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

*Mohammed Hadish*

## **Abstract**

In this chapter, we present a study of the inaccuracy estimation of a function ~*ζ* conjugate of a function *<sup>ζ</sup>* (2*π*-periodic) in weighted Lipschitz space *W Lp* ð Þ , *<sup>p</sup>*<sup>≥</sup> 1, *ξ ω*ð Þ , by Cesàro-Matrix (*C<sup>δ</sup> T*) 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.

**Keywords:** weighted Lipschitz class, error approximation, Cesàro (*C<sup>δ</sup>* ) means, Matrix (T) means *C<sup>δ</sup> T* product means, conjugate Fourier series, generalized Minkowski's inequality

## **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 [1–4] 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 [5–12] 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 ~*ζ* conjugate of a function *ζ* ð Þ 2*π* � periodic by trigonometric polynomial of degree more than *λ*. 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 *Cδ T* of product operators which is developed here in order to work using more generalized operator. It is important to mention here that *C<sup>δ</sup> T* is the more generalized product operator than the product operators Cesàro-Harmonic *C<sup>δ</sup> H* , Cesàro-Nörlund *C<sup>δ</sup> Np* , Cesàro-Riesz *C<sup>δ</sup> Np* , Cesàro-generalized Nörlund *C<sup>δ</sup> Npq* and

<sup>1</sup> CFS denotes Conjugate Fourier series and we use this abbreviation throughout the paper.

Cesàro-Euler *C<sup>δ</sup> H* � � and furthermore *C*<sup>1</sup> *H*, *C*<sup>1</sup> *Np*, *C*<sup>1</sup> *Npq*, *C*<sup>1</sup> *Eq* and *C*<sup>1</sup> *E*<sup>1</sup> product operators are the special cases of *C<sup>δ</sup> T* for *δ* ¼ 1.

Therefore, we establish two theorems so obtain best inaccuracy estimation of a function <sup>~</sup>*ζ*, conjugate to a 2*π*-periodic function *<sup>ζ</sup>* in weighted *W Lp* ð Þ , *ξ ω*ð Þ space of its CFS. Here we shall consider the two cases (i) *p*>1 and (ii) *p* ¼ 1 in order to get the Hölder's inequality satisfied. Our theorems generalizes six previously known results. Thus, the results of [5, 8–12] 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.*<sup>2</sup>

**Example 1** *The series*

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

conjugate to the FS

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

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*<sup>π</sup>* is a Banach space of all periodic functions with period 2*π* and continuous on the interval 0≤*x*≤2*π* under the supremum norm.

The best *λ*-order error approximation of a function ~*ζ* ∈*C*2*<sup>π</sup>* is defined by

$$E\_{\vec{\lambda}}(\tilde{\zeta}) = \inf\_{t\_{\vec{\lambda}}} \| \tilde{\zeta} - t\_{\vec{\lambda}} \|,$$

where *t<sup>λ</sup>* is a trigonometric polynomial of degree *λ* (Bernstein [14]). Let us define the *Lp* space of all 2*π*-periodic and integrable functions as

$$L^p[0,2\pi] := \left\{ \tilde{\zeta} : [0,2\pi] \to \mathbb{R} : \int\_0^{2\pi} \left| \tilde{\zeta}(\varkappa) \right|^p d\varkappa < \infty \right\}, p \ge 1.$$

Now, ∥*:*∥*<sup>p</sup>* is defined as

$$\|\tilde{\zeta}\|\_{p} = \begin{cases} \left\{ \frac{1}{2\pi} \int\_{0}^{2\pi} |\tilde{\zeta}(\mathbf{x})|^{p} \, d\mathbf{x} \right\}^{\frac{1}{p}} & \text{for } \mathbf{1} \le p < \infty, \\\ \text{ess}\, \sup\_{\mathbf{x} \in (0, 2\pi)} |\tilde{\zeta}(\mathbf{x})| & \text{for } p = \infty. \end{cases}$$

<sup>2</sup> FS denotes Fourier series and we use this abbreviation throughout the paper.

We define the following Lipschitz classes of function

$$\zeta \in Lipa \text{ if } Lipa := \{ \zeta : [0, 2\pi] \to \mathbb{R} : |\zeta(\infty + o) - \zeta(\infty)| = O(o^a) \} $$

for 0< *α*≤1;

$$\zeta \in \operatorname{Lip}(a, p) \text{ if } \operatorname{Lip}(a, p) \coloneqq \left\{ \zeta \in L^p[\mathbf{0}, 2\pi] : \|\zeta(\mathbf{x} + a) - \zeta(\mathbf{x})\|\_p = \mathcal{O}(a^a) \right\}$$

for *p*≥ 1, 0< *α*≤1;

