A Study on Approximation of a Conjugate Function Using Cesàro-Matrix Product Operator

Abstract

In this chapter, we present a study of the inaccuracy estimation of a function ζ∼ conjugate of a function ζ (2π-periodic) in weighted Lipschitz space WLpp≥1ξω, by Cesàro-Matrix (CδT) product means of its CFS1. This chapter is divided into seven section. The first section contains introduction of our chapter, the second section, we introduce some basic definitions and notations. In the third section lemmas and the fourth section contains our main theorems and proofs. In the fifth section, we introduce corollaries, the sixth section contains particular cases of our results and the last section contains exercise of our chapter.

1. Introduction

The studies of estimations of conjugate of functions in different Lipschitz classes and Hölder classes using single summability operators, have been made by the researchers like [1, 2, 3, 4] etc. in past few decades. The studies of estimation of error of cojugate of functions in different Lipschitz classes and Hölder classes using different product operator, have been made by the researchers like [5, 6, 7, 8, 9, 10, 11, 12] etc. in recent past.

In this problem, we andeavour consider more sophisticated class of function in contemplation of reach at the best estimation of function $\stackrel{\sim }{\zeta }$ conjugate of a function $\zeta$ $\left(2\pi -\text{periodic}\right)$ by trigonometric polynomial of degree more than $\lambda$. It can be paid attention the results procure thus far in the route of present work could not lay out the best approximation of the function also, in this work, we have used Cesàro-Matrix $\left(C^{\delta }T\right)$ of product operators which is developed here in order to work using more generalized operator. It is important to mention here that $C^{\delta }T$ is the more generalized product operator than the product operators Cesàro-Harmonic $\left(C^{\delta }H\right)$, Cesàro-Nörlund $\left(C^{\delta }{N}_p\right)$, Cesàro-Riesz $\left(C^{\delta }{\overline{N}}_p\right)$, Cesàro-generalized Nörlund $\left(C^{\delta }{N}_{\mathit{pq}}\right)$ and Cesàro-Euler $\left(C^{\delta }H\right)$ and furthermore $C^{1}H$, $C^1{N}_p$, $C^1{N}_{\mathit{pq}}$, $C^1{E}^q$ and $C^1{E}^1$ product operators are the special cases of $C^{\delta }T$ for $\delta =1$.

Therefore, we establish two theorems so obtain best inaccuracy estimation of a function $\stackrel{\sim }{\zeta }$, conjugate to a $2\pi$-periodic function $\zeta$ in weighted $W\left(L^p,\xi \left(\omega \right)\right)$ space of its CFS. Here we shall consider the two cases (i) $p>1$ and (ii) $p=1$ in order to get the Hölder’s inequality satisfied. Our theorems generalizes six previously known results. Thus, the results of [5, 8, 9, 10, 11, 12] becomes the special cases of our theorem. Some inportant corollaries are also obtained from our theorems.

Note 1 The CFS is not necessarily a FS.²

Example 1 The series

$$\sum _{\lambda =2}^{\infty }‍\left(\frac{sin\left(\mathit{\lambda x}\right)}{log\lambda }\right)$$

conjugate to the FS

$$\sum _{\lambda =2}^{\infty }‍\left(\frac{cos\left(\mathit{\lambda x}\right)}{log\lambda }\right)$$

is not a FS (Zygmund [13], p. 186).

From above example, we conclude that, a separate study of conjugate series in the present direction of work is quite essential.

2. Definitions and notations

2.1 Lipschitz class

Let $C_{2\pi }$ is a Banach space of all periodic functions with period $2\pi$ and continuous on the interval $0\le x\le 2\pi$ under the supremum norm.

The best $\lambda$-order error approximation of a function $\stackrel{\sim }{\zeta }\in C_{2\pi }$ is defined by

$$E_{\lambda }\left(\stackrel{\sim }{\zeta }\right)=\underset{t_{\lambda }}{inf}\parallel \stackrel{\sim }{\zeta }-t_{\lambda }\parallel ,$$

where $t_{\lambda }$ is a trigonometric polynomial of degree $\lambda$ (Bernstein [14]).

Let us define the $L^p$ space of all $2\pi$-periodic and integrable functions as

$$L^p\left[0,2\pi \right]≔\left\{\stackrel{\sim }{\zeta }:\left[0,2\pi \right]\to \mathbb{R}:{\int }_0^{2\pi }‍{\left|\stackrel{\sim }{\zeta }\left(x\right)\right|}^p\mathit{dx}<\infty \right\},p\ge 1.$$

Now, $\parallel .{\parallel }_p$ is defined as

$$\parallel \stackrel{\sim }{\zeta }{\parallel }_p=\left\{\begin{array}{ll}{\left\{\frac{1}{2\pi }{\int }_0^{2\pi }‍{\left|\stackrel{\sim }{\zeta }\left(x\right)\right|}^p\mathit{dx}\right\}}^{\frac{1}{p}}& \text{for}1\le p<\infty \\ \mathrm{ess}\underset{x\in \left(0,2\pi \right)}{sup}\mid \stackrel{\sim }{\zeta }\left(x\right)\mid & \text{for}p=\infty .\end{array}\right.$$

We define the following Lipschitz classes of function

$$\zeta \in \mathit{\text{Lipα}}\text{if}\mathit{\text{Lipα}}≔\left\{\zeta :\left[0,2\pi \right]\to \mathbb{R}:|\zeta \left(x+\omega \right)-\zeta \left(x\right)|=O\left({\omega }^{\alpha }\right)\right\}$$

for $0<\alpha \le 1$;

$$\zeta \in \mathit{Lip}\left(\alpha ,p\right)\text{if}\mathit{Lip}\left(\alpha ,p\right)≔\left\{\zeta \in L^p\left[0,2\pi \right]:\parallel \zeta \left(x+\omega \right)-\zeta \left(x\right){\parallel }_p=O\left({\omega }^{\alpha }\right)\right\}$$

for $p\ge 1,0<\alpha \le 1$;

$$\zeta \in \mathit{Lip}\left(\alpha ,\xi \left(\omega \right)\right)\text{if}\mathit{Lip}\left(\alpha ,\xi \left(\omega \right)\right)≔\left\{\zeta \in L^p\left[0,2\pi \right]:\parallel \zeta \left(x+\omega \right)-\zeta \left(x\right){\parallel }_p=O\left(\xi \left(\omega \right)\right)\right\}$$

for $p\ge 1,0<\alpha \le 1\&\beta \ge 0$;

$$\begin{array}{l}\zeta \in W\left(L^p,\xi \left(\omega \right)\right)\text{if}W\left(L^p,\xi \left(\omega \right)\right)≔\{\zeta \in L^p\left[0,2\pi \right]:{‖\left(\zeta \left(x+\omega \right)-\zeta \left(x\right)\right){sin}^{\beta }\left(\frac{\omega }{2}\right)‖}_p=O\left(\xi \left(\omega \right)\right)\}\end{array}$$

where $\xi \left(\omega \right)>0$ and increasing with $\omega >0$ and $L^p$ space of all $2\pi$-periodic and integrable functions. Under above assumptions for $\alpha \in \left(0,1\right],p\ge 1,\omega >0$, we observed that

$$W\left(L^p,\xi \left(\omega \right)\right)\stackrel{\beta =0}{\to }\mathit{Lip}\left(\xi \left(\omega \right),p\right)\stackrel{\xi \left(\omega \right)={\omega }^{\alpha }}{\to }\mathit{Lip}\left(\alpha ,p\right)\stackrel{p\to \infty }{\to }\mathit{\text{Lipα}}.$$

Remark 1 If $\frac{\xi \left(\omega \right)}{\omega }$is non-increasing; then $\frac{\xi \left(\frac{\pi }{\lambda +1}\right)}{\frac{\pi }{\lambda +1}}\le \frac{\xi \left(\frac{1}{\lambda +1}\right)}{\frac{1}{\lambda +1}}$ i.e., $\xi \left(\frac{\pi }{\lambda +1}\right)\le \mathit{\pi \xi }\left(\frac{1}{\lambda +1}\right)$.

