Approximation of Periodic Functions Belonging to $W(L^{r},\xi (t),(\beta \ge 0))$-Class By $(C^{1}\cdot T)$ Means of Fourier Series

Abstract

Various investigators such as Khan [3], Qureshi [8–10], Qureshi and Nema [11], Leindler [6] and Chandra [1] have determined the degree of approximation of functions belonging to the classes $W(L^{r},\xi (t)), Lip (\xi (t), r), Lip (\alpha , r)$ and $Lip \alpha$ using different summability methods with monotonocity conditions. Recently, Lal [5] has determined the degree of approximation of the functions belonging to $Lip \alpha$ and $W(L^{r},\xi (t))$ classes by using Cesàro-Nörlund $(C^{1}\cdot N_{p})$—summability with non-increasing weights $\{p_{n}\}$. In this paper, we shall determine the degree of approximation of 2$\pi$-periodic function f belonging to the function classes $Lip\alpha$ and $W(L^{r},\xi (t))$ by $C^{1}\cdot T$—means of Fourier series of f. Our theorems generalize the results of Lal [5], and we also improve these results in the light of [7, 12, 13]. From our results, we derive some corollaries also.

1 Introduction

For a given signal $f \in L^{r}: = L^{r} [0,2\pi ], r \ge 1,$ let

$$\begin{aligned} s_{n}(f):= s_{n}(f;x)=\frac{a_{0}}{2} +\displaystyle \sum _{k=1}^{n}(a_{k}\cos kx + b_{k}\sin kx) = \displaystyle \sum _{k=0}^{n}u_{k}(f;x), \end{aligned}$$

(1)

denote the partial sums, called trigonometric polynomial of degree (or order) n, of the first $(n+1)$ terms of the Fourier series of f. The matrix means of (1) are defined by

$$ t_{n}(f):= t_{n}(f;x)= \displaystyle \sum _{k=0}^{n}a_{n,k}s_{k} ,\,\,\,n=0,1,2,..., $$

where $T\equiv (a_{n,k})$ is a lower triangular matrix with non-negative entries such that $a_{n,-1}=0, A_{n,k}=\sum _{r=k}^{n}a_{n,r}$ so that $A_{n,0}=1, \forall n\ge 0.$ The Fourier series of f is said to be T-summable to s, if $t_{n}(f) \rightarrow s$ as $n \rightarrow \infty .$

By superimposing $C^{1}$ summability upon T summability, we get the $C^{1}\cdot T$ summability. Thus the $C^{1}\cdot T$ means of $\{s_{n}(f)\}$ denoted by $t_{n}^{C^{1}\cdot T}(f)$ are given by

$$ t_{n}^{C^{1}\cdot T}(f):=(n+1)^{-1} \sum _{r=0}^{n}\bigg (\sum _{k=0}^{r}a_{r,k}s_{k}(f)\bigg ). $$

If $t_{n}^{C^{1}\cdot T}\rightarrow s_{1}$ as $n\rightarrow \infty$, then the Fourier series of f is said to be ${C^{1}\cdot T}$—summable to the sum $s_{1}.$ The regularity of methods $C^{1}$ and T implies regularity of method $C^{1}\cdot T.$

A function $f \in Lip \alpha$ if $|f(x\,+\,t)-f(x)|=O(|t|^{\alpha })$, for $0< \alpha \le 1,$ $f \in Lip (\alpha , r)$ if $\left( \int _{0}^{2 \pi } | f(x\,+\,t)-f(x)| ^{r} dx\right) ^{1/r}=O(|t|^{\alpha }), 0<\alpha \le 1, r\ge 1,$

$f \in Lip (\xi (t), r)$ if $\left( \int _{0}^{2 \pi } | f(x\,+\,t)-f(x)| ^{r} dx\right) ^{1/r}=O(\xi (t))$ and

$f \in W(L^{r}, \xi (t))$ if $\left( \int _{0}^{2 \pi } | (f(x+t)-f(x))\sin ^{\beta }(x/2)|^{r} dx\right) ^{1/r}=O(\xi (t)), \,\,$

$\beta \ge 0, r \ge 1,$ where $\xi (t)$ is a positive increasing function of t.