$$\mathcal{L} \in \text{Lip}(a, \xi(\omega)) \text{ if } \text{Lip}(a, \xi(\omega)) \coloneqq \left\{ \zeta \in L^p[0, 2\pi] : \|\zeta(\mathbf{x} + \boldsymbol{\omega}) - \zeta(\mathbf{x})\|\_p = O(\xi(\boldsymbol{\omega})) \right\}.$$

$$\text{for } p \ge 1, 0 < a \le 1 \text{ & } \beta \ge 0;$$

$$\mathcal{L} \subset \mathcal{W}(L^p \xi(\omega))\\ \text{if } \mathcal{W}(L^p \xi(\omega)) \coloneqq \{ \zeta \in L^p[02\pi] : \left\| \left( \zeta(\mathbf{x} + \omega) - \zeta(\mathbf{x}) \right) \sin^\beta \left( \frac{\theta^\flat}{2} \right) \right\|\_p = O(\xi(\omega)) \} $$

where *ξ ω*ð Þ<sup>&</sup>gt; 0 and increasing with *<sup>ω</sup>*>0 and *<sup>L</sup><sup>p</sup>* space of all 2*π*-periodic and integrable functions. Under above assumptions for *α* ∈ð � 0, 1 , *p* ≥1,*ω*>0, we observed that

$$W(L^p, \xi(o)) \xrightarrow{\beta=0} Lip(\xi(o), p) \xrightarrow{\xi(o)=o^a} Lip(a, p) \xrightarrow{p \to \infty} Lipa.$$

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

#### **2.2 Some important single summability**

Let

$$\sum\_{\lambda=0}^{\infty} v\_{\lambda} \tag{1}$$

be an infinite series such that *sk* <sup>¼</sup> <sup>P</sup>*<sup>k</sup> <sup>m</sup>*¼<sup>0</sup> *vm*. Let

$$\sigma\_r^\eta = \sum\_{k=0}^r \frac{\binom{r-k+\eta-1}{r-k}}{\binom{r+\eta}{r}} s\_k \text{, for } \eta > -1. \tag{2}$$

If lim *<sup>λ</sup>*!<sup>∞</sup>*σ<sup>η</sup> <sup>λ</sup>* ¼ *s* then we say that the series (1) is ð Þ *C*, *η* summable to *s* or summable by Cesàro mean of order *η*.If we take *η* ¼ 0 in (2), ð Þ *C*, *η* summability reduces to an ordinary sum and if we take *η* ¼ 1, then ð Þ *C*, *η* summability reduces to ð Þ *C*, 1 summability or Cesàro summability of order 1.

Let

$$t\_{\lambda}^{E^{\mathbb{N}}} = \frac{1}{(1+q)^{\lambda}} \sum\_{k=0}^{\lambda} \binom{\lambda}{k} \frac{1}{q^{k-\lambda}} s\_k, q > 0.$$

If lim *<sup>λ</sup>*!<sup>∞</sup>*t Eq <sup>λ</sup>* ¼ *s* then we say that the series (1) is ð Þ *E*, *q* summable to *s* or summable by Euler mean ð Þ *E*, *q* (Hardy [15]). If *q* ¼ 0, ð Þ *E*, *q* method reduces to an ordinary sum and if *q* ¼ 1, ð Þ *E*, *q* means reduces to ð Þ *E*, 1 means.

An infinite series (1) with the sequence f g *s<sup>λ</sup>* of its partial sums is said to be summable by harmonic method (Riesz [16] or simply summable *N*, <sup>1</sup> *λ*þ1 � � to sum *<sup>s</sup>*, where *s* is a finite number, if the sequence to sequence transformation

$$t\_{\vec{\lambda}} = \frac{1}{\log \vec{\lambda}} \sum\_{v=0}^{\vec{\lambda}} \frac{s\_v}{\vec{\lambda} - v + 1} \text{ as } \vec{\lambda} \to \infty.$$

Let *p<sup>λ</sup>* � � be a sequence of constants, real or complex and let

$$P\_{\vec{\lambda}} = \sum\_{k=0}^{\vec{\lambda}} p\_k, (P\_{\vec{\lambda}} \neq \mathbf{0}).$$

Let

$$t\_{\lambda}^{N\_p} = \frac{1}{P\_{\lambda}} \sum\_{k=0}^{\lambda} p\_{\lambda-k} s\_k = \frac{1}{P\_{\lambda}} \sum\_{k=0}^{\lambda} p\_k s\_{\lambda-k}. \tag{3}$$

If

$$\lim\_{\vec{\lambda}\to\infty} t\_{\vec{\lambda}}^{N\_p} = s$$

then we say that the series (1) is *N*, *p<sup>λ</sup>* � � summable to *s* or summable by Nörlund *N*, *p<sup>λ</sup>* � � means.

