Trigonometric approximation of functions belonging to Lipschitz class by matrix ($C^1\cdot N_p$) operator of conjugate series of Fourier series

Abstract

In the present paper, a new theorem on the degree of approximation of a function $\tilde{f}$, conjugate to a 2π periodic function f belonging to the Lipα ($0<\alpha \le 1$) class without the monotonicity condition on the generating sequence $\{p_n\}$ has been established, which in turn generalizes the results of Lal (Appl. Math. Comput. 209: 346-350, 2009) on a Fourier series.

MSC:40G05, 41A10, 42B05, 42B08.

1 Introduction

The degree of approximation of functions belonging to Lipα, $Lip(\alpha ,r)$, $Lip(\xi (t),r)$ and $W(L_r,\xi (t))$, $(r\ge 1)$-classes through trigonometric Fourier approximation using different summability matrices with monotone rows has been proved by various investigators like Khan [1], Mittal et al. [2, 3], Mittal, Rhoades and Mishra [4], Qureshi [5], Chandra [6], Leindler [7], Rhoades et al. [8]. Recently Lal [9] has proved a theorem on the degree of approximation of a function f belonging to the Lipα ($0<\alpha \le 1$) class by $C^1\cdot N_p$ summability method of its Fourier series. Lal [9] has assumed monotonicity on the generating sequence $\{p_n\}$. The approximation of a function $\tilde{f}(x)$, conjugate to a 2π periodic function to $f\in Lip\alpha$ ($0<\alpha \le 1$) using product ($C^1\cdot N_p$)-summability has not been studied so far. In this paper, we obtain a new theorem on the degree of approximation of a function $\tilde{f}$, conjugate to a 2π periodic function $f\in Lip\alpha$ ($0<\alpha \le 1$) class without monotonicity condition on the generating sequence $\{p_n\}$.

Let ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$ be a given infinite series with the sequence of n th partial sums $\{s_n\}$. Let $\{p_n\}$ be a non-negative sequence of constants, real (R) or complex, and let us write

$$P_n=\sum _{k=0}^n{p}_k\ne 0\mathrm{\forall }n\ge 0,p_{-1}=0=P_{-1}\text{ and }{P}_n\to \mathrm{\infty }\text{ as }n\to \mathrm{\infty }.$$

The sequence to sequence transformation $t_n^N={\sum }_{\nu =0}^n{p}_{n-\nu }{s}_{\nu }/P_n$ defines the sequence $\{t_n^N\}$ of Nörlund means of the sequence $\{s_n\}$, generated by the sequence of coefficients $\{p_n\}$. The series ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$ is said to be $N_p$ summable to the sum s if ${lim}_{n\to \mathrm{\infty }}{t}_n^N$ exists and is equal to a finite number s. In the special case, in which

$$p_n=\left(\begin{array}{c}n+\alpha -1\\ \alpha -1\end{array}\right)=\frac{(n+\alpha )}{(n+1)(\alpha )}(\alpha >0),$$

the Nörlund summability $N_p$ reduces to the familiar $C^{\alpha }$ summability.

The product of $C^1$ summability with a $N_p$ summability defines $C^1\cdot N_p$ summability. Thus the $C^1\cdot N_p$ mean is given by $t_n^{CN}=\frac{1}{n+1}{\sum }_{k=0}^n{P}_k^{-1}{\sum }_{\nu =0}^k{p}_{k-\nu }{s}_{\nu }$.

If $t_n^{CN}\to s$ as $n\to \mathrm{\infty }$, then the infinite series ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$ or the sequence $\{s_n\}$ is said to be summable $C^1\cdot N_p$ to the sum s if ${lim}_{n\to \mathrm{\infty }}{t}_n^{CN}$ exists and is equal to s.

$$\begin{array}{rl}{s}_n\to s& \Rightarrow N_p(s_n)=t_n^N=P_n^{-1}\sum _{\nu =0}^n{p}_{n-\nu }{s}_{\nu }\to s,\text{as }n\to \mathrm{\infty },N_p\text{ method is regular,}\\ \Rightarrow C^1(N_p(s_n))=t_n^{CN}\to s,\text{as }n\to \mathrm{\infty },C^1\text{ method is regular,}\\ \Rightarrow C^1\cdot N_p\text{ method is regular.}\end{array}$$