2.2 Some important single summability

Let

$$\sum _{\lambda =0}^{\infty }‍v_{\lambda }$$E1

be an infinite series such that $s_k={\sum }_{m=0}^k‍v_m$. Let

$${\sigma }_r^{\eta }=\sum _{k=0}^r‍\frac{\left(\begin{array}{l}r-k+\eta -1\\ r-k\end{array}\right)}{\left(\begin{array}{l}r+\eta \\ r\end{array}\right)}{s}_k,\text{for}\eta >-1.$$E2

If ${lim}_{\lambda \to \infty }{\sigma }_{\lambda }^{\eta }=s$ then we say that the series (1) is $\left(C,\eta \right)$ summable to $s$ or summable by Cesàro mean of order $\eta$.If we take $\eta =0$ in (2), $\left(C,\eta \right)$ summability reduces to an ordinary sum and if we take $\eta =1$, then $\left(C,\eta \right)$ summability reduces to $\left(C,1\right)$ summability or Cesàro summability of order 1.

Let

$$t_{\lambda }^{E^q}=\frac{1}{{\left(1+q\right)}^{\lambda }}\sum _{k=0}^{\lambda }‍\left(\begin{array}{l}\lambda \\ k\end{array}\right)\frac{1}{q^{k-\lambda }}{s}_k,q>0.$$

If ${lim}_{\lambda \to \infty }{t}_{\lambda }^{E^q}=s$ then we say that the series (1) is $\left(E,q\right)$ summable to $s$ or summable by Euler mean $\left(E,q\right)$ (Hardy [15]). If $q=0$, $\left(E,q\right)$ method reduces to an ordinary sum and if $q=1$, $\left(E,q\right)$ means reduces to $\left(E,1\right)$ means.

An infinite series (1) with the sequence $\left\{s_{\lambda }\right\}$ of its partial sums is said to be summable by harmonic method (Riesz [16] or simply summable $\left(N,\frac{1}{\lambda +1}\right)$ to sum $s$, where $s$ is a finite number, if the sequence to sequence transformation

$$t_{\lambda }=\frac{1}{log\lambda }\sum _{v=0}^{\lambda }‍\frac{s_v}{\lambda -v+1}\mathrm{as}\lambda \to \infty .$$

Let $\left\{p_{\lambda }\right\}$ be a sequence of constants, real or complex and let

$$P_{\lambda }=\sum _{k=0}^{\lambda }‍p_k,\left(P_{\lambda }\ne 0\right).$$

Let

$$t_{\lambda }^{N_p}=\frac{1}{P_{\lambda }}\sum _{k=0}^{\lambda }‍p_{\lambda -k}{s}_k=\frac{1}{P_{\lambda }}\sum _{k=0}^{\lambda }‍p_k{s}_{\lambda -k}.$$E3

If

$$\underset{\lambda \to \infty }{lim}{t}_{\lambda }^{N_p}=s$$

then we say that the series (1) is $\left(N,p_{\lambda }\right)$ summable to $s$ or summable by Nörlund $\left(N,p_{\lambda }\right)$ means.

Let $\left\{p_{\lambda }\right\}$ and $\left\{q_{\lambda }\right\}$, be two sequences of constants, real or complex such that

$$P_{\lambda }=p_0+p_1+\dots +p_{\lambda };P_{-1}=p_{-1}=0,$$E4

$$Q_{\lambda }=q_0+q_1+\dots +q_{\lambda };Q_{-1}=q_{-1}=0.$$E5

$$R_{\lambda }=\sum _{k=0}^{\lambda }‍p_k{q}_{\lambda -k}\ne 0\text{for}\mathrm{all}\lambda .$$E6

Convolution of the two sequences $\left\{p_{\lambda }\right\}$ and $\left\{q_{\lambda }\right\}$, is defined as

$$R_{\lambda }={\left(p\ast q\right)}_{\lambda }=\sum _{k=0}^{\lambda }‍p_k{q}_{\lambda -k}.$$

We write

$$t_{\lambda }^{N_{\mathit{pq}}}=\frac{1}{R_{\lambda }}\sum _{k=0}^{\lambda }‍p_{\lambda -k}{q}_k{s}_k;$$

then the generalized Nörlund means $\left(N_{p,q}\right)$ of the sequence $\left\{s_{\lambda }\right\}$ is denoted by the sequence $t_{\lambda }^{\mathit{pq}}$. If $t_{\lambda }^{\mathit{pq}}\to s,\mathrm{as}\lambda \to \infty$ then, the series (1) is said to be summable to $s$ by $N_{p,q}$ method and is denoted by $s_{\lambda }\to s\left(N_{p,q}\right)$ ([17]).

Let $\left\{p_{\lambda }\right\}$ be a sequence of real constants such that $p_0>0,p_{\lambda }\ge 0$ and $P_{\lambda }={\sum }_{v=0}^{\lambda }‍p_v\ne 0$, such that $P_{\lambda }\to \infty$ as $\lambda \to \infty$.

If

$$t_{\lambda }=\frac{1}{P_{\lambda }}\sum ‍p_v{s}_v\to s,\mathrm{as}\lambda \to \infty ,$$

then we say that $\left\{s_{\lambda }\right\}$ is summable by $\left(\overline{N},p_{\lambda }\right)$ means and we write

$$s_{\lambda }=s\left(\overline{N},p_{\lambda }\right),$$

where $\left\{s_{\lambda }\right\}$ is the sequence of ${\lambda }^{\mathit{th}}$ partial sum of the series (1).

Let $T=\left(l_{\lambda ,k}\right)$ be an infinite triangular matrix satisfying the conditions of regularity [18] i.e.,

$$\left\{\begin{array}{l}\sum _{k=0}^{\lambda }‍l_{\lambda ,k}=1\mathit{as}\lambda \to \infty \\ l_{\lambda ,k}=0\mathit{\text{for}}k>\lambda \\ \sum _{k=0}^{\lambda }‍\mid l_{\lambda ,k}\mid \le M,a\mathit{\text{finite constant}}\\ & \end{array}\right.$$E7

The sequence-to-sequence transformation

$$t_{\lambda }^T\left(\stackrel{\sim }{\zeta };x\right)≔\sum _{k=0}^{\lambda }‍l_{\lambda ,k}{s}_k=\sum _{k=0}^{\lambda }‍l_{\lambda ,\lambda -k}{s}_{\lambda -k}$$

defines the sequence $t_{\lambda }^T\left(\zeta ;x\right)$ of triangular matrix means of the sequence $\left\{s_{\lambda }\right\}$ generated by the sequence of coefficients $\left(l_{\lambda ,k}\right)$.

If $t_{\lambda }^T\left(\stackrel{\sim }{\zeta };x\right)\to s$ as $\lambda \to \infty$ then the infinite series ${\sum }_{\lambda =0}^{\infty }‍v_{\lambda }$ or the sequence $\left\{s_{\lambda }\right\}$ is summable to $s$ by triangular matrix ($T$-method) [13].

2.3 ${\mathit{C}}^{\mathit{\delta }}\mathit{T}$ product means

we define $C^{\delta }T$ means as

$$t_{\lambda }^{C^{\delta }T}\left(\stackrel{\sim }{\zeta };x\right)≔\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}\lambda -r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,k}{s}_k\left(\stackrel{\sim }{\zeta };x\right)$$E8

If $t_{\lambda }^{C^{\delta }T}\left(\stackrel{\sim }{\zeta };x\right)\to s$ as $\lambda \to \infty$, then ${\sum }_{\lambda =0}^{\infty }‍v_{\lambda }$ is summable to $s$ by $C^{\delta }T$ method.