Let *p<sup>λ</sup>* � � and *<sup>q</sup><sup>λ</sup>* � �, be two sequences of constants, real or complex such that

$$P\_{\lambda} = p\_0 + p\_1 + \dots + p\_{\lambda}; P\_{-1} = p\_{-1} = 0,\tag{4}$$

$$Q\_{\lambda} = q\_0 + q\_1 + \dots + q\_{\lambda}; Q\_{-1} = q\_{-1} = \mathbf{0}.\tag{5}$$

$$R\_{\lambda} = \sum\_{k=0}^{\lambda} p\_k q\_{\lambda-k} \neq \mathbf{0} \text{ for all } \lambda. \tag{6}$$

Convolution of the two sequences *p<sup>λ</sup>* � � and *<sup>q</sup><sup>λ</sup>* � �, is defined as

$$R\_{\vec{\lambda}} = (p \ast q)\_{\vec{\lambda}} = \sum\_{k=0}^{\vec{\lambda}} p\_k q\_{\vec{\lambda}-k}.$$

We write

$$t\_{\lambda}^{N\_{\text{PV}}} = \frac{1}{R\_{\lambda}} \sum\_{k=0}^{\lambda} p\_{\lambda-k} q\_k s\_k;$$

then the generalized Nörlund means *Np*,*<sup>q</sup>* � � of the sequence f g *<sup>s</sup><sup>λ</sup>* is denoted by the sequence *t pq <sup>λ</sup>* . If *t pq <sup>λ</sup>* ! *s*, as *λ* ! ∞ then, the series (1) is said to be summable to *s* by *Np*,*<sup>q</sup>* method and is denoted by *s<sup>λ</sup>* ! *s Np*,*<sup>q</sup>* � � ([17]).

*A Study on Approximation of a Conjugate Function Using Cesàro-Matrix Product Operator DOI: http://dx.doi.org/10.5772/intechopen.103015*

Let *p<sup>λ</sup>* � � be a sequence of real constants such that *<sup>p</sup>*<sup>0</sup> <sup>&</sup>gt;0, *<sup>p</sup><sup>λ</sup>* <sup>≥</sup>0 and *<sup>P</sup><sup>λ</sup>* <sup>¼</sup> <sup>P</sup>*<sup>λ</sup> <sup>v</sup>*¼<sup>0</sup> *pv* 6¼ 0, such that *<sup>P</sup><sup>λ</sup>* ! <sup>∞</sup> as *<sup>λ</sup>* ! <sup>∞</sup>. If

$$t\_{\vec{\lambda}} = \frac{1}{P\_{\vec{\lambda}}} \sum p\_v s\_v \to s \text{, as } \vec{\lambda} \to \infty,$$

then we say that f g *s<sup>λ</sup>* is summable by *N*, *p<sup>λ</sup>* � � means and we write

$$s\_{\lambda} = s(\overline{N}, p\_{\lambda}),$$

where f g *<sup>s</sup><sup>λ</sup>* is the sequence of *<sup>λ</sup>th* partial sum of the series (1).

Let *T* ¼ *l*ð Þ *<sup>λ</sup>*,*<sup>k</sup>* be an infinite triangular matrix satisfying the conditions of regularity [18] i.e.,

$$\begin{cases} \sum\_{k=0}^{\lambda} l\_{\lambda,k} = \mathbf{1} \text{ as } \lambda \to \infty \\\ l\_{\lambda,k} = \mathbf{0} \text{ for } k > \lambda \\\ \sum\_{k=0}^{\lambda} |l\_{\lambda,k}| \le M, a \text{ finite constant} \end{cases} \tag{7}$$

The sequence-to-sequence transformation

$$t\_{\lambda}^{T}(\check{\zeta}; \varkappa) := \sum\_{k=0}^{\lambda} l\_{\lambda,k} s\_k = \sum\_{k=0}^{\lambda} l\_{\lambda,k-k} s\_{\lambda-k}$$

defines the sequence *t T <sup>λ</sup>* ð Þ *ζ*; *x* of triangular matrix means of the sequence f g *s<sup>λ</sup>* generated by the sequence of coefficients *l*ð Þ *<sup>λ</sup>*,*<sup>k</sup>* .

If *t T <sup>λ</sup>* <sup>~</sup>*ζ*; *<sup>x</sup>* � � ! *<sup>s</sup>* as *<sup>λ</sup>* ! <sup>∞</sup> then the infinite series <sup>P</sup><sup>∞</sup> *<sup>λ</sup>*¼<sup>0</sup> *<sup>v</sup><sup>λ</sup>* or the sequence f g *<sup>s</sup><sup>λ</sup>* is summable to *s* by triangular matrix (*T*-method) [13].