Let $f(x)$ be a 2π-periodic function and Lebesgue integrable. The Fourier series of $f(x)$ is given by

$$f(x)\sim \frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}(a_{n}cosnx+b_{n}sinnx)\equiv \sum _{n=0}^{\mathrm{\infty }}{A}_n(x),\mathrm{\forall }n\ge 0,$$

(1.1)

with $(n+1)$th partial sum $s_n(f;x)$ called the trigonometric polynomial of degree (order) n of the first ($n+1$) terms of the Fourier series of f.

The conjugate series of Fourier series (1.1) is given by

$$\sum _{n=1}^{\mathrm{\infty }}(b_{n}cosnx-a_{n}sinnx)\equiv \sum _{n=1}^{\mathrm{\infty }}{B}_n(x).$$

(1.2)

A function $f(x)\in Lip\alpha$ if

$$f(x+t)-f(x)=O\left(\right|t^{\alpha }\left|\right)\text{for }0<\alpha \le 1,t>0.$$

$L_{\mathrm{\infty }}$-norm of a function $f:R\to R$ is defined by ${\parallel f\parallel }_{\mathrm{\infty }}=sup\{|f(x)|:x\in R\}$.

The degree of approximation of a function $f:R\to R$ by the trigonometric polynomial $t_n$ of order n under the sup norm ${\parallel \parallel }_{\mathrm{\infty }}$ is defined by [10]

$${\parallel t_n-f\parallel }_{\mathrm{\infty }}=sup\left\{\right|t_n(x)-f(x)|:x\in R\}$$

and $E_n(f)$ of a function $f\in L_r$ is given by $E_n(f)={min}_n{\parallel t_n-f\parallel }_r$.

The conjugate function $\tilde{f}(x)$ is defined for almost every x by

$$\begin{array}{rl}\tilde{f}(x)& =-\frac{1}{2\pi }{\int }_0^{\pi }\psi (t)cott/2dt\\ =\underset{h\to 0}{lim}(-\frac{1}{2\pi }{\int }_h^{\pi }\psi (t)cott/2dt)\text{(see [11, Definition~1.10])}.\end{array}$$

We note that $t_n^N$ and $t_n^{CN}$ are also trigonometric polynomials of degree (or order) n.

Abel’s transformation: The formula

$$\sum _{k=m}^n{u}_k{v}_k=\sum _{k=m}^{n-1}{U}_k(v_k-v_{k+1})-U_{m-1}{v}_m+U_n{v}_n,$$

(1.3)

where $0\le m\le n$, $U_k=u_0+u_1+u_2+\cdots +u_k$, if $k\ge 0$, $U_{-1}=0$, which can be verified, is known as Abel’s transformation and will be used extensively in what follows.

If $v_m,v_{m+1},\dots ,v_n$ are non-negative and non-increasing, the left-hand side of (1.3) does not exceed $2v_m{max}_{m-1\le k\le n}|U_k|$ in absolute value. In fact,

$$\begin{array}{rl}\left|\sum _{k=m}^n{u}_k{v}_k\right|& \le max|U_k|\{\sum _{k=m}^{n-1}(v_k-v_{k+1})+v_m+v_n\}\\ =2v_{m}max|U_k|.\end{array}$$

(1.4)

We write throughout the paper

$$\begin{array}{r}{\psi }_x(t)=\psi (t)=f(x+t)-f(x-t),\\ {(\widetilde{CN})}_n(t)=\frac{1}{2\pi (n+1)}\sum _{k=0}^n{P}_k^{-1}\sum _{\nu =0}^k{p}_{\nu }\frac{cos(k-v+1/2)t}{sint/2},\end{array}$$

(1.5)

$\tau =[1/t]$, where τ denotes the greatest integer not exceeding $1/t$, $P_{\tau }=P[1/t]$, $\mathrm{\Delta }{p}_k=p_k-p_{k+1}$.