Note 2 Since $C^{\delta }$ and $T$ both are regular then $C^{\delta }T$ method is also regular.

Remark 2 The special cases of $C^{\delta }T$ means: $C^{\delta }T$ transform reduces to

1. $C^{\delta }H$ transform if $l_{\lambda ,k}=\frac{1}{\left(\lambda -k+1\right)log\left(\lambda +1\right)}$;

2. $C^{\delta }{N}_p$ transform if $l_{\lambda ,k}=\frac{p_{\lambda -k}}{P_{\lambda }}$ where $P_{\lambda }={\sum }_{k=0}^{\lambda }‍p_k\ne 0$;

3. $C^{\delta }{\overline{N}}_p$ transform if $l_{\lambda ,k}=\frac{p_k}{P_{\lambda }}$;

4. $C^{\delta }{E}^q$ transform when $a_{\lambda ,k}=\frac{1}{{\left(1+q\right)}^{\lambda }}\left(\begin{array}{l}\lambda \\ k\end{array}\right)q^{\lambda -k}$;

5. $C^{\delta }{E}^1$ when $l_{\lambda ,k}=\frac{1}{2^{\lambda }}\left(\begin{array}{l}\lambda \\ k\end{array}\right)$;

6. $C^{\delta }{N}_{\mathit{pq}}$ transform if $l_{\lambda ,k}=\frac{p_{\lambda -k}{q}_k}{R_{\lambda }}$ where $R_{\lambda }={\sum }_{k=0}^{\lambda }‍p_k{q}_{\lambda -k}$.

In above special case (ii), (iii), and (vi) $p_{\lambda }$ and $q_{\lambda }$ are two non-negative monotonic non-increasing sequences of real constants.

Remark 3 $C^{1}H$, $C^1{N}_p$, $C^1{N}_{\mathit{pq}}$, $C^1{E}^q$ and $C^1{E}^1$ transforms are also the special cases of $C^{\delta }T$ for $\delta =1$.

Example 2 we consider

$$1-1574\sum _{\lambda =1}^{\infty }‍{\left(-1573\right)}^{\lambda -1}$$E9

The ${\lambda }^{\mathit{th}}$ partial sum of the series (9) is given by

$$s_{\lambda }={\left(-1573\right)}^{\lambda },\forall \lambda \in {\mathbb{N}}_0$$

we take $l_{\lambda ,k}=\frac{1}{{\left(787\right)}^{\lambda }}\left(\begin{array}{l}\lambda \\ k\end{array}\right){\left(786\right)}^{\lambda -k}$, then

$$\begin{array}{cc}{t}_{\lambda }^T& =l_{\lambda ,0}{s}_0+l_{\lambda ,1}{s}_1+\dots +l_{\lambda ,\lambda }{s}_{\lambda }\\ & =\frac{1}{{\left(787\right)}^{\lambda }}\left[\left(\begin{array}{l}\lambda \\ 0\end{array}\right){\left(786\right)}^{\lambda }-\left(\begin{array}{l}\lambda \\ 1\end{array}\right){\left(786\right)}^{\lambda -1}.1573+\dots +\left(\begin{array}{l}\lambda \\ \lambda \end{array}\right){\left(-1573\right)}^{\lambda }\right]\\ & =\frac{1}{{\left(787\right)}^{\lambda }}{\left(-787\right)}^{\lambda }\\ & =\left\{\begin{array}{ll}1,& \lambda \mathit{\text{is even}}\\ -1,& \lambda \mathit{\text{is}}\mathit{odd}\end{array}\right.\end{array}$$E10

in above example, we see the series is summable neither by Cesàro means nor Matrix means, but summable by Cesàro-Matrix.

Thus, $C^{\delta }T$ means is more powerfull and effective than single $C^{\delta }$ and $T$ means.

Example 3 we consider another infinite series

$$1-6+30-150+750-3750+18750-\dots$$E11

The ${\lambda }^{\mathit{th}}$ partial sum of the series (11) is given by

$$s_{\lambda }={\left(-5\right)}^{\lambda },\forall \lambda \in {\mathbb{N}}_0$$

we take $l_{\lambda ,k}=\frac{1}{3^{\lambda }}\left(\begin{array}{l}\lambda \\ k\end{array}\right)2^{\lambda -k}$, then

$$\begin{array}{cc}{t}_{\lambda }^T& =l_{\lambda ,0}{s}_0+l_{\lambda ,1}{s}_1+\dots +l_{\lambda ,\lambda }{s}_{\lambda }\\ & =\frac{1}{3^{\lambda }}\left[\left(\begin{array}{l}\lambda \\ 0\end{array}\right)2^{\lambda }-\left(\begin{array}{l}\lambda \\ 1\end{array}\right)2^{\lambda -1}.5+\dots +\left(\begin{array}{l}\lambda \\ \lambda \end{array}\right){\left(-5\right)}^{\lambda }\right]\\ & =\frac{1}{3^{\lambda }}{\left(-3\right)}^{\lambda }\\ & =\left\{\begin{array}{ll}1,& \lambda \in \left\{2m:m\in Z\right\}\\ -1,& \lambda \in \left\{2m+1:m\in Z\right\}\end{array}\right.\end{array}$$E12

in above example, we see the series is summable neither by Cesàro means of order one nor Matrix means, but summable by Cesàro-Matrix.

2.4 Notations

$${\stackrel{\sim }{K}}_{\lambda }^{C^{\delta }T}=\frac{1}{2\pi }\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,r-k}\frac{cos\left(r-k+\frac{1}{2}\right)\omega }{sin\frac{\omega }{2}}$$E13

$$\varrho =\text{integral part of}\left(\frac{1}{\omega }\right)$$

$$\psi \left(x,\omega \right)=\zeta \left(x+\omega \right)-\zeta \left(x-\omega \right)$$

We use the following in our work.

$$\frac{1}{sin\left(\frac{\omega }{2}\right)}\le \frac{\pi }{\omega },0<\omega \le \pi$$E14

$$sin\omega \le \omega ,\omega \ge 0$$E15

$$\mid cos\mathit{\lambda \omega }\mid \le 1,\forall \omega \in \mathbb{R}$$E16

Zygmund ([13]).

Note 3 Following conditions are used in the proof of the main results

$$\left\{\begin{array}{l}{l}_{\lambda ,\lambda -k}-l_{\lambda +1,\lambda +1-k}\ge 0\text{for}0\le k\le \lambda \\ L_{\lambda ,k}=\sum _{r=k}^{\lambda }‍l_{\lambda ,\lambda -r}\text{and}{L}_{\lambda ,0}=1,\forall \lambda \in {\mathbb{N}}_0\end{array}\right..$$E17

Remark 4 Considering the matrix $T=\left(l_{\lambda ,k}\right)$ as

$$l_{\lambda ,k}=\left\{\begin{array}{ll}\frac{2018\times {\left(2019\right)}^k}{{\left(2019\right)}^{\lambda +1}-1},& 0\le k\le \lambda \\ 0,& k>\lambda \end{array}\right.,$$

we can observe that (7, 17) satisfied.

Remark 5 Function $\overline{\zeta }$ denotes a conjugate to a $2\pi$-period and Lebesgue integrable function and this notation is used throughout the chapter.