If $\beta =0, \,W(L^{r},\xi (t),) \equiv Lip (\xi (t), r)$ and for $\xi (t)=t^{\alpha } (\alpha >0), Lip(\xi (t),r)\equiv Lip (\alpha , r).$ $Lip (\alpha , r)\rightarrow Lip \alpha$ as $r\rightarrow \infty .$ Thus

$$ Lip \alpha \subseteq Lip (\alpha , r) \subseteq Lip (\xi (t), r) \subseteq W(L^{r}, \xi (t)). $$

The $L^{r}$-norm of $f\in L^{r}[0,2\pi ]$ is defined by

$\,\,\,\,\, \Vert f \Vert _{r} = \left\{ \frac{1}{2 \pi }\int _{0}^{2 \pi } |f(x)| ^{r} dx \right\} ^{1/r}$ $(1 \le r < \infty )$ and $\Vert f \Vert _{\infty } = \displaystyle \sup _{x\in [0,2\pi ]} |f(x)|.$

The degree of approximation of $f\in L^{r}$ denoted by $E_{n}(f)$ is given by

$$ E_{n}(f) =\displaystyle \min _{T_{n}}\parallel f(x)-T_{n}(x)\parallel _{r}, $$

in terms of n , where $T_{n}(x)$ is a trigonometric polynomial of degree n.

This method of approximation is called trigonometric Fourier approximation.

We also write

$$\begin{aligned} \phi (t)&= f(x\,+\,t)+f(x\,-\,t)-2f(x),\\ (C^{1}\cdot T)_{n}(t)&= \frac{1}{2\pi (n\,+\,1)}\displaystyle \sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k}\frac{\sin (r-k+1/2)t}{\sin (t/2)}, \end{aligned}$$

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

2 Known Results

Various investigators such as Khan [3], Qureshi [8–10], Qureshi and Nema [11], Leindler [6] and Chandra [1] have determined the degree of approximation of functions belonging to the classes $W(L^{r},\xi (t)), Lip (\xi (t), r), Lip (\alpha , r)$ and $Lip \alpha$ with $r\ge 1$ and $0<\alpha \le 1$ using different summability methods with monotonocity conditions on the rows of summability matrices. Recently, Lal [5] has determined the degree of approximation of the functions belonging to $Lip \alpha$ and $W(L^{r},\xi (t))$ classes by using Cesáro-Nörlund $(C^{1}\cdot N_{p})$—summability with non-increasing weights $\{p_{n}\}$. He proved:

Theorem 1

Let $N_{p}$ be a regular Nörlund method defined by a sequence $\{p_{n}\}$ such that

$$\begin{aligned} P_{\tau }\displaystyle \sum _{v=\tau }^{n} P_v^{-1} = O (n+1). \end{aligned}$$

(2)

Let $f \in L^{1}[0,2\pi ]$ be a $2\pi$-periodic function belonging to $Lip\, \alpha \,(0 < \alpha \le 1),$ then the degree of approximation of f by $C^{1}\cdot N_{p}$ means of its Fourier series is given by

$$\begin{aligned} \displaystyle \sup _{0 \le x \le 2\pi } | t_{n}^{CN}(x)\, -\, f(x)| =\Vert t_{n}^{CN}\,-\,f\Vert _{\infty } = \left\{ \begin{array}{ll} O((n+1)^{-\alpha }), &{} 0 < \alpha < 1 , \\ O\left( \log (n+1)\pi e/(n+1)\right) ,&{} \,\,\,\,\,\,\,\,\,\, \alpha = 1. \end{array} \right. \end{aligned}$$

Theorem 2

If f is a $2\pi$-periodic function and Lebesgue integrable on $[0,2\pi ]$ and is belonging to $W(L^{r},\xi (t))$ class then its degree of approximation by $C^{1}\cdot N_{p}$ means of its Fourier series is given by

$$\begin{aligned} \displaystyle \Vert t_{n}^{CN} -f \Vert _{r} = O\left( (n+1)^{\beta + 1/r} \xi \left( 1/(n+1)\right) \right) ,\end{aligned}$$

provided $\xi (t)$ satisfies the following conditions:

$$\begin{aligned} \{\xi (t)/t\}\, \text{ be } \text{ a } \text{ decreasing } \text{ sequence }, \end{aligned}$$