2 Known results

In a recent paper Lal [9] obtained a theorem on the degree of approximation for a function belonging to the Lipschitz class Lipα using Cesàro-Nörlund $(C^1\cdot N_p)$-summability means of its Fourier series with non-increasing weights $\{p_n\}$. He proved the following theorem.

Theorem 2.1 Let $N_p$ be a regular Nörlund method defined by a sequence $\{p_n\}$ such that

$$P_{\tau }\sum _{\nu =\tau }^n{P}_{\nu }^{-1}=O(n+1).$$

(2.1)

Let $f\in L^1[0,2\pi ]$ be a 2π-periodic function belonging to Lipα ($0<\alpha \le 1$), then the degree of approximation of f by $C^1\cdot N_p$ means of its Fourier series (1.1) is given by

$$\underset{0\le x\le 2\pi }{sup}|t_n^{CN}(x)-f(x)|={\parallel t_n^{CN}-f\parallel }_{\mathrm{\infty }}=\{\begin{array}{ll}O({(n+1)}^{-\alpha }),& 0<\alpha <1,\\ O(log(n+1)\pi e/(n+1)),& \alpha =1.\end{array}$$

(2.2)

Remark 1 In the proof of Theorem 2.1 of Lal [5, p.349], the estimate for the case $\alpha =1$ is obtained as

$$O\left(\frac{1}{n+1}\right)+O\left(\frac{log(n+1)\pi }{n+1}\right)=O\left(\frac{loge}{n+1}\right)+O\left(\frac{log(n+1)\pi }{n+1}\right)=O\left(\frac{log(n+1)\pi e}{n+1}\right).$$

Since $1/(n+1)\le log((n+1)\pi )/(n+1)$, the e is not needed in (2.2) for the case $\alpha =1$ (cf. [[8], p.6870]).

Remark 2 Lal [9] has used the monotonicity condition on the generating sequence $\{p_n\}$ in the proof of Theorem 2.1 but has not mentioned it in the statement.

3 Main theorem

The theory of approximation is a very extensive field and the study of theory of trigonometric approximation is of great mathematical interest and of great practical importance. It is well known that the theory of approximations, i.e., TFA, which originated from a well-known theorem of Weierstrass, has become an exciting interdisciplinary field of study for the last 130 years. These approximations have assumed important new dimensions due to their wide applications in signal analysis [12] in general and in digital signal processing [13] in particular, in view of the classical Shannon sampling theorem. Mittal et al. [2–4, 14] have obtained many interesting results on TFA using summability methods without monotonicity on the rows of the matrix T: a digital filter. Broadly speaking, signals are treated as functions of one variable and images are represented by functions of two variables. But till now, nothing seems to have been done so far to obtain the degree of approximation of conjugate of a function using $C^1\cdot N_p$ product summability method of its conjugate series of a Fourier series. The observations of Remarks 1 and 2 motivated us to determine a proper set of conditions to prove Theorem 2.1 on the conjugate series of its Fourier series. The series, conjugate to a Fourier series, is not necessarily a Fourier series. Hence a separate study of conjugate series is desirable, which attracted the attention of researchers.

Therefore, the purpose of present paper is to establish a quite new theorem on the degree of approximation of a function $\tilde{f}(x)$, conjugate to a 2π-periodic function f belonging to the Lipα ($0<\alpha \le 1$) class by $C^1\cdot N_p$ means of conjugate series of its Fourier series without monotonicity on the generating sequence $\{p_n\}$ (that is, weakening the conditions on the filter, we improve the quality of a digital filter [[2], p.4485]). More precisely, we prove the following theorem.