3. Lemmas

For the proof of our theorems, following lemmas are required:

Lemma 3.1 If conditions (7, 17) hold for $\left\{l_{\lambda ,k}\right\}$, then

$$\mid {\stackrel{\sim }{K}}_{\lambda }^{C^{\delta }T}\left(\omega \right)\mid =O\left(\frac{1}{\omega }\right)\forall \delta \ge 1,0<\omega \le \frac{\pi }{\lambda +1}.$$

Proof. For $0<\omega \le \frac{\pi }{\lambda +1}$, using (14, 15, 16)

$$\begin{array}{cc}\mid {\stackrel{\sim }{K}}_{\lambda }^{C^{\mathit{\delta T}}}\left(\omega \right)\mid & =\left|\frac{1}{2\pi }\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,r-k}\frac{cos\left(r-k+\frac{1}{2}\right)\omega }{sin\frac{\omega }{2}}\right|\\ & \le \frac{1}{2\pi }\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,r-k}\frac{\mid cos\left(r-k+\frac{1}{2}\right)\omega \mid }{\mid sin\frac{\omega }{2}\mid }\\ & \le \frac{1}{2\omega }\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,r-k}\\ & =\frac{1}{2\omega }\sum _{r=0}^{\omega }‍\frac{\left(r+\delta -1\right)!}{\left(\delta -1\right)!r!}\times \frac{\delta !\lambda !}{\left(\delta +\lambda \right)!}{L}_{r,0}\\ & =\frac{\lambda !\delta }{2\omega \left(\delta +\lambda \right)!}\sum _{r=0}^{\lambda }‍\frac{\left(r+\delta -1\right)!}{r!}\text{since}{\mathrm{L}}_{\mathrm{r},0}=1\\ & =\frac{\lambda !\delta }{2\omega \delta !\left(\delta +1\right)\cdots \left(\delta +\lambda \right)}\left[\frac{\left(\delta -1\right)!}{0!}+\frac{\delta !}{1!}\cdots +\frac{\left(\lambda +\delta -1\right)!}{\lambda !}\right]\\ & \le \frac{\lambda !\delta }{2\omega \delta !\left(\delta +1\right)\cdots \left(\delta +\lambda \right)}\times \left(\lambda +1\right)\frac{\delta !\left(\delta +1\right)\cdots \left(\delta +\lambda -1\right)}{\lambda !}\\ & =\frac{\delta }{2\omega }\times \frac{\lambda +1}{\delta +\lambda }\\ & \le \frac{\delta }{2\omega }\\ & =O\left(\frac{1}{\omega }\right)\text{for}\mathrm{all}\delta \ge 1.\end{array}$$

Lemma 3.2 If conditions (7, 17) holds for $\left\{l_{\lambda ,k}\right\}$, then

$$\left|{\stackrel{\sim }{K}}_{\lambda }^{C^{\delta }T}\left(\omega \right)\right|=O\left(\frac{1}{\left(\lambda +1\right){\omega }^2}\right)\forall \delta \ge 1,\frac{\pi }{\lambda +1}\le \omega \le \pi .$$

Proof. For $\frac{\pi }{\lambda +1}\le \omega \le \pi$, using (14), $l_{r,r-k}\ge l_{r+1,r+1-k}\ge l_{r+1,r-k}$ and $L_{\varrho +1,0}=1$.

$$\begin{array}{cc}\left|{\stackrel{\sim }{K}}_{\lambda }^{C^{\mathit{\delta T}}}\left(\omega \right)\right|& =\left|{\left(2\pi \right)}^{-1}\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,r-k}\frac{cos\left(r-k+\frac{1}{2}\right)\omega }{sin\frac{\omega }{2}}\right|\\ & =O\left(\frac{1}{\omega }\right)\left|Re\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,r-k}{e}^{i\left(r-k+\frac{1}{2}\right)\omega }\right|\end{array}$$E18

Now we consider

$$\begin{array}{cc}\left|\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,r-k}{e}^{i\left(r-k+\frac{1}{2}\right)\omega }\right|& \le \left|\sum _{r=0}^{\varrho }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,r-k}{e}^{\iota \left(r-k\right)\omega }\right|\\ & +\left|\sum _{r=\varrho +1}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^{\varrho }‍l_{r,r-k}{e}^{i\left(r-k\right)\omega }\right|\\ & +\left|\sum _{r=\varrho +1}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=\varrho +1}^r‍l_{r,r-k}{e}^{i\left(r-k\right)\omega }\right|\\ & ={\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3,\left(\mathrm{say}\right).\end{array}$$E19

Now,

$$\begin{array}{cc}{\mathrm{\Lambda }}_1& \le \sum _{r=0}^{\varrho }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,r-k}\left|e^{i\left(r-k\right)\omega }\right|\\ & \le \sum _{r=0}^{\varrho }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}{L}_{r,0}\\ & =\sum _{r=0}^{\varrho }‍\frac{\left(r+\delta -1\right)!}{\left(\delta -1\right)!r!}\times \frac{\delta !\lambda !}{\left(\delta +\lambda \right)!}\text{since},{\mathrm{L}}_{\mathrm{r},0}=1\\ & =\frac{\lambda !}{\left(\delta +1\right)\dots \left(\delta +\lambda \right)\delta !}\sum _{r=0}^{\varrho }‍\frac{\left(r+\delta -1\right)!\delta }{r!}\\ & =\frac{\lambda !}{\left(\delta +1\right)\dots \left(\delta +\varrho \right)\dots \left(\delta +\lambda \right)}\left[1+\delta +\frac{\delta \left(\delta +1\right)}{2!}+\dots +\frac{\left(\varrho +\delta -1\right)!}{\varrho !\left(\delta -1\right)!}\right]\\ & \le \frac{\lambda !}{\left(\delta +1\right)\dots \left(\delta +\varrho \right)\dots \left(\delta +\lambda \right)}\left[\left(\varrho +1\right)\frac{\delta \left(\delta +1\right)\dots \left(\delta +\varrho -1\right)}{\varrho !}\right]\\ & =\frac{\varrho !\left(\varrho +1\right)\dots \lambda }{\left(\delta +\varrho \right)\dots \left(\delta +\lambda -1\right)}\times \frac{\delta }{\delta +\lambda }\times \frac{\varrho +1}{\varrho !}\\ & \le \frac{\delta }{\delta +\lambda }\times \left(\varrho +1\right)\\ & \le \frac{\delta }{\delta +\lambda }\times \left(\frac{1}{\omega }+1\right)\\ & =O\left(\frac{1}{\omega \left(\lambda +1\right)}\left(1+\omega \right)\right)\text{for all}\delta \ge 1.\end{array}$$

Changing the order of summation and applying Abel’s transformation in ${\mathrm{\Lambda }}_2$, we have