#### **2.3** *C<sup>δ</sup> T* **product means**

we define *C<sup>δ</sup> T* means as

$$t\_{\lambda}^{C^{\delta}T}(\tilde{\zeta};\mathbf{x}) \coloneqq \sum\_{r=0}^{\lambda} \frac{\binom{\lambda-r+\delta-1}{\delta-1}}{\binom{\delta+\lambda}{\delta}} \sum\_{k=0}^{r} l\_{r,k} s\_k(\tilde{\zeta};\mathbf{x}) \tag{8}$$

If *t Cδ T <sup>λ</sup>* <sup>~</sup>*ζ*; *<sup>x</sup>* � � ! *<sup>s</sup>* as *<sup>λ</sup>* ! <sup>∞</sup>, then <sup>P</sup><sup>∞</sup> *<sup>λ</sup>*¼<sup>0</sup> *<sup>v</sup><sup>λ</sup>* is summable to *<sup>s</sup>* by *<sup>C</sup><sup>δ</sup> T* method. **Note 2** *Since C<sup>δ</sup> and T both are regular then C<sup>δ</sup> T method is also regular.* **Remark 2** *The special cases of C<sup>δ</sup> T means: C<sup>δ</sup> T transform reduces to*

i. *C<sup>δ</sup> <sup>H</sup>* transform if *<sup>l</sup><sup>λ</sup>*,*<sup>k</sup>* <sup>¼</sup> <sup>1</sup> ð Þ *<sup>λ</sup>*�*k*þ<sup>1</sup> log ð Þ *<sup>λ</sup>*þ<sup>1</sup> ;

$$\text{iii. } C^{\lambda}N\_p \text{ transform if } l\_{\lambda,k} = \frac{p\_{\lambda-k}}{p\_{\lambda}} \text{ where } P\_{\lambda} = \sum\_{k=0}^{\lambda} p\_k \neq 0;$$

iii. *C<sup>δ</sup> Np* transform if *<sup>l</sup><sup>λ</sup>*,*<sup>k</sup>* <sup>¼</sup> *pk Pλ* ;

iv. *C<sup>δ</sup> Eq* transform when *<sup>a</sup><sup>λ</sup>*,*<sup>k</sup>* <sup>¼</sup> <sup>1</sup> ð Þ <sup>1</sup>þ*<sup>q</sup> <sup>λ</sup> λ k* � �*q<sup>λ</sup>*�*k*;

$$\text{v. C}^{\delta}E^1 \text{ when } l\_{\lambda,k} = \frac{1}{2^{\lambda}} \binom{\lambda}{k};$$

vi. *C<sup>δ</sup> Npq* transform if *<sup>l</sup><sup>λ</sup>*,*<sup>k</sup>* <sup>¼</sup> *<sup>p</sup>λ*�*<sup>k</sup>qk <sup>R</sup><sup>λ</sup>* where *<sup>R</sup><sup>λ</sup>* <sup>¼</sup> <sup>P</sup>*<sup>λ</sup> <sup>k</sup>*¼<sup>0</sup> *pkq<sup>λ</sup>*�*k*.

In above special case (ii), (iii), and (vi) *p<sup>λ</sup>* and *q<sup>λ</sup>* are two non-negative monotonic non-increasing sequences of real constants.

**Remark 3** *C*<sup>1</sup> *H, C*<sup>1</sup> *Np, C*<sup>1</sup> *Npq, C*<sup>1</sup> *Eq and C*<sup>1</sup> *E*<sup>1</sup> *transforms are also the special cases of Cδ T for δ* ¼ 1*.*

**Example 2** *we consider*

$$(1 - 1574\sum\_{k=1}^{\bullet} \left(-1573\right)^{k-1}$$

The *λth* partial sum of the series (9) is given by

$$\mathfrak{s}\_{\mathbb{A}} = (-\mathbf{1}\mathbf{573})^{\mathbb{A}}, \forall \mathbb{A} \in \mathbb{N}^{0}$$

we take *<sup>l</sup>λ*,*<sup>k</sup>* <sup>¼</sup> <sup>1</sup> ð Þ <sup>787</sup> *<sup>λ</sup> λ k* � �ð Þ <sup>786</sup> *<sup>λ</sup>*�*<sup>k</sup>* , then *t T <sup>λ</sup>* ¼ *lλ*,0*s*<sup>0</sup> þ *lλ*,1*s*<sup>1</sup> þ … þ *lλ*,*λs<sup>λ</sup>* <sup>¼</sup> <sup>1</sup> ð Þ <sup>787</sup> *<sup>λ</sup> λ* 0 !ð Þ <sup>786</sup> *<sup>λ</sup>* � *λ* 1 !ð Þ <sup>786</sup> *<sup>λ</sup>*�<sup>1</sup> *:*1573 þ … þ *λ λ* !ð Þ �<sup>1573</sup> *<sup>λ</sup>* " # <sup>¼</sup> <sup>1</sup> ð Þ <sup>787</sup> *<sup>λ</sup>* ð Þ �<sup>787</sup> *<sup>λ</sup>* ¼ 1, *λ is even* �1, *<sup>λ</sup> is odd* ( (10)