Theorem 3.1 Let $N_p$ be the regular Nörlund summability matrix generated by the non-negative $\{p_n\}$ such that

$$(n+1)p_n=O(P_n),\mathrm{\forall }n\ge 0.$$

(3.1)

Let $f\in L^1[0,2\pi ]$ be a 2π-periodic signal (function). Then the degree of approximation of $\tilde{f}(x)$, conjugate to $f\in Lip\alpha$ ($0<\alpha \le 1$) by $C^1\cdot N_p$ means of conjugate series of its Fourier series, is given by

$$\begin{array}{rl}{\parallel {\tilde{t}}_n^{CN}(f;x)-\tilde{f}(x)\parallel }_{\mathrm{\infty }}& =\underset{0\le x\le 2\pi }{sup}|{\tilde{t}}_n^{CN}(f;x)-\tilde{f}(x)|\\ =\{\begin{array}{ll}O({(n+1)}^{-\alpha }),& 0<\alpha <1,\\ O(\frac{log(n+1)}{n+1}),& \alpha =1.\end{array}\end{array}$$

(3.2)

Remark 3 For a non-increasing sequence $\{p_n\}$, we get

$$P_n=\sum _{k=0}^n{p}_k\ge p_n\sum _{k=0}^{n}1=(n+1)p_n,\mathit{\text{i.e.}}(n+1)p_n=O(P_n).$$

Thus the condition (3.1) holds for a non-increasing sequence $\{p_n\}$. Hence our Theorem 3.1 generalizes Theorem 2.1 on conjugate series of its Fourier series.

Note 1 The product transform $C^1\cdot N_p$ plays an important role in signal theory as a double digital filter [14].

4 Lemmas

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

Lemma 1 If $P_n$ is positive and $P_n^{-1}\ge P_{n+1}^{-1}$ $\mathrm{\forall }n\ge 0$, then for $0\le a<b\le \mathrm{\infty }$, $0<t\le \pi$ and for any n, we have

$$\left|\sum _{k=a}^b{P}_k^{-1}{e}^{i(n-k)t}\right|=\{\begin{array}{ll}O(t^{-1})& \mathit{\text{for any}}a,\\ O(t^{-1}{P}_a^{-1})& \mathit{\text{for}}a\ge [t^{-1}].\end{array}$$

Proof Let $\tau =[t^{-1}]$. Then

$$\left|\sum _{k=a}^b{P}_k^{-1}{e}^{i(n-k)t}\right|=\left|e^{int}\sum _{k=a}^b{P}_k^{-1}{e}^{-ikt}\right|\le \left|\sum _{k=a}^{\tau -1}{P}_k^{-1}{e}^{-ikt}\right|+\left|\sum _{k=\tau }^b{P}_k^{-1}{e}^{-ikt}\right|;$$

but

$$\left|\sum _{k=a}^{\tau -1}{P}_k^{-1}{e}^{-ikt}\right|\le \left|e^{-iat}\right|\left|\sum _{k=a}^{\tau -1}{P}_k^{-1}\right|\le \left|\sum _{k=0}^{\tau -1}{P}_k^{-1}\right|\le \frac{\tau }{p_0}=O\left(t^{-1}\right),$$

and, by (1.4), we have

$$\begin{array}{rl}\left|\sum _{k=\tau }^b{P}_k^{-1}{e}^{-ikt}\right|& \le 2P_{\tau }^{-1}\underset{\tau +1\le k\le b}{max}\left|\frac{1-e^{-i(k+1)t}}{1-e^{-it}}\right|\le 4P_{\tau }^{-1}\left|\frac{e^{it/2}}{e^{it/2}-e^{-it/2}}\right|\\ \le 2P_{\tau }^{-1}\left(\frac{1}{sin(t/2)}\right)=O\left(t^{-1}{P}_{\tau }^{-1}\right).\end{array}$$

Since $P_n>0$ and $P_n^{-1}\ge P_{n+1}^{-1}$ $\mathrm{\forall }n\ge 0$, we have

$$t^{-1}{P}_{t^{-1}}^{-1}\le (\left[t^{-1}\right]+1)P_{[t^{-1}]}^{-1}\le P_{[t^{-1}]}^{-1}\le P^{-1}\left(t^{-1}\right),$$

and, in case $a\ge [t^{-1}]$, we would have

$$\left|\sum _{k=a}^b{P}_k^{-1}{e}^{-ikt}\right|\le 2P_a^{-1}\underset{a\le k\le b}{max}\left|\frac{1-e^{-i(k+1)t}}{1-e^{-it}}\right|\le C{t}^{-1}{P}_a^{-1}=O\left(t^{-1}{P}_a^{-1}\right).$$

This completes the proof of Lemma 1. □

Lemma 2 $|{(\widetilde{CN})}_n(t)|=O[1/t]$ for $0<t\le \pi /(n+1)$.