$$\begin{array}{cc}{\mathrm{\Lambda }}_2& =\left|\sum _{k=0}^{\varrho }‍\sum _{r=\varrho +1}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}{l}_{r,r-k}{e}^{i\left(r-k\right)\omega }\right|\\ & =\frac{1}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}|\sum _{k=0}^{\varrho }‍[\left\{\sum _{r=\varrho +1}^{\lambda -1}‍\left(\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)l_{r,r-k}-\left(\begin{array}{l}r+\delta \\ \delta -1\end{array}\right)l_{r+1,r+1-k}\right)\sum _{\nu =0}^r‍e^{i\left(\nu -k\right)\omega }\right\}\\ & +\left(\begin{array}{l}\lambda +\delta -1\\ \delta -1\end{array}\right)l_{\lambda ,\lambda -k}\sum _{\nu =0}^{\lambda }‍e^{i\left(\nu -k\right)\omega }-\left(\begin{array}{l}\delta +\varrho \\ \delta -1\end{array}\right)l_{\varrho +1,\varrho +1-k}\left]\right|\\ & =O\left({\omega }^{-1}\right)\frac{1}{\left(\begin{array}{l}\eta +\lambda \\ \eta \end{array}\right)}\sum _{k=0}^{\varrho }‍\left[\right|\sum _{r=\varrho +1}^{\lambda -1}‍\left(\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)l_{r,r-k}-\left(\begin{array}{l}r+\delta \\ \delta -1\end{array}\right)l_{r+1,r+1-k}\right)|\\ & +\left|\left(\begin{array}{l}\lambda +\delta -1\\ \delta -1\end{array}\right)l_{\lambda ,\lambda -k}\right|+\left|\left(\begin{array}{l}\delta +\varrho \\ \delta -1\end{array}\right)l_{\varrho +1,\varrho +1-k}\right|]\\ & =O\left({\omega }^{-1}\right)\frac{1}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^{\varrho }‍[\left(\begin{array}{l}\delta +\varrho \\ \delta -1\end{array}\right)l_{\varrho +1,\varrho +1-k}+\left(\begin{array}{l}\delta +\lambda -1\\ \delta -1\end{array}\right)l_{\lambda ,\lambda -k}+\left(\begin{array}{l}\delta +\lambda -1\\ \delta -1\end{array}\right)l_{\lambda ,\lambda -k}\\ & +\left(\begin{array}{l}\delta +\varrho \\ \delta -1\end{array}\right)l_{\varrho +1,\varrho +1-k}]\\ & =O\left({\omega }^{-1}\right)\frac{1}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^{\varrho }‍\left[\left(\begin{array}{l}\delta +\varrho \\ \delta -1\end{array}\right)l_{\varrho +1,\varrho +1-k}+\left(\begin{array}{l}\delta +\lambda -1\\ \delta -1\end{array}\right)l_{\lambda ,\lambda -k}\right]\\ & =O\left({\omega }^{-1}\right)\frac{\lambda !}{\left(\delta +1\right)\dots \left(\delta +\lambda \right)}\sum _{k=0}^{\varrho }‍[\frac{\delta \left(\delta +1\right)\dots \left(\delta +\varrho \right)\dots \left(\delta +\lambda \right)}{\left(\delta +\varrho +1\right)\dots \left(\delta +\lambda \right)\left(\varrho +1\right)!}{l}_{\varrho +1,\varrho +1-k}\\ & +\frac{\delta \left(\delta +1\right)\dots \left(\delta +\lambda -1\right)\left(\delta +\lambda \right)}{\left(\delta +\lambda \right)\lambda !}{l}_{\lambda ,\lambda -k}]\\ & =O\left({\omega }^{-1}\right)\frac{\lambda !}{\left(\delta +1\right)\dots \left(\delta +\lambda \right)}\times \frac{\delta \left(\delta +1\right)\dots \left(\delta +\lambda \right)}{\left(\lambda +1\right)!}\sum _{k=0}^{\varrho }‍\left(l_{\varrho ,\varrho -k}+l_{\lambda ,\lambda -k}\right)\\ & =O\left(\frac{1}{\omega \left(\lambda +1\right)}\left(L_{\varrho ,0}+L_{\lambda ,0}\right)\right)\\ & =O\left(\frac{1}{\omega \left(\lambda +1\right)}\right).\end{array}$$

Applying Abel’s transformation in ${\mathrm{\Lambda }}_3$, we have

$$\begin{array}{rc}{\mathrm{\Lambda }}_3& =|\sum _{r=\varrho +1}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}[\sum _{k=\varrho +1}^{r-1}‍\left(l_{r,r-k}-l_{r,r-k+1}\right)\sum _{\nu =0}^k‍e^{i\left(r-\nu \right)\omega }+l_{r,0}\sum _{\nu =0}^r‍e^{\iota \left(r-\nu \right)\omega }\\ & -l_{r,r-\tau -1}\sum _{\nu =0}^{\varrho }‍e^{i\left(r-\nu \right)\omega }\left|\right]\hfill \\ & =O\left({\omega }^{-1}\right)\sum _{r=\varrho +1}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\left[\left|\sum _{k=\varrho +1}^{r-1}‍\left(l_{r,r-k}-l_{r,r-k+1}\right)\right|+l_{r,0}+l_{r,r-\varrho -1}\right]\\ & =O\left({\omega }^{-1}\right)\sum _{r=\varrho +1}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\left[|-l_{r,r-\varrho }+l_{r,1}|+l_{r,0}+l_{r,r-\varrho -1}\right]\\ & =O\left({\omega }^{-1}\right)\frac{1}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{r=\varrho +1}^{\lambda }‍\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)l_{r,r-\varrho }\\ & =\begin{array}{l}O\left({\omega }^{-1}\right)\frac{\lambda !}{\left(\delta +1\right)\dots \left(\delta +\lambda \right)}\\ \times \frac{\delta \left(\delta +1\right)\dots \left(\delta +\lambda \right)}{\lambda !\left(\delta +\lambda \right)}\left[\frac{\delta \dots \left(\delta +\varrho \right)}{\left(\varrho +1\right)!}{l}_{\varrho +1,1}+\dots +\frac{\delta \dots \left(\delta +\lambda -1\right)}{\lambda !}{l}_{\lambda ,\lambda -\varrho }\right]\end{array}\\ & =O\left({\omega }^{-1}\right)\frac{\lambda !}{\left(\delta +1\right)\dots \left(\delta +\lambda \right)}\times \frac{\delta \left(\delta +1\right)\dots \left(\delta +\lambda \right)}{\lambda !\left(\delta +\lambda \right)}\left[l_{\varrho +1,1}+l_{\varrho +2,2}+\dots l_{\lambda ,\lambda -\varrho }\right]\\ & =O\left({\omega }^{-1}\right)\frac{\delta }{\delta +\lambda }\left[l_{\varrho +1,1}+l_{\varrho +1,2}+\dots +l_{\varrho +1,\lambda -\varrho }\right]\\ & =O\left(\frac{1}{\omega \left(\lambda +1\right)}\right)L_{\varrho +1,1}\\ & =O\left(\frac{1}{\omega \left(\lambda +1\right)}\right).\end{array}$$

Combining ${\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2$ and ${\mathrm{\Lambda }}_3$ we have,

$$\begin{array}{cc}{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3& =O\left[\frac{1}{\omega \left(\lambda +1\right)}\times \left(1+\omega \right)\right]+O\left[\frac{1}{\omega \left(\lambda +1\right)}\right]+O\left[\frac{1}{\omega \left(\lambda +1\right)}\right]\\ & =O\left[\frac{1}{\left(\lambda +1\right)}\left(1+\frac{3}{\omega }\right)\right]\\ & =O\left(\frac{1}{\lambda +1}\times \frac{3+\pi }{\omega }\right)\end{array}$$E20

(Let 1+$\frac{3}{\omega }\le \frac{k}{\omega }$ for $\omega$ fixed $k^{\mathit{min}}=3+\pi$)

Now, from (19, 20) we get

$$\left|{\stackrel{\sim }{K}}_{\lambda }^{C^{\delta }T}\left(\omega \right)\right|=O\left(\frac{1}{\left(\lambda +1\right){\omega }^2}\right)$$