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

Thus, *C<sup>δ</sup> T* means is more powerfull and effective than single *C<sup>δ</sup>* and *T* means. **Example 3** *we consider another infinite series*

$$1 - 6 + 30 - 150 + 750 - 3750 + 18750 - \dots \tag{11}$$

The *λth* partial sum of the series (11) is given by

$$\mathfrak{s}\_{\mathbb{A}} = (-\mathfrak{s})^{\mathbb{A}}, \forall \mathbb{A} \in \mathbb{N}\_0.$$

*A Study on Approximation of a Conjugate Function Using Cesàro-Matrix Product Operator DOI: http://dx.doi.org/10.5772/intechopen.103015*

$$\begin{aligned} \text{we take } l\_{\lambda,k} &= \frac{1}{3^{\lambda}} \binom{\lambda}{k} 2^{\lambda-k}, \text{ then} \\ &t\_{\lambda}^{T} = l\_{\lambda,0} \mathfrak{s}\_{0} + l\_{\lambda,1} \mathfrak{s}\_{1} + \dots + l\_{\lambda,k} \mathfrak{s}\_{\lambda} \\ &= \frac{1}{3^{\lambda}} \left[ \binom{\lambda}{0} 2^{\lambda} - \binom{\lambda}{1} 2^{\lambda-1} . 5 + \dots + \binom{\lambda}{\lambda} (-5)^{\lambda} \right] \\ &= \frac{1}{3^{\lambda}} (-3)^{\lambda} \\ &= \begin{cases} 1, & \lambda \in \{2m: m \in \mathbb{Z}\} \\ -1, & \lambda \in \{2m+1: m \in \mathbb{Z}\} \end{cases} \end{aligned} \tag{12}$$

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**

$$\tilde{K}\_{\lambda}^{C^{l}T} = \frac{1}{2\pi} \sum\_{r=0}^{\lambda} \frac{\left(\frac{r+\delta-1}{\delta-1}\right)}{\left(\frac{\delta+\lambda}{\delta}\right)} \sum\_{k=0}^{r} l\_{r,r-k} \frac{\cos\left(r-k+\frac{1}{2}\right)\alpha}{\sin\frac{\omega}{2}} \tag{13}$$

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

$$\wp(\mathbf{x}, \alpha) = \zeta(\mathbf{x} + \alpha) - \zeta(\mathbf{x} - \alpha)$$

We use the following in our work.

$$\frac{1}{\sin\left(\frac{\alpha}{2}\right)} \le \frac{\pi}{\alpha}, 0 < \alpha \le \pi \tag{14}$$

$$
\sin \alpha \le \alpha, \alpha \ge 0 \tag{15}
$$

$$|\cos \lambda o| \le \mathbf{1}, \forall o \in \mathbb{R} \tag{16}$$

Zygmund ([13]).

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

$$\begin{cases} l\_{\lambda, \lambda - k} - l\_{\lambda + 1, \lambda + 1 - k} \ge 0 \text{ for } 0 \le k \le \lambda \\\ L\_{\lambda, k} = \sum\_{r=k}^{\lambda} l\_{\lambda, \lambda - r} \text{ and } L\_{\lambda, 0} = 1, \forall \lambda \in \mathbb{N}\_0 \end{cases} \tag{17}$$

**Remark 4** *Considering the matrix T* ¼ *l*ð Þ *<sup>λ</sup>*,*<sup>k</sup> as*

$$l\_{\lambda,k} = \begin{cases} \frac{2018 \times (2019)^k}{(2019)^{k+1} - 1}, & 0 \le k \le \lambda \\\ 0, & k > \lambda \end{cases},$$

we can observe that (7, 17) satisfied.

**Remark 5** *Function ζ denotes a conjugate to a* 2*π-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 l* f g *<sup>λ</sup>*,*<sup>k</sup> , then*

$$|\tilde{K}\_{\lambda}^{C^{\tilde{\lambda}}T}(\boldsymbol{\alpha})| = O\left(\frac{1}{\boldsymbol{\alpha}}\right) \forall \delta \ge 1, \boldsymbol{0} < \boldsymbol{\alpha} \le \frac{\pi}{\lambda + 1}.$$

*Proof.* For 0 <*ω*≤ *<sup>π</sup> λ*þ1 , using (14, 15, 16)