Proof For $0<t\le \pi /(n+1)$, $sin(t/2)\ge (t/\pi )$ and $|cosnt|\le 1$.

$$\begin{array}{rl}|{(\widetilde{CN})}_n(t)|& =\left|\frac{1}{2\pi (n+1)}\sum _{k=0}^n{P}_k^{-1}\sum _{v=0}^k{p}_{\nu }\frac{cos(k-v+1/2)t}{sint/2}\right|\\ \le \frac{1}{2\pi (n+1)}\sum _{k=0}^n{P}_k^{-1}\sum _{v=0}^k{p}_{\nu }\frac{|cos(k-v+1/2)t|}{|sint/2|}\\ \le \frac{1}{2t(n+1)}\sum _{k=0}^n{P}_k^{-1}\sum _{v=0}^k{p}_{\nu }\\ =\frac{1}{2t(n+1)}\sum _{k=0}^n{P}_k^{-1}{P}_k\\ =O[\tau ].\end{array}$$

This completes the proof of Lemma 2. □

Lemma 3 Let $\{p_n\}$ be a non-negative sequence satisfying (3.1), then

$$|{(\widetilde{CN})}_n(t)|=O(\frac{{\tau }^2}{(n+1)}+\tau ),\mathit{\text{uniformly in}}\frac{\pi }{(n+1)}<t\le \pi .$$

Proof For $\pi /(n+1)<t\le \pi$, we have

$$\begin{array}{rl}{(\widetilde{CN})}_n(t)& =\frac{1}{2\pi (n+1)}\sum _{k=0}^n{P}_k^{-1}\sum _{v=0}^k{p}_{\nu }\frac{cos(k-v+1/2)t}{sint/2}\\ =\frac{1}{2\pi (n+1)}(\sum _{k=0}^{\tau }+\sum _{k=\tau +1}^n)P_k^{-1}\sum _{v=0}^k{p}_{\nu }\frac{cos(k-v+1/2)t}{sint/2}\\ ={\tilde{J}}_1(n,t)+{\tilde{J}}_2(n,t),\text{say,}\end{array}$$

(4.1)

where

$$\begin{array}{rl}|{\tilde{J}}_1(n,t)|& =\left|\frac{1}{2\pi (n+1)}\sum _{k=0}^{\tau }{P}_k^{-1}\sum _{v=0}^k{p}_{\nu }\frac{cos(k-v+1/2)t}{sint/2}\right|\\ \le \frac{1}{2\pi (n+1)}\sum _{k=0}^{\tau }{P}_k^{-1}\sum _{v=0}^k{p}_{\nu }\frac{|cos(k-v+1/2)t|}{|sint/2|}\\ \le \frac{1}{2t(n+1)}\sum _{k=0}^{\tau }{P}_k^{-1}\sum _{v=0}^k{p}_{\nu }\\ =O\left(\frac{{\tau }^2}{(n+1)}\right),\end{array}$$

(4.2)

in view of ${(sint/2)}^{-1}\le \pi /t$, for $0<t\le \pi$.

Again, using ${(sint/2)}^{-1}\le \pi /t$, for $0<t\le \pi$ and changing the order of summation, we find

$$\begin{array}{rl}|{\tilde{J}}_2(n,t)|=& \left|\frac{1}{2\pi (n+1)}\sum _{k=\tau +1}^n{P}_k^{-1}\sum _{v=0}^k{p}_{\nu }\frac{cos(k-v+1/2)t}{sint/2}\right|\\ \le & \frac{1}{2t(n+1)}|\sum _{k=\tau +1}^n{P}_k^{-1}\sum _{\nu =0}^k{p}_{\nu }cos(k-\nu +1/2)t|\\ =& O\left(\frac{\tau }{n+1}\right)\left|\right(\sum _{\nu =0}^{\tau +1}{p}_{\nu }\sum _{k=\tau +1}^n{P}_k^{-1}cos(k-\nu +1/2)t\\ +\sum _{\nu =\tau +1}^n{p}_{\nu }\sum _{k=\nu }^n{P}_k^{-1}cos(k-\nu +1/2)t\left)\right|.\end{array}$$