4. Main theorems

Theorem 4.1 The error approximation of $\stackrel{\sim }{\zeta }$ in $W\left(L^p,\xi \left(\omega \right)\right)$, $\left(p>1\right)$, by $C^{\delta }T$ means of its CFS is given by

$$\parallel t_{\lambda }^{C^{\mathit{\delta T}}}\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{\zeta }\left(x\right){\parallel }_p=O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right],$$

where $0\le \beta <\frac{1}{p}$ and condition (17) holds and positive increasing function $\xi \left(\omega \right)$ satisfies the following conditions:

$$\frac{\xi \left(\omega \right)}{{\omega }^{\beta +1-\sigma }}\text{is non}‐\text{decreasing};$$E21

$${\left\{{\int }_0^{\frac{\pi }{\lambda +1}}‍{\left(\frac{{\lambda }^{-\sigma }\mid \psi \left(x,\omega \right)\mid {sin}^{\beta }\left(\frac{\omega }{2}\right)}{\xi \left(\omega \right)}\right)}^p\mathit{d\omega }\right\}}^{\frac{1}{p}}=O\left({\left(\lambda +1\right)}^{\sigma -\frac{1}{p}}\right),\text{for}\beta <\sigma <\frac{1}{p};$$E22

$$\frac{\xi \left(\omega \right)}{\omega }\text{is non}‐\text{decreasing};$$E23

$$\text{and}{\left\{{\int }_{\frac{\pi }{\lambda +1}}^{\pi }‍{\left(\frac{{\omega }^{-\eta }\mid \psi \left(x,\omega \right)\mid {sin}^{\beta }\left(\frac{\omega }{2}\right)}{\xi \left(\omega \right)}\right)}^p\mathit{d\omega }\right\}}^{\frac{1}{p}}=O\left({\left(\lambda +1\right)}^{\eta -\frac{1}{p}}\right),$$E24

where $\frac{1}{p}<\eta <\beta +\frac{1}{p}$ for $\eta$ being an arbitrary number and $p+q=\mathit{pq}$. Conditions (22, 24) hold uniformly in $x$.

Conditions (22, 24) can be verified by using the fact that $\psi \left(x,\omega \right)\in W\left(L_p,\xi \left(\omega \right)\right)$ and $\frac{\psi \left(x,\omega \right)}{\xi \left(\omega \right)}$ is bounded function.

Proof. The ${\lambda }^{\mathit{th}}$ partial sums of the CFS is denoted by $s_{\lambda }\left(\stackrel{\sim }{\zeta };x\right)$, and is given by

$$s_{\lambda }\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{\zeta }\left(x\right)=\frac{1}{2\pi }{\int }_0^{\pi }‍\psi \left(x,\omega \right)\frac{cos\left(\lambda +\frac{1}{2}\right)\omega }{sin\frac{\omega }{2}}\mathit{d\omega },$$

one can consult [13] for detailed work on FS and CFS.

Denoting $C^{\delta }T$ means of $\left\{s_{\lambda }\left(\stackrel{\sim }{\zeta }:x\right)\right\}$ by $t_{\lambda }^{C^{\delta }T}\left(\stackrel{\sim }{\zeta }:x\right)$, we get

$$\begin{array}{cc}{t}_{\lambda }^{C^{\mathit{\delta T}}}\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{\zeta }\left(x\right)& =\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}\lambda -r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,k}\left[s_k\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{\zeta }\left(x\right)\right]\\ & =\frac{1}{2\pi }{\int }_0^{\pi }‍\psi \left(x,\omega \right)\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,r-k}\frac{cos\left(r-k+\frac{1}{2}\right)\omega }{sin\left(\frac{\omega }{2}\right)}\mathit{d\omega }\end{array}$$E25

$$\begin{array}{l}={\int }_0^{\pi }‍\psi \left(x,\omega \right){\stackrel{\sim }{K}}_{\lambda }^{C^{\delta }T}\left(\omega \right)\mathit{d\omega }\left(\mathrm{By}\text{the notation}\left(13\right)\right)\\ ={\int }_0^{\frac{\pi }{\lambda +1}}‍\psi \left(x,\omega \right){\stackrel{\sim }{K}}_{\lambda }^{C_{\delta }T}\left(\xi \right)\mathit{d\omega }+{\int }_{\frac{\pi }{\lambda +1}}^{\pi }‍\psi \left(x,\omega \right){\stackrel{\sim }{K}}_{\lambda }^{C_{\delta }T}\left(\omega \right)\mathit{d\omega }\\ =I_1+I_2,\mathrm{say}\end{array}$$E26

Applying (14), Lemma 3.1, Hölder’s inequality and second mean value theorem for integral, we have

$$\begin{array}{ll}{I}_1& =O\left(1\right){\left\{{\int }_0^{\frac{\pi }{\lambda +1}}‍{\left(\frac{{\omega }^{-\sigma }\mid \psi \left(x,\omega \right)\mid {sin}^{\beta }\left(\frac{\omega }{2}\right)}{\xi \left(\omega \right)}\right)}^p\mathit{d\omega }\right\}}^{\frac{1}{p}}\times {\left\{{\int }_0^{\frac{\pi }{\lambda +1}}‍{\left(\frac{\xi \left(\omega \right)}{{\omega }^{-\sigma +1}{sin}^{\beta }\left(\frac{\omega }{2}\right)}\right)}^q\mathit{d\omega }\right\}}^{\frac{1}{q}}\\ & =O\left[{\left(\lambda +1\right)}^{\sigma -\frac{1}{p}}\times {\left\{{\int }_0^{\frac{\pi }{\lambda +1}}‍{\left(\frac{\xi \left(\omega \right)}{{\omega }^{\beta +1-\sigma }}\right)}^q\mathit{d\omega }\right\}}^{\frac{1}{q}}\right]\\ & =O\left[{\left(\lambda +1\right)}^{\sigma -\frac{1}{p}}{\left(\lambda +1\right)}^{\beta +\frac{1}{p}-\sigma }\xi \left(\frac{\pi }{\lambda +1}\right)\right]\\ & =O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right]\end{array}$$E27

in view of condition (22) and $p^{-1}+q^{-1}=1$ and Remark 1.

Again using Lemma 3.2, Hölder’s inequality and (14), we have

$$\begin{array}{ll}{I}_2& =O\left(\frac{1}{\lambda +1}\right){\int }_{\frac{\pi }{\lambda +1}}^{\pi }‍\frac{\mid \psi \left(x,\omega \right)\mid }{{\omega }^2}\mathit{d\omega }\\ & =O{\left\{\frac{1}{\lambda +1}{\int }_{\frac{\pi }{\lambda +1}}^{\pi }‍{\left(\frac{\omega -\eta \mid \psi \left(x,\omega \right)\mid {sin}^{\beta }\left(\frac{\omega }{2}\right)}{\xi \left(\omega \right)}\right)}^p\mathit{d\omega }\right\}}^{\frac{1}{p}}\times {\left\{{\int }_{\frac{\pi }{\lambda +1}}^{\pi }‍{\left(\frac{{\omega }^{-1}\xi \left(\omega \right)}{{\omega }^{-\eta +1+\beta }}\right)}^q\mathit{d\omega }\right\}}^{\frac{1}{q}}\\ & =O\left[{\left(\lambda +1\right)}^{-1+\eta -\frac{1}{p}}\xi \left(\frac{\pi }{\lambda +1}\right)\left(\frac{\lambda +1}{\pi }\right){\left({\int }_{\frac{\pi }{\lambda +1}}^{\pi }‍{\omega }^{-\left(\beta +1-\eta \right)q}\mathit{d\omega }\right)}^{\frac{1}{q}}\right]\\ & =O\left[{\left(\lambda +1\right)}^{\eta -\frac{1}{p}}\xi \left(\frac{\pi }{\lambda +1}\right){\left(\lambda +1\right)}^{\beta +1-\eta -\frac{1}{q}}\right]\\ & =O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right]\end{array}$$E28

in view of (23, 24) the second mean value theorem for integrals, $0<\eta <\beta +\frac{1}{p}$, $p+q=\mathit{pq}$ and Remark 1.