(3)

$$\begin{aligned} \left( {\displaystyle \int _{0}^{\pi /(n+1)} \Big (t|\phi (t)| \sin ^{\beta }(t)/ \xi (t)\Big )^{r}dt}\right) ^{1/r}= O((n+1)^{-1}), \end{aligned}$$

(4)

$$\begin{aligned} \left( {\displaystyle \int _{\pi /(n+1)}^{\pi } \Big (t^{-\delta }|\phi (t)|/ \xi (t)\Big )^{r}dt} \right) ^{1/r}= O((n+1)^{\delta }), \end{aligned}$$

(5)

where $\delta$ is an arbitrary number such that $s(1-\delta )-1>0, r^{-1}+ s^{-1}=1, r\ge 1,$ conditions (4) and (5) hold uniformly in x.

The improved version of above theorems with their generalization to non-monotone weights $\{p_{n}\}$ can be seen in [13].

3 Main Results

In this paper, we generalize Theorems 1 and 2 by replacing matrix $N_{p}$ with matrix T in the light of Remarks 2.3 and 2.4 of [13, pp. 3–4]. More precisely, we prove:

Theorem 3

If $T\equiv (a_{n,k})$ is a lower triangular regular matrix with non-negative and non-decreasing (with respect to k) entries which satisfy

$$\begin{aligned} \displaystyle \sum _{r=\tau }^{n}A_{r,r-\tau } = O(n+1), \end{aligned}$$

(6)

hold uniformly in $\tau =[1/t],$ then the degree of approximation of a $2\pi$-periodic function $f \in Lip \alpha$ $(0<\alpha \le 1)\subset L^{1}[0,2\pi ]$ by $C^{1}\cdot T$ means of its Fourier series is given by

$$\begin{aligned} \Vert t_{n}^{C^{1}\cdot T}(f)-f (x)\Vert _{\infty } = \left\{ \begin{array}{ll} O((n+1)^{-\alpha }), &{} 0 < \alpha < 1 ,\\ O\left( (\log (n+1))/(n+1) \right) ,&{} \,\,\,\,\,\,\,\,\,\,\,\,\alpha = 1. \end{array} \right. \end{aligned}$$

(7)

Theorem 4

If $T\equiv (a_{n,k})$ be a lower triangular regular matrix with non-negative and non-decreasing (with respect to k) entries which satisfy condition (6), then the degree of approximation of a $2\pi$-periodic function with $r>1$ and $0<\beta \, s<1$ by $C^{1}\cdot T$ means of its Fourier series is given by

$$\begin{aligned} \Vert t_{n}^{C^{1}\cdot T}(f) - f(x)\Vert _{r} = O\left( (n+1)^{\beta + 1/r} \xi \left( 1/(n+1)\right) \right) , \end{aligned}$$

(8)

provided positive increasing function $\xi (t)$ satisfies the conditions:

$$\begin{aligned} \xi (t)/t \, \text{ be } \text{ a } \text{ decreasing } \text{ function }, \end{aligned}$$

(9)

$$\begin{aligned} \left( {\displaystyle \int _{0}^{\pi /(n+1)} \left( |\phi (t)| \sin ^{\beta }(t/2)/\xi (t) \right) ^{r}dt}\right) ^{1/r}= O((n+1)^{-1/r}), \end{aligned}$$

(10)

$$\begin{aligned} \left( {\int _{\pi /(n+1)}^{\pi } \left( t^{-\delta }|\phi (t)|\sin ^{\beta }(t/2)/\xi (t) \right) ^{r}dt} \right) ^{1/r}= O((n+1)^{\delta -1/r}), \end{aligned}$$

(11)

where $\delta$ is a real number such that $\beta +1/r>\delta >r^{-1}, r^{-1}+ s^{-1}=1,$ $r> 1.$ Also, conditions (10) and (11) hold uniformly in x.