∣*K*~*<sup>C</sup>δ<sup>T</sup> <sup>λ</sup>* ð Þ *<sup>ω</sup>* <sup>∣</sup> <sup>¼</sup> <sup>1</sup> 2*π* X *λ r*¼0 *r* þ *δ* � 1 *δ* � 1 � � *δ* þ *λ δ* � � <sup>X</sup>*<sup>r</sup> k*¼0 *lr*,*r*�*<sup>k</sup>* cos *r* � *k* þ 1 2 � �*<sup>ω</sup>* sin *<sup>ω</sup>* 2 � � � � � � � � � � � � � � � � � � ≤ 1 2*π* X *λ r*¼0 *r* þ *δ* � 1 *δ* � 1 ! *δ* þ *λ δ* ! <sup>X</sup>*<sup>r</sup> k*¼0 *lr*,*r*�*<sup>k</sup>* ∣ cos *r* � *k* þ 1 2 � �*ω*<sup>∣</sup> <sup>∣</sup>sin *<sup>ω</sup>* 2 ∣ ≤ 1 2*ω* X *λ r*¼0 *r* þ *δ* � 1 *δ* � 1 ! *δ* þ *λ δ* ! <sup>X</sup>*<sup>r</sup> k*¼0 *lr*,*r*�*<sup>k</sup>* ¼ 1 2*ω* X*ω r*¼0 ð Þ *r* þ *δ* � 1 ! ð Þ *<sup>δ</sup>* � <sup>1</sup> !*r*! � *δ*!*λ*! ð Þ *δ* þ *λ* ! *Lr*,0 <sup>¼</sup> *<sup>λ</sup>*!*<sup>δ</sup>* 2*ω δ*ð Þ þ *λ* ! X *λ r*¼0 ð Þ *r* þ *δ* � 1 ! *<sup>r</sup>*! since Lr,0 <sup>¼</sup> <sup>1</sup> <sup>¼</sup> *<sup>λ</sup>*!*<sup>δ</sup>* 2*ω δ*!ð Þ *δ* þ 1 ⋯ð Þ *δ* þ *λ* ð Þ *δ* � 1 ! 0! þ *δ*! 1! <sup>⋯</sup> <sup>þ</sup> ð Þ *<sup>λ</sup>* <sup>þ</sup> *<sup>δ</sup>* � <sup>1</sup> ! *λ*! � � ≤ *λ*!*δ* 2*ω δ*!ð Þ *δ* þ 1 ⋯ð Þ *δ* þ *λ* � ð Þ *λ* þ 1 *δ*!ð Þ *δ* þ 1 ⋯ð Þ *δ* þ *λ* � 1 *λ*! ¼ *δ* 2*ω* � *λ* þ 1 *δ* þ *λ* ≤ *δ* 2*ω* <sup>¼</sup> *<sup>O</sup>* <sup>1</sup> *ω* � � for all *<sup>δ</sup>*≥1*:*

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

$$\left| \dot{\boldsymbol{K}}\_{\lambda}^{\mathcal{C}^{\mathcal{C}}}(\boldsymbol{\alpha}) \right| = \left| (2\pi)^{-1} \sum\_{r=0}^{\lambda} \frac{\binom{r+\delta-1}{\delta-1}}{\binom{\delta+\lambda}{\delta}} \sum\_{k=0}^{r} l\_{r,r-k} \frac{\cos\left(r-k+\frac{1}{2}\right) \boldsymbol{\alpha}}{\sin\frac{\boldsymbol{\alpha}}{2}} \right|$$

$$\left| \boldsymbol{\alpha} \right| \tag{18}$$

$$=O\left(\frac{1}{a}\right)\left|\operatorname{Re}\sum\_{r=0}^{\lambda}\frac{\binom{r+\delta-1}{r}}{\binom{\delta-1}{\delta+\lambda}}\sum\_{k=0}^{r}l\_{r,r-k}e^{i\left(r-k+\frac{1}{2}\right)a}\right|$$

$$\left| \sum\_{r=0}^{\lambda} \frac{\binom{r+\delta-1}{\delta-1}}{\binom{\delta+\lambda}{\delta}} \sum\_{k=0}^{r} l\_{r,r-k} e^{i\binom{r-k+\frac{1}{2}}{2}o} \right| \le \left| \sum\_{r=0}^{\varrho} \frac{\binom{r+\delta-1}{\delta-1}}{\binom{\delta+\lambda}{\delta}} \sum\_{k=0}^{r} l\_{r,r-k} e^{i(r-k)o} \right| $$

$$+\left|\sum\_{r=q+1}^{\lambda} \frac{\binom{r+\delta-1}{\delta-1}}{\binom{\delta+\lambda}{\delta}} \sum\_{k=0}^{\varrho} l\_{r,r-k} e^{i(r-k)\alpha} \right|$$

$$+\left|\sum\_{r=q+1}^{\lambda} \frac{\binom{r+\delta-1}{\delta-1}}{\binom{\delta+\lambda}{\delta}} \sum\_{k=q+1}^{r} l\_{r,r-k} e^{i(r-k)\alpha} \right|$$