(4.3)

Using Lemma 1, we have

$$\begin{array}{r}|\sum _{\nu =0}^{\tau +1}{p}_{\nu }\sum _{k=\tau +1}^n{P}_k^{-1}cos(k-\nu +1/2)t|\\ \le \left(\sum _{\nu =0}^{\tau +1}{p}_{\nu }\right|\sum _{k=\tau +1}^n{P}_k^{-1}{e}^{i(k-\nu )t}\cdot e^{it/2}\left|\right)\\ =\sum _{\nu =0}^{\tau +1}{p}_{\nu }\left|\sum _{k=\tau +1}^n{P}_k^{-1}{e}^{i(k-\nu )t}\right|=\sum _{\nu =0}^{\tau +1}{p}_{\nu }O\left(\tau P_{\tau +1}^{-1}\right)=O(\tau ).\end{array}$$

(4.4)

Using Abel’s transformation, we obtain

$$\begin{array}{rl}\sum _{k=\nu }^n{P}_k^{-1}cos(k-\nu +1/2)t=& \sum _{k=\nu }^{n-1}\left(\mathrm{\Delta }{P}_k^{-1}\right)\sum _{\gamma =0}^{k}cos(k-\gamma +1/2)t+P_n^{-1}\sum _{\gamma =0}^{n}cos(k-\gamma +1/2)t\\ -P_{\nu }^{-1}\sum _{\gamma =0}^{\nu -1}cos(k-\gamma +1/2)t.\end{array}$$

Therefore, we have

$$\begin{array}{r}|\sum _{k=\nu }^n{P}_k^{-1}cos(k-\nu +1/2)t|\\ \le \sum _{k=\nu }^{n-1}\left|\mathrm{\Delta }{P}_k^{-1}\right||\sum _{\gamma =0}^{k}cos(k-\gamma +1/2)t|+P_n^{-1}|\sum _{\gamma =0}^{n}cos(k-\gamma +1/2)t|\\ +P_{\nu }^{-1}|\sum _{\gamma =0}^{\nu -1}cos(k-\gamma +1/2)t|\\ =O(\tau )\left(\sum _{k=\nu }^{n-1}\right|\mathrm{\Delta }{P}_k^{-1}|+P_n^{-1}+P_{\nu }^{-1})=O(\tau )(P_n^{-1}+P_{\nu }^{-1}),\end{array}$$

(4.5)

by virtue of the fact that ${\sum }_{k=\lambda }^{\mu }exp(-ikt)=O(\tau )$, $0\le \lambda \le k\le \mu$, and $P_n\ge P_{n-1}$ $\mathrm{\forall }n\ge 0$.

On combining (4.3) to (4.5), we get

$$\begin{array}{rl}|{\tilde{J}}_2(n,t)|& =O\left(\frac{{\tau }^2}{n+1}\right)(1+\sum _{\nu =\tau +1}^n{p}_{\nu }(P_n^{-1}+P_{\nu }^{-1}))\\ =O\left(\frac{{\tau }^2}{n+1}\right)(1+P_n^{-1}\sum _{\nu =0}^n{p}_{\nu }+\sum _{\nu =\tau +1}^n\frac{p_{\nu }}{P_{\nu }})\\ =O\left(\frac{{\tau }^2}{n+1}\right)(1+1+\sum _{\nu =\tau +1}^n\frac{1}{\nu +1})=O\left(\frac{{\tau }^2}{n+1}\right)(2+O\left(\frac{n-\tau }{\tau +1}\right))\\ =O\left(\frac{{\tau }^2}{n+1}\right)+O(\left(\frac{{\tau }^2}{n+1}\right)\cdot \left(\frac{n}{\tau +1}\right))=O(\frac{{\tau }^2}{n+1}+\tau ),\end{array}$$

(4.6)

in view of (3.1) and $\tau \le 1/t<\tau +1$.