Remark 1

If we take $a_{n,k}=p_{n-k}/P_{n}$ for $k\le n$ and $a_{n,k}=0$ for $k > n$ such that $P_{n}(= \sum _{k=0}^{n}p_{k}\ne 0)\rightarrow \infty$ as $n\rightarrow \infty$ and $P_{-1} = 0 = p_{-1}$, then $C^{1}\cdot T$ means reduce to $C^{1}\cdot N_{p}$ means and

$$ \sum _{r=\tau }^{n}A_{r,r-\tau } = \sum _{r=\tau }^{n} \sum _{k=r-\tau }^{r}a_{r,k} = \sum _{r=\tau }^{n} \sum _{k=r-\tau }^{r}(p_{r-k}/P_{r})= \sum _{r=\tau }^{n} ( P_{\tau }/P_{r} )=P_{\tau } \sum _{r=\tau }^{n}P_{r}^{-1}. $$

Therefore, condition (6) reduces to condition (2) and $t_{n}^{C^{1}\cdot T}$ means reduce to $t_{n}^{CN} means.$ Hence our Theorems 3 and 4 are generalization of Theorems 1 and 2, respectively.

4 Lemmas

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

Lemma 1

Let $\{a_{r,k}\}$ be a non-negative sequence of real numbers, then

$$(C^{1}\cdot T)_{n}(t)= O(n+1),\, \text{ for }\,\, 0 < t \le \pi /(n+1).$$

Proof

Using $\sin nt \le nt$ and $\sin (t/2) \ge t/ \pi$ for $0< t \le \pi /(n+1),$ we have

$$\begin{aligned} \left| (C^{1}\cdot T)_{n}(t)\right|= & {} (2\pi (n+1))^{-1}\left| \sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k} \, \sin ((r-k+1/2)t) / \sin (t/2) \right| \\= & {} (2\pi (n+1))^{-1}\sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k}\left| \sin ((r-k+1/2)t) / \sin (t/2) \right| \\\le & {} (2\pi (n+1))^{-1}\sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k} (r-k+1/2)t / (t/ \pi )\\\le & {} (4(n+1))^{-1}\sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k}(2r-2k+1)\\\le & {} (4(n+1))^{-1}\sum _{r=0}^{n}(2r+1)\sum _{k=0}^{r}a_{r,r-k}\\= & {} (4(n+1))^{-1}\sum _{r=0}^{n}(2r+1) A_{r,0}=O(n+1). \end{aligned}$$

Lemma 2

[4] If $\{a_{r,k}\}$ is a non-negative and non-decreasing (with respect to k) sequence, then for $0\le a<b\le \infty ,$ $0 < t\le \pi$ and for every r

$$\left| \sum _{k=a}^{b}a_{r,r-k} e^{i(r-k)t}\right| = O(A_{r,r-\tau }).$$

Lemma 3

If $\{a_{r,k}\}$ is non-negative and non-decreasing (with respect to k) sequence, then for $0 < t \le \pi$

$$\left| \displaystyle \sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k} e^{i(r-k)t}\right| = O(t^{-1})+O\left( \displaystyle \sum _{r=\tau }^{n} A_{r,r-\tau }\right) ,$$