Now,

$$\Lambda\_1 \le \sum\_{r=0}^{\varrho} \frac{\binom{r+\delta-1}{}}{\binom{\delta-1}{\delta+\lambda}} \sum\_{k=0}^r \, \_{l,r-k} |e^{i(r-k)\alpha}|$$

$$\leq \sum\_{r=0}^{\varrho} \frac{\binom{r+\delta-1}{}}{\binom{\delta-1}{\delta+\lambda}} L\_{r,0}$$

$$=\sum\_{r=0}^{\varrho} \frac{(r+\delta-\mathbf{1})!}{(\delta-\mathbf{1})!r!} \times \frac{\delta!\lambda!}{(\delta+\lambda)!} \text{ since, } \mathbf{L}\_{\mathbf{r},0} = \mathbf{1}$$

$$=\frac{\lambda!}{\left(\delta+\mathbf{1}\right)\ldots\left(\delta+\lambda\right)\delta!}\sum\_{r=0}^{\varrho}\frac{\left(r+\delta-\mathbf{1}\right)!\delta!}{r!}$$

$$\begin{aligned} &= \frac{\lambda!}{\left(\delta+1\right)\dots\left(\delta+\phi\right)\dots\left(\delta+\lambda\right)} \left[1+\delta+\frac{\delta(\delta+1)}{2!}+\dots+\frac{\left(\rho+\delta-1\right)!}{\varrho!\left(\delta-1\right)!}\right] \\\\ &\leq \frac{\lambda!}{\left(\delta+1\right)\dots\left(\delta+\varrho\right)\dots\left(\delta+\lambda\right)} \left[\left(\varrho+1\right)\frac{\delta(\delta+1)\dots\left(\delta+\varrho-1\right)}{\varrho!}\right] \\\\ &= \frac{\varrho!\left(\varrho+1\right)\dots\lambda}{\left(\delta+\varrho\right)\dots\left(\delta+\lambda-1\right)} \times \frac{\delta}{\delta+\lambda} \times \frac{\varrho+1}{\varrho!} \\\\ &\leq \frac{\delta}{\delta+\lambda} \times \left(\varrho+1\right) \\\\ &\leq \frac{\delta}{\delta+\lambda} \times \left(\frac{1}{\varrho!}+1\right) \\\\ &= O\left(\frac{1}{\varrho(\lambda+1)}(1+\varkappa)\right) \text{ for all } \delta \geq 1. \end{aligned}$$

*A Study on Approximation of a Conjugate Function Using Cesàro-Matrix Product Operator DOI: http://dx.doi.org/10.5772/intechopen.103015*

Combining Λ1, Λ<sup>2</sup> and Λ<sup>3</sup> we have,

$$
\Lambda\_1 + \Lambda\_2 + \Lambda\_3 = O\left[\frac{1}{o(\lambda+1)} \times (1+o)\right] + O\left[\frac{1}{o(\lambda+1)}\right] + O\left[\frac{1}{o(\lambda+1)}\right]
$$

$$
= O\left[\frac{1}{(\lambda+1)} \left(1 + \frac{3}{o}\right)\right] \tag{20}
$$

$$
= O\left(\frac{1}{\lambda+1} \times \frac{3+\pi}{o}\right)
$$

(Let 1+<sup>3</sup> *<sup>ω</sup>* ≤ *<sup>k</sup> <sup>ω</sup>* for *<sup>ω</sup>* fixed *<sup>k</sup>min* <sup>¼</sup> <sup>3</sup> <sup>þ</sup> *<sup>π</sup>*) Now, from (19, 20) we get

$$\left| \bar{\mathcal{K}}\_{\lambda}^{C^{\delta}T}(a) \right| = O\left( \frac{1}{(\lambda+1)a^2} \right)$$

## **4. Main theorems**

**Theorem 4.1** *The error approximation of* <sup>~</sup>*<sup>ζ</sup> in W L<sup>p</sup>* ð Þ , *ξ ω*ð Þ *, p*ð Þ <sup>&</sup>gt;<sup>1</sup> *, by C<sup>δ</sup> T means of its CFS is given by*

$$\|t\_{\lambda}^{\mathcal{C}^{\mathcal{S}}}\left(\tilde{\zeta};\varkappa\right) - \tilde{\zeta}(\varkappa)\|\_{p} = \mathcal{O}\left[ (\lambda + \mathbf{1})^{\beta} \xi\left(\frac{\mathbf{1}}{\lambda + \mathbf{1}}\right) \right],$$

where 0 ≤*β* < <sup>1</sup> *<sup>p</sup>* and condition (17) holds and positive increasing function *ξ ω*ð Þ satisfies the following conditions:

$$\frac{\xi(\alpha)}{\alpha^{\beta+1-\sigma}}\text{ is non-decreasing};\tag{21}$$

$$\left\{ \int\_{0}^{\frac{\pi}{l+1}} \left( \frac{\lambda^{-\sigma} |\psi(\mathbf{x}, a)| \sin^{\beta} \left( \frac{\omega}{2} \right)}{\xi(a)} \right)^{p} da \alpha \right\}^{\frac{1}{p}} = O\left( \left( \lambda + 1 \right)^{\sigma - \frac{1}{p}} \right), \text{for } \beta < \sigma < \frac{1}{p};\tag{22}$$

$$\frac{\xi(o)}{o} \text{ is non-decreasing};\tag{23}$$

$$\text{and} \left\{ \int\_{\frac{\pi}{\lambda+1}}^{\pi} \left( \frac{o^{-\eta} |\psi(\mathbf{x}, a)| \sin^{\beta} \left(\frac{\omega}{2}\right)}{\xi(a)} \right)^{p} dao \right\}^{\frac{1}{p}} = O\left( (\lambda+1)^{\eta-\frac{1}{p}} \right),\tag{24}$$

where <sup>1</sup> *<sup>p</sup>* <sup>&</sup>lt; *<sup>η</sup>*<sup>&</sup>lt; *<sup>β</sup>* <sup>þ</sup> <sup>1</sup> *p* for *η* being an arbitrary number and *p* þ *q* ¼ *pq*. Conditions (22, 24) hold uniformly in *x*.