Finally, collecting (4.1), (4.2) and (4.6) yields Lemma 3.

This completes the proof of Lemma 3. □

5 Proof of the theorem

Let ${\tilde{s}}_n(f;x)$ denote the partial sum of series (1.2), then we have

$${\tilde{s}}_n(f;x)-\tilde{f}(x)=\frac{1}{2\pi }{\int }_0^{\pi }{\psi }_x(t)\frac{cos(n+1/2)t}{sint/2}dt.$$

Denoting $C^1\cdot N_p$ means of $\{\widetilde{s_n}(f;x)\}$ by ${\tilde{t}}_n^{CN}$, we write

$$\begin{array}{rl}{\tilde{t}}_n^{CN}(f;x)-\tilde{f}(x)& ={\int }_0^{\pi }{\psi }_x(t)\frac{1}{2\pi (n+1)}\sum _{k=0}^n{P}_k^{-1}\sum _{\nu =0}^k{p}_{\nu }\frac{cos(k-v+1/2)t}{sint/2}dt\\ ={\int }_0^{\pi }{\psi }_x(t){(\widetilde{CN})}_n(t)dt\\ =[{\int }_0^{\pi /(n+1)}+{\int }_{\pi /(n+1)}^{\pi }]{\psi }_x(t){(\widetilde{CN})}_n(t)dt=I_1+I_2\text{(say)}\end{array}$$

(5.1)

If $f(x)\in Lip\alpha$, then

$$\begin{array}{rl}|{\psi }_x(t+h)-{\psi }_x(t)|& =|f(x+t+h)-f(x+t)+f(x-t)-f(x-t-h)|\\ \le |f(x+t+h)-f(x+t)|+|f(x-t)-f(x-t-h)|\\ \le C|h{|}^{\alpha }.\end{array}$$

Therefore ${\psi }_x(t)\in Lip\alpha$.

Now, using Lemma 2, we have

$$|I_1|=O({\int }_0^{\pi /(n+1)}{t}^{\alpha }\cdot \frac{1}{t}dt)=O\left({(n+1)}^{-\alpha }\right).$$

(5.2)

Using Lemma 3, we obtain

$$|I_2|=O\left\{{\int }_{\pi /(n+1)}^{\pi }{t}^{\alpha }(\frac{{\tau }^2}{n+1}+\tau )dt\right\}=O(I_{21})+O(I_{22}),\text{say,}$$

(5.3)

where

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

(5.4)

and

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

(5.5)

On combining (5.1) with (5.5) and using the inequality $1/(n+1)\le log(n+1)/(n+1)$, for higher values of n, we have

$$|{\tilde{t}}_n^{CN}(f;x)-\tilde{f}(x)|=\{\begin{array}{ll}O({(n+1)}^{-\alpha }),& 0<\alpha <1,\\ O(\frac{log(n+1)}{n+1}),& \alpha =1.\end{array}$$

Hence,

$${\parallel {\tilde{t}}_n^{CN}(f;x)-\tilde{f}(x)\parallel }_{\mathrm{\infty }}=\underset{0\le x\le 2\pi }{sup}|{\tilde{t}}_n^{CN}(f;x)-\tilde{f}(x)|=\{\begin{array}{ll}O({(n+1)}^{-\alpha }),& 0<\alpha <1,\\ O(\frac{log(n+1)}{n+1}),& \alpha =1.\end{array}$$

This completes the proof of Theorem 3.1.

6 Conclusion

Several results concerning the degree of approximation of periodic signals (functions) belonging to the Lipschitz class by Matrix Operator have been reviewed and the condition of monotonicity on the generating sequence $\{p_n\}$ has been relaxed. Further, a proper set of conditions has been discussed to rectify the errors and applications pointed out in Remarks 1 and 2. Some interesting applications of the operator used in this paper were pointed out in Note 1.