holds uniformly in $\tau =[1/t].$

Proof

For $0 < t \le \pi ,$ we have

$$\begin{aligned} \left| \displaystyle \sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k} e^{i(r-k)t}\right|\le & {} \left| \sum _{r=0}^{\tau -1}\sum _{k=0}^{r}a_{r,r-k} e^{i(r-k)t}+\sum _{r=\tau }^{n}\sum _{k=0}^{r}a_{r,r-k} e^{i(r-k)t}\right| \\\le & {} \sum _{r=0}^{\tau -1}\sum _{k=0}^{r}a_{r,r-k}| e^{i(r-k)t}| +\left| \sum _{r=\tau }^{n}\sum _{k=0}^{r}a_{r,r-k} e^{i(r-k)t}\right| \\\le & {} \sum _{r=0}^{\tau -1}\sum _{k=0}^{r}a_{r,r-k}+\sum _{r=\tau }^{n} \left| \sum _{k=0}^{r}a_{r,r-k} e^{i(r-k)t}\right| \\\le & {} \sum _{r=0}^{\tau -1}1 + \sum _{r=\tau }^{n}O(A_{r,r-\tau }) = (\tau -1+1)+ O\left( \sum _{r=\tau }^{n}A_{r,r-\tau }\right) \\= & {} O(t^{-1})+O\left( \sum _{r=\tau }^{n}A_{r,r-\tau }\right) , \end{aligned}$$

in view of Lemma 2.

Lemma 4

If $\{a_{r,k}\}$ is non-negative and non-decreasing (with respect to k) sequence and satisfies the condition (6), then

$$ |(C^{1}\cdot T)_{n}(t)| = O\left( t^{-2}/(n+1)\right) +O(t^{-1}),\, \,\text{ for }\,\, \pi /(n+1) < t \le \pi . $$

Proof

Using $\sin (t/2) \ge t/ \pi$, for $\pi /(n+1) < t \le \pi$ and Lemma 3, we have

$$\begin{aligned} |(C^{1}\cdot T)_{n}(t)|= & {} (2\pi (n+1))^{-1}\left| \sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k} \sin ((r-k+1/2)t)/ \sin (t/2)\right| \\\le & {} (2\pi (n+1))^{-1}\left| \sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k} \sin ((r-k+1/2)t)/ ( t/ \pi ) \right| \\= & {} (2t(n+1))^{-1}\left| \sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k}\sin (r-k+1/2)t\right| \\\le & {} (2t(n+1))^{-1}\left| \sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k} e^{i(r-k+1/2)t}\right| \\= & {} (2t(n+1))^{-1}\left| e^{it/2}\sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k} e^{i(r-k)t}\right| \\= & {} (2t(n+1))^{-1}\left| \sum _{r=0}^{n}\sum _{k=0}^{r}a_{r,r-k} e^{i(r-k)t}\right| \\= & {} (2t(n+1))^{-1}\left| O(t^{-1})+O\left( \sum _{r=\tau }^{n}A_{r,r-\tau }\right) \right| = O\left( t^{-2}/(n+1)\right) +O(t^{-1}), \end{aligned}$$

in view of condition (6).

5 Proof of Theorem 3

Following Titchmarsh [14], we have

$$s_{n}(f;x)-f(x) = \frac{1}{2\pi }\int _{0}^{\pi } \phi (t)(\sin (n+1/2)t / \sin (t/2)) dt$$

Denoting $C^{1}\cdot T$ means of $\{s_{n}(f;x)\}$ by $t_{n}^{C^{1}\cdot T}(f),$ we write

$$\begin{aligned} t_{n}^{C^{1}\cdot T}(f)-f(x)= & {} \int _{0}^{\pi } \phi (t) (2\pi (n+1))^{-1} \sum _{r=0}^{n} \sum _{k=0}^{r}a_{r,r-k} \sin ((r-k+1/2)t)/ \sin (t/2) dt\nonumber \\= & {} \int _{0}^{\pi /(n+1)} \phi (t)(C^{1}\cdot T)_{n}(t)dt+\int _{\pi /(n+1)}^{\pi } \phi (t)(C^{1}\cdot T)_{n}(t)dt\nonumber \\ {}= & {} I_{1} + I_{2},\, \text{ say }. \end{aligned}$$

(12)

Using Lemma 1 and the fact that $f \in Lip \,\alpha \Rightarrow \phi \in Lip\,\alpha$ {[2], Lemma 5.27}, we have

$$\begin{aligned} |I_{1}|\le & {} \int _{0}^{\pi /(n+1)} |\phi (t)| |(C^{1}\cdot T)_{n}(t)| dt = O(n+1) \int _{0}^{\pi /(n+1)} t^{\alpha } dt\nonumber \\= & {} O(n+1) ((n+1)^{-\alpha - 1}) = O((n+1)^{-\alpha }). \end{aligned}$$