Collecting (26)-(28), we get

$$\left|t_{\lambda }^{C^{\delta }T}\left(\stackrel{\sim }{\zeta },x\right)-\stackrel{\sim }{\zeta }\left(x\right)\right|=O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right].$$

Now, using $L_p$-norm of a function, we get

$${‖t_{\lambda }^{C^{\delta }T}\left(\stackrel{\sim }{\zeta },x\right)-\stackrel{\sim }{\zeta }\left(x\right)‖}_p=O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right]$$

Now, we establish the following theorem for the case $p=1$:

Theorem 4.2 The inaccuracy estimation of $\stackrel{\sim }{\zeta }\in W\left(L^1,\xi \left(\omega \right)\right)$, by $C^{\delta }T$ product operatior of its CFS is given by

$$\parallel t_{\lambda }^{C^{\mathit{\delta T}}}\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{\zeta }\left(x\right){\parallel }_1=O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right],$$

where $0\le \beta <1$, provided (17) holds and increasing function $\xi \left(\omega \right)>0$ satisfies conditions (21) to (24) of Theorem 4.1 for $p=1$, $\beta <\sigma <1$ and $1<\eta <\beta +1$.

Proof. Following the proof of Theorem 4.1, for $p=1$, i.e., $q=\infty$, we have

$$\begin{array}{ll}{I}_1& =O\left\{{\int }_0^{\frac{\pi }{\lambda +1}}‍\left(\frac{{\omega }^{-\sigma }\mid \psi \left(x,\omega \right)\mid {sin}^{\beta }\left(\frac{\omega }{2}\right)}{\xi \left(\omega \right)}\right)\mathit{d\omega }\times \mathrm{ess}\underset{0<\omega \le \frac{\pi }{\lambda +1}}{sup}\left|\frac{\xi \left(\omega \right)}{{\omega }^{-\sigma +1}{sin}^{\beta }\left(\frac{\omega }{2}\right)}\right|\right\}\\ & =O\left({\left(\lambda +1\right)}^{\sigma -1}\right)\mathrm{ess}\underset{0<\omega \le \frac{\pi }{\lambda +1}}{sup}\left|\frac{\xi \left(\omega \right)}{{\omega }^{\beta -\sigma +1}}\right|\end{array}$$

$$\begin{array}{ll}& =O\left({\left(\lambda +1\right)}^{\sigma -1}\right)\left\{\frac{\xi \left(\frac{\pi }{\lambda +1}\right)}{{\left(\frac{\pi }{\lambda +1}\right)}^{\beta -\sigma +1}}\right\}\\ & =O\left({\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right)\end{array}$$E29

in view of conditions (21, 22) for $p=1$,

$$\begin{array}{ll}{I}_2& =O\left(\frac{1}{\lambda +1}\right){\int }_{\frac{\pi }{\lambda +1}}^{\pi }‍\frac{\mid \psi \left(x,\omega \right)\mid }{{\omega }^2}\mathit{d\omega }\\ & =O\left\{\frac{1}{\lambda +1}{\int }_{\frac{\pi }{\lambda +1}}^{\pi }‍\frac{{\omega }^{-\eta }\mid \psi \left(x,\omega \right)\mid {sin}^{\beta }\left(\frac{\omega }{2}\right)}{\xi \left(\omega \right)}\mathit{d\omega }\right\}\times \mathrm{ess}\underset{\frac{\pi }{\lambda +1}\le \omega \le \pi }{sup}\left|\frac{\xi \left(\omega \right)}{{\omega }^{-\eta +\beta +2}}\right|\\ & =O\left[{\left(\lambda +1\right)}^{\eta -2}\xi \left(\frac{\pi }{\lambda +1}\right)\left(\frac{{\left(\lambda +1\right)}^{2+\beta -\eta }}{{\pi }^{2+\beta -\eta }}\right)\right]\\ & =O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{\pi }{\lambda +1}\right)\right]\end{array}$$E30

in view of (21, 22). Collecting (28) and (29), we get

$$\mid t_{\lambda }^{C^{\mathit{\delta T}}}\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{\zeta }\left(x\right)\mid =O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right]$$E31

finally from (31),

$$\parallel t_{\lambda }^{C^{\mathit{\delta T}}}\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{\zeta }\left(x\right){\parallel }_1=O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right]$$

in view of Remark 1.

5. Corollaries

Corollary 5.1 The inaccuracy estimation of $\stackrel{\sim }{\zeta }\in \mathit{Lip}\left(\xi \left(\omega \right),p\right)$ class by $C^{\delta }T$ means of its CFS is given by

$${‖t_{\lambda }^{C^{\delta }T}\left(\stackrel{\sim }{\zeta },x\right)-\stackrel{\sim }{\zeta }\left(x\right)‖}_p=O\left[\xi \left(\frac{1}{\lambda +1}\right)\right]$$

where, $C^{\delta }T$ is as defined in (8).

Proof. Considering $\beta =0$ in Theorem 4.1, we can obtain the proof.

Corollary 5.2 The inaccuracy estimation of $\stackrel{\sim }{\zeta }\in \mathit{Lip}\left(\alpha ,p\right)$ space by $C^{\delta }T$ product means of its CFS is given by

$${‖t_{\lambda }^{C^{\delta }T}\left(\stackrel{\sim }{\zeta },x\right)-\stackrel{\sim }{\zeta }\left(x\right)‖}_p=O\left[{\left(\lambda +1\right)}^{-\alpha }\right]$$

where, $C^{\delta }T$ is as defined in (8).

Proof. If we consider $\beta =0\&\xi \left(\omega \right)={\omega }^{\alpha }$ in Theorem 4.1, we can obtain the proof.

Corollary 5.3 The error estimate of $\stackrel{\sim }{\zeta }$ in $\mathit{\text{Lipα}}\left(0<\alpha <1\right)$ class by $C^{\delta }T$ product means of its CFS is given by

$${‖t_{\lambda }^{C^{\delta }T}\left(\stackrel{\sim }{\zeta },x\right)-\stackrel{\sim }{\zeta }\left(x\right)‖}_p=O\left[{\left(\lambda +1\right)}^{-\alpha }\right]$$

where, $C^{\delta }T$ is as defined in (8).

Proof. If we take $\beta =0\&\xi \left(\omega \right)={\omega }^{\alpha }\&p\to \infty$ in Theorem 4.1, we can obtain the proof.

For $\alpha =1$, we can write an independent proof to obtain

$${‖t_{\lambda }^{C^{\delta }T}\left(\stackrel{\sim }{\zeta },x\right)-\stackrel{\sim }{\zeta }\left(x\right)‖}_{\infty }=O\left[\frac{log\left(\lambda +1\right)}{\lambda +1}\right]$$

Corollary 5.4 The error estimate of $\stackrel{\sim }{\zeta }\in W\left(L^p,\xi \left(\omega \right)\right)$ class by $C^{\delta }H$ means

$$t_{\lambda }^{C^{\delta }H}=\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}\lambda -r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}{\left(log\left(r+1\right)\right)}^{-1}\sum _{k=0}^r‍\frac{1}{\left(r-k+1\right)}{s}_k,$$

of the CFS is given by

$$\parallel t_{\lambda }^{C^{\mathit{\delta H}}}\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{\zeta }\left(x\right){\parallel }_p=O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right]$$

provided $C^{\delta }T$ defined in (8) and $\xi \left(\omega \right)$ satisfies the conditions (21) to (24).

Corollary 5.5 The error estimate of $\stackrel{\sim }{\zeta }\in W\left(L^p,\xi \left(\omega \right)\right)$ class by $C^{\delta }{N}_p$ means

$$t_{\lambda }^{C^{\delta }{N}_p}=\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}\lambda -r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\frac{1}{P_r}\sum _{k=0}^r‍p_{r-k}{s}_k,$$

of the CFS is given by

$$\parallel t_{\lambda }^{C^{\delta N_p}}\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{\zeta }\left(x\right){\parallel }_p=O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right]$$

provided $C^{\delta }T$ defined in (8) and $\xi \left(\omega \right)$ satisfies the conditions (21) to (24).

