left({\left(n+1\right)}^{-\alpha +1/r}\right).\end{array}$
(5.3)
Combining (5.1)-(5.3), we have
$|{C}_{n}^{1}{E}_{n}^{q}-\stackrel{˜}{f}|=O\left({\left(n+1\right)}^{-\alpha +1/r}\right).$
Hence,
${\parallel {C}_{n}^{1}{E}_{n}^{q}-\stackrel{˜}{f}\parallel }_{r}={\left({\int }_{0}^{2\pi }{|{C}_{n}^{1}{E}_{n}^{q}-\stackrel{˜}{f}\left(x\right)|}^{r}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r}=O\left({n}^{-\alpha +1/r}\right).$

This completes the proof of Theorem 2.

Remark 2 The proof of Theorem 2 for $r=1$, i.e., $s=\mathrm{\infty }$, can be written by using sup norm while using Hölder’s inequality.

## 6 Corollaries

Corollary 1 When $q=1$, then $\left(C,1\right)\left(E,q\right)$ means reduces to $\left(C,1\right)\left(E,1\right)$ means.

Hence, Theorem  2 reduces to Theorem  1.

Corollary 2 If $f:R\to R$ is a 2π-periodic, Lebesgue integrable and belonging to the Lipα ($0<\alpha \le 1$) class, then the degree of approximation of $\stackrel{˜}{f}\left(x\right)$, the conjugate of $f\left(x\right)\in Lip\alpha$, with $0<\alpha \le 1$ by $\left(C,1\right)\left(E,q\right)$ means of its Fourier series is given by
${\parallel {C}_{n}^{1}{E}_{n}^{q}-\stackrel{˜}{f}\parallel }_{\mathrm{\infty }}=O\left({n}^{-\alpha }\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for}}n=0,1,2,\dots .$
Proof If $r\to \mathrm{\infty }$ in Theorem 2, then for $0<\alpha <1$,
${\parallel {C}_{n}^{1}{E}_{n}^{q}-\stackrel{˜}{f}\left(x\right)\parallel }_{\mathrm{\infty }}=O\left({n}^{-\alpha }\right).$

For $\alpha =1$, we can write an independent proof by using $\alpha =1$ and $\psi \left(t\right)=O\left(t\right)$ in ${I}_{1}$ and ${I}_{2}$. □

## Declarations

### Acknowledgements

The authors would like to thank the referee for his valuable comments and suggestions for the improvement of the manuscript.

## Authors’ Affiliations

(1)
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

## References 