(13)

Now, using Lemma 4 and the fact that $f \in Lip \,\alpha \Rightarrow \phi \in Lip\, \alpha ,$

$$\begin{aligned} |I_{2}|\le & {} \int _{\pi /(n+1)}^{\pi } | \phi (t)|\,\, \left| (C^{1}\cdot T)_{n}(t)\right| dt \le \int _{\pi /(n+1)}^{\pi } | \phi (t)| O\left[ (t^{-2}/(n+1))+t^{-1}\right] dt\nonumber \\= & {} O(I_{21}) + O(I_{22}),\, \text{ say, } \end{aligned}$$

(14)

where

$$\begin{aligned} I_{21}= (n+1)^{-1} \int _{\pi /(n+1)}^{\pi } t^{\alpha -2} dt = \left\{ \begin{array}{ll} O((n+1)^{-\alpha }), &{} 0 < \alpha < 1 , \\ O \left( \log (n+1) /(n+1)\right) ,&{} \,\,\,\,\,\,\,\,\,\,\,\, \alpha = 1. \end{array} \right. \end{aligned}$$

(15)

and

$$\begin{aligned} I_{22} = O\left( \int _{\pi /(n+1)}^{\pi } t^{\alpha -1} dt\right) = O((n+1)^{-\alpha }). \end{aligned}$$

(16)

Collecting (12)–(16), we get

$$ t_{n}^{C^{1}\cdot T}(f)-f(x) = \left\{ \begin{array}{ll} O((n+1)^{-\alpha }), &{} 0 < \alpha < 1,\\ O(\log (n+1) /(n+1)),&{} \,\,\,\,\,\,\,\,\,\,\,\, \alpha = 1. \end{array} \right. $$

Thus

$$ \Vert t_{n}^{C^{1}\cdot T}(f) - f \Vert _{\infty } = \displaystyle \sup _{0 \le x \le 2\pi }\{ | t_{n}^{C^{1}\cdot T}(x) - f(x)|\} = \left\{ \begin{array}{ll} O((n+1)^{-\alpha }), &{} 0 < \alpha < 1 ,\\ O((\log (n+1))/(n+1)),&{} \,\,\,\,\,\,\,\,\,\,\,\,\alpha = 1. \end{array} \right. $$

6 Proof of Theorem 4

Following the proof of Theorem 3,

$$\begin{aligned} t_{n}^{C^{1}\cdot T}(f) - f(x)= & {} \int _{0}^{\pi /(n+1)} \phi (t)(C^{1}\cdot T)_{n}(t)dt+\int _{\pi /(n+1)}^{\pi } \phi (t)(C^{1}\cdot T)_{n}(t)dt\nonumber \\= & {} I_{1}^{'} + I_{2}^{'}, \,\text{ say }. \end{aligned}$$

(17)

Using Hölder’s inequality, $\phi (t) \in W(L^{r},\xi (t)),$ condition (10), Lemma 1 and mean value theorem for integrals, we have

$$\begin{aligned} |I_{1}^{'}|&= \left| \lim _{ \varepsilon \rightarrow 0} \int _{ \varepsilon }^{\pi /(n+1)} \left[ (\phi (t)\sin ^{\beta } (t/2) /\xi (t))\cdot (\xi (t) (C^{1}\cdot T)_{n}(t))/(\sin ^{\beta } (t/2) )\right] dt\right| \nonumber \\&\le \left[ \int _{0}^{\pi / (n+1)} \left( |\phi (t)| \sin ^{\beta } (t/2) / \xi (t)\right) ^{r}dt\right] ^{1/r}\nonumber \\&\cdot \left[ \lim _{ \varepsilon \rightarrow 0} \int _{ \varepsilon }^{\pi /(n+1)} \left( \xi (t)|(C^{1}\cdot T)_{n}(t)| / ( \sin ^{\beta } (t/2) \right) ^{s}dt\right] ^{1/s}\nonumber \\&= O((n+1)^{-1/r})\left[ \displaystyle \lim _{\varepsilon \rightarrow 0} \int _{\varepsilon }^{\pi /(n+1)} \left| \xi (t) (n+1) /(\sin ^{\beta }(t/2)) \right| ^{s}dt \right] ^{1/s}\nonumber \\&= O(n+1)^{1-1/r}(\xi (\pi /(n+1)) \left[ \lim _{ \varepsilon \rightarrow 0} \int _{\varepsilon }^{\pi /(n+1)} t^{-\beta s} dt \right] ^{1/s} \nonumber \\&= O(\xi (1/(n\,+\,1) (n\,+\,1)^{\beta +1-1/r-1/s}) =O((n+1)^{\beta } \xi (1/(n+1)), \end{aligned}$$