Corollary 5.6 The error estimate of $\stackrel{\sim }{\zeta }\in W\left(L^p,\xi \left(\omega \right)\right)$ class by $C^{\delta }{N}_{\mathit{pq}}$ means

$$t_{\lambda }^{C^{\delta }{N}_{\mathit{pq}}}=\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}\lambda -r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\frac{1}{R_r}\sum _{k=0}^r‍p_{r-k}{q}_k{s}_k,$$

of the CFS is given by

$$\parallel t_{\lambda }^{C^{\delta N_{\mathit{pq}}}}\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{\zeta }\left(x\right){\parallel }_p=O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right]$$

provided $C^{\delta }T$ defined in (8) and $\xi \left(\omega \right)$ satisfies the conditions (21) to (24).

Corollary 5.7 The error approximation of $\stackrel{\sim }{\zeta }\in W\left(L^p,\xi \left(\omega \right)\right)$ class by $C^{\delta }{\overline{N}}_p$ means

$$t_{\lambda }^{C^{\delta }{\overline{N}}_p}=\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}\lambda -r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\frac{1}{P_r}\sum _{k=0}^r‍p_k{s}_k,$$

of the CFS is given by

$$\parallel t_{\lambda }^{C^{\delta }{\overline{N}}_p}\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{\zeta }\left(x\right){\parallel }_p=O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right]$$

provided $C^{\delta }T$ defined in (8) and $\xi \left(\omega \right)$ satisfies the conditions (21) to (24).

Corollary 5.8 The error estimate of $\stackrel{\sim }{\zeta }\in W\left(L^p,\xi \left(\omega \right)\right)$ class by $C^{\delta }{E}^q$ means

$$t_{\lambda }^{C^{\delta }{E}^q}=\sum _{r=0}^{\delta }‍\frac{\left(\begin{array}{l}\delta -r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\delta \\ \delta \end{array}\right)}\frac{1}{{\left(1+q\right)}^r}\sum _{k=0}^r‍\left(\begin{array}{l}r\\ k\end{array}\right)q^{r-k}{s}_k,$$

of the CFS is given by

$$\parallel t_{\lambda }^{C^{\delta E^q}}\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{\zeta }\left(x\right){\parallel }_p=O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right]$$

provided $C^{\delta }T$ defined in (8) and $\xi \left(\omega \right)$ satisfies the conditions (21) to (24).

Corollary 5.9 The error estimate of $\zeta \in W\left(L^p,\xi \left(\omega \right)\right)$ class by $C^{\delta }{E}^1$ means

$$t_{\lambda }^{C^{\delta }{E}^1}=\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}\lambda -r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\frac{1}{2^r}\sum _{k=0}^r‍\left(\begin{array}{l}r\\ k\end{array}\right)s_k,$$

of the FS is given by

$$\parallel t_{\lambda }^{C^{\delta E^1}}\left(\stackrel{\sim }{\zeta };x\right)-\stackrel{\sim }{h}\left(x\right){\parallel }_p=O\left[{\left(\lambda +1\right)}^{\beta }\xi \left(\frac{1}{\lambda +1}\right)\right]$$

provided $C^{\delta }T$ defined in (8) and $\xi \left(\omega \right)$ satisfies the conditions (21) to (24).

Remark 6 The corollaries for 5.1 to 5.9 can also be obtained for the special cases $C^{1}H,C^1{N}_p$, $C^1{N}_{\mathit{pq}}$, $C^1{\stackrel{\sim }{N}}_p,C^1{E}^q\mathit{\text{and}}{C}^1{E}^1$ all things considered Remark 3.

6. Particular cases

The following special cases of our theorems for $\delta =1$ are.

6.1. If we take Remark $1\left(\mathit{iv}\right)$ and $\beta =0,\xi \left(\omega \right)={\omega }^{\alpha },0<\alpha \le 1$ in our theorem, then the Theorem 2 of [8] become a special case of our theorem.

6.2. If $\beta =0,\xi \left(\omega \right)={\omega }^{\alpha },0<\alpha \le 1\&p\to \infty$ in our theorem, then the Theorem 3.3 of [9] become a special case of our result.

6.3. If we consider Remark 2$\left(\mathit{iv}\right)$ then the main Theorem 2.2. of [5] become a special case of our result.

6.4. The Theorem 2 of [10] become a special case of our result.

6.5. If we consider Remark 2$\left(\mathit{iv}\right)$ then the Theorem 3.1 of [11] become a special case of our result.

6.6. If we consider Remark 2$\left(\mathit{ii}\right)$ then the main Theorem 1 of [12] become a special case of our result.

7. Exercise

Q. 7.1. Prove that the infinite series $1-4038{\sum }_{j=1}^{\infty }‍{\left(-4038\right)}^{j-1}$ is neither summable by matrix means(T) nor Cesáro means of order one $\left(C^1\right)$ but it summable by $C^{\delta }T$ means for $\delta =1$.

Q. 7.2. Prove that a function $f$ is $2\pi$-periodic and Lebesgue integrable then the error approximation of $f$ in Lip$\alpha$ class by $C^{\delta }T$ product means of its Fourier series is given by

$$E_n\left(f\right)=\left\{\begin{array}{ll}O\left[{\left(n+1\right)}^{-\alpha }\right],& 0\le \alpha <1\\ O\left[{\left(n+1\right)}^{-1}\left\{log\left(n+1\right)\right\}\right],& \alpha =1,\end{array}\right.$$

where $C^{\delta }T$ is as defined in (8) and provided (17) holds.

{Hint: see [19]}.

Q. 7.3. Consider the matrix $T=\left(a_{n,k}\right)$ as

$$a_{n,k}=\left\{\begin{array}{ll}\frac{2\times 3^k}{3^{n+1}-1},& 0\le k\le j\\ 0,& k>n\end{array}\right.,$$

check all conditions of $T$ method as defined in (7) and also satisfies condition (17). [Hint: see [19]].

Q. 7.4. If the conditions of (7) and (17) holds for $\left\{a_{\lambda ,k}\right\}$, then prove that

$$\left|{\left(2\pi \right)}^{-1}\sum _{r=0}^{\lambda }‍\frac{\left(\begin{array}{l}r+\delta -1\\ \delta -1\end{array}\right)}{\left(\begin{array}{l}\delta +\lambda \\ \delta \end{array}\right)}\sum _{k=0}^r‍l_{r,r-k}\frac{cos\left(r-k+\frac{1}{2}\right)\omega }{sin\frac{\omega }{2}}\right|=\left\{\begin{array}{ll}& O\left(\lambda +1\right),\forall \delta \ge 1,0<\omega \le \frac{\pi }{\lambda +1}\\ & O\left(\frac{1}{\left(\lambda +1\right){\omega }^2}\right),\forall \delta \ge 1,\frac{\pi }{\lambda +1}\le \omega \le \pi .\end{array}\right.$$

Acknowledgments

My heart goes out to acknowledge my indebtedness to my reverend parents for their blessing, sacrifice, affection and giving me enthusiasm at every stage of my study. I am also grateful to all my family and friends specially Mrs. Anshu Rani, Dr. Pradeep Kumar and Niraj Pal for their timely help and giving me constant encouragements.

I also would like to thank the reviewers for their thoughtful and efforts towards improving my chapter.

AMS classification

40C10, 40G05, 40G10, 42A10, 42A24, 40C05, 41A10, 41A25, 42B05, 42A50