(18)

in view of condition (9), i.e. $(\xi (\pi /(n\,+\,1))/(\pi /(n\,+\,1))) \le (\xi ( 1/(n\,+\,1))/(1/(n\,+\,1))).$

Using Lemma 4, we have

$$\begin{aligned} | I_{2}^{'}|= & {} \left[ \int _{\pi /(n+1)}^{\pi } |\phi (t)| \left[ O \left( t^{-2} / (n+1)\right) + O\left( t^{-1}\right) \right] dt\right] \nonumber \\= & {} O\left[ \int _{\pi /(n+1)}^{\pi }t^{-2} | \phi (t)|/ (n+1) dt\right] + O \left[ \int _{\pi /(n+1)}^{\pi }t^{-1}| \phi (t)| dt\right] \nonumber \\= & {} O(I_{21}^{'})+ O(I_{22}^{'}). \end{aligned}$$

(19)

Using Hölder’s inequality, $| \sin t| \le 1,\, \sin (t/2) \ge (t/\pi )$ and condition (11), we have

$$\begin{aligned} |I_{21}^{'}|&= (n+1)^{-1} \left[ \int _{\pi /(n+1)}^{\pi }\left\{ (t^{-\delta }| \phi (t)| \sin ^{\beta }(t/2) / \xi (t)) \,\cdot (\xi (t) / (t^{-\delta + 2}\,\sin ^{\beta }(t/2))) \right\} dt \right] \nonumber \\&\le ((n+1)^{-1}) \left[ \int _{\pi /(n+1)}^{\pi }\left| t^{-\delta } | \phi (t)| \sin ^{\beta }(t/2) / \xi (t)\right| ^{r} dt \right] ^{1/r}\nonumber \\&\quad \quad \cdot \left[ \int _{\pi /(n+1)}^{\pi }\left| \xi (t)/\left( t^{-\delta + 2}\sin ^{\beta }(t/2)\right) \right| ^{s}dt\right] ^{1/s}\nonumber \\&= O((n+1)^{-1}) \left[ \int _{\pi /(n+1)}^{\pi }\left| t^{-\delta } | \phi (t)| \sin ^{\beta }(t/2)/ \xi (t)\right| ^{r} dt \right] ^{1/r} \nonumber \\&\cdot \left[ \int _{\pi /(n+1)}^{\pi }\left| \xi (t) /\left( t^{-\delta + 2}\sin ^{\beta }(t/2)\right) \right| ^{s}dt\right] ^{1/s}\nonumber \\&= O((n+1)^{-1}) O\left( (n+1)^{\delta -1/r}\right) \left[ \int _{\pi /(n+1)}^{\pi }\left| \xi (t)/ \left( t^{-\delta + 2} \sin ^{\beta }(t/2)\right) \right| ^{s}dt \right] ^{1/s}\nonumber \\&= O((n+1)^{\delta -1-1/r})\left[ \int _{\pi /(n+1)}^{\pi }\left( \xi (t) / t^{-\delta + 2 + \beta } \right) ^{s}dt\right] ^{1/s}\nonumber \\&= O((n+1)^{\delta -1/r})\xi (\pi /(n+1)) \left[ \int _{\pi /(n+1)}^{\pi } t^{-(-\delta + 1 + \beta )s}dt\right] ^{1/s}\nonumber \\&= O((n+1)^{\delta -1/r})\xi (\pi /(n+1)) \left[ (n+1)^{(-\delta +1 + \beta )-1/s}dt\right] \nonumber \\&= O(\xi (1/(n+1) (n+1)^{\beta }) \end{aligned}$$

(20)

in view of decreasing nature of $\xi (t)/t$ and $r^{-1}+s^{-1}=1.$

Similarly, as above, we have

$$\begin{aligned} |I_{22}^{'}|= & {} \int _{\pi /(n+1)}^{\pi } t^{-1}| \phi (t)| dt = \int _{\pi /(n+1)}^{\pi }\left( t^{-\delta }|\phi (t)| \sin ^{\beta }(t/2) / \xi (t)\right) \left( \xi (t) / ( t^{1-\delta } \sin ^{\beta }(t/2) ) \right) dt\nonumber \\\le & {} \left[ \int _{\pi /(n+1)}^{\pi } \left| t^{-\delta }| \phi (t)|\sin ^{\beta }(t/2) / \xi (t) \right| ^{r}dt\right] ^{1/r}\left[ \int _{\pi /(n+1)}^{\pi } \left| \xi (t) / \left( t^{1-\delta } \sin ^{\beta }(t/2) \right) \right| ^{s} dt\right] ^{1/s}\nonumber \\= & {} O\left( (n+1)^{\delta -1/r}\right) \left[ \int _{\pi /(n+1)}^{\pi } \left( \xi (t) / t^{1-\delta + \beta } \right) ^{s} dt\right] ^{1/s}\nonumber \\= & {} O\left( (n+1)^{\delta +1-1/r}\right) \xi (1/(n+1)) \left[ \int _{\pi /(n+1)}^{\pi }t^{(\delta -\beta )s }dt\right] ^{1/s}\nonumber \\= & {} O\left( (n+1)^{\delta +1-1/r}\right) \xi (1/(n+1)) (n+1)^{(-\delta +\beta )-1/s }\nonumber \\= & {} O(\xi (1/(n+1)) (n+1)^{ \beta +1-1/r-1/s }\nonumber \\= & {} O(\xi (1/(n+1)) (n+1)^{\beta }. \end{aligned}$$

(21)

Collecting (17)–(21), we have

$$ |t_{n}^{C^{1}\cdot T}(f)-f(x)|= O\left( (n+1)^{\beta }\xi (1/(n+1))\right) . $$

Hence,

$$\begin{aligned} \Vert t_{n}^{C^{1}\cdot T}(f) - f(x)\Vert _{r}= & {} \left( 1/2\pi \int _{0}^{2\pi }| t_{n}^{C^{1}\cdot T}(f) - f(x)|^{r}dx\right) ^{1/r} = O\left( (n+1)^{\beta }\xi \left( 1/(n+1 \right) \right) . \end{aligned}$$

Remark 2

The proof of Theorem 3, for $r=1,$ i.e. $s=\infty$ can be written by using sup norm while using Hölder’s inequality.

7 Corollaries

The following corollaries can be derived from Theorem 4

1. If $\beta =0$ , then for $f\in Lip (\xi (t),r), \Vert t_{n}^{C^{1}\cdot T}(f) - f(x)\Vert _{r} = O \left( \xi (1/n)\right) .$

2. If $\beta =0, \xi (t)=t^{\alpha }(0 < \alpha \le 1),$ then for $f \in Lip(\alpha ,r)(\alpha > 1/r),$

$$\begin{aligned} \Vert t_{n}^{C^{1}\cdot T}(f) - f(x)\Vert _{r} = O\left( n^{-\alpha }\right) . \end{aligned}$$

(22)

3. If $r\rightarrow \infty$ in Corollary 2, then for $f\in Lip \alpha (0 < \alpha < 1),$ (22) gives

$$\Vert t_{n}^{C^{1}\cdot T}(f) - f(x)\Vert _{\infty } = O(n^{-\alpha }).$$

Remark 3

In view of Remark 2, corollaries of Lal [5, p. 350] are particular cases of our Corollaries 2 and 3, respectively.