Conditions (22, 24) can be verified by using the fact that *<sup>ψ</sup>*ð Þ *<sup>x</sup>*, *<sup>ω</sup>* <sup>∈</sup>*W Lp*, *ξ ω*ð Þ � � and *<sup>ψ</sup>*ð Þ *<sup>x</sup>*,*<sup>ω</sup> ξ ω*ð Þ is bounded function.

*Proof.* The *λth* partial sums of the CFS is denoted by *s<sup>λ</sup>* ~*ζ*; *x* � �, and is given by

$$s\_{\lambda}(\check{\zeta};\mathfrak{x}) - \check{\zeta}(\mathfrak{x}) = \frac{1}{2\pi} \int\_{0}^{\pi} \psi(\mathfrak{x}, \alpha) \frac{\cos\left(\lambda + \frac{1}{2}\right)\alpha}{\sin\frac{\alpha}{2}} \, d\alpha,$$

one can consult [13] for detailed work on FS and CFS. Denoting *C<sup>δ</sup> T* means of *s<sup>λ</sup>* ~*ζ* : *x* � � � � by *t Cδ T <sup>λ</sup>* <sup>~</sup>*<sup>ζ</sup>* : *<sup>x</sup>* � �, we get

$$\begin{split} \mathbf{f}\_{\boldsymbol{\lambda}}^{\mathcal{C}^{\text{ur}}} \left( \tilde{\boldsymbol{\zeta}}; \mathbf{x} \right) - \tilde{\boldsymbol{\zeta}} (\mathbf{x}) &= \sum\_{r=0}^{\dot{\lambda}} \frac{\binom{\dot{\lambda} - r + \delta - 1}{\delta - 1}}{\binom{\delta + \dot{\lambda}}{\delta}} \sum\_{k=0}^{r} l\_{r,k} \left[ s\_k \left( \tilde{\boldsymbol{\zeta}}; \mathbf{x} \right) - \tilde{\boldsymbol{\zeta}} (\mathbf{x}) \right] \\ &= \frac{1}{2\pi} \int\_{0}^{\pi} \Psi(\mathbf{x}, \boldsymbol{\omega}) \sum\_{r=0}^{\dot{\lambda}} \frac{\binom{r + \delta - 1}{\delta - 1}}{\binom{\delta + \dot{\lambda}}{\delta}} \sum\_{k=0}^{r} l\_{r,r-k} \frac{\cos \left( r - k + \frac{1}{2} \right)\_{\delta 0}}{\sin \left( \frac{\delta \nu}{2} \right)} d\boldsymbol{\omega} \,\boldsymbol{\xi} \,\tag{25} \end{split} \tag{25}$$

$$\begin{aligned} &= \int\_0^\pi \psi(\mathbf{x}, a) \tilde{K}\_\lambda^{\mathbf{C}^\dagger T}(a) \, da \, \left( \text{By the notation } (\mathbf{13}) \right) \\ &= \int\_0^\mathbf{\tilde{f}} \psi(\mathbf{x}, a) \tilde{K}\_\lambda^{\mathbf{C} \tilde{\iota}^T}(\xi) \, da + \int\_{\tilde{\iota}^1}^\mathbf{\tilde{r}} \psi(\mathbf{x}, a) \tilde{K}\_\lambda^{\mathbf{C} \tilde{\iota}^T}(a) \, da \\ &= I\_1 + I\_2, \text{say} \end{aligned} \tag{26}$$

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

$$\begin{split} I\_{1} &= O(1) \left\{ \int\_{0}^{\frac{\pi}{\lambda+1}} \left( \frac{\boldsymbol{\alpha}^{-\sigma} |\boldsymbol{\varphi}(\boldsymbol{x},\boldsymbol{\alpha})| \sin^{\beta}\left(\frac{\boldsymbol{\alpha}}{2}\right)}{\xi(\boldsymbol{\alpha})} \right)^{p} d\boldsymbol{\alpha} \right)^{\frac{1}{p}} \times \left\{ \left[ \int\_{0}^{\frac{\pi}{\lambda+1}} \left( \frac{\xi(\boldsymbol{\alpha})}{\boldsymbol{\alpha}^{-\sigma+1} \sin^{\beta}\left(\frac{\boldsymbol{\alpha}}{2}\right)} \right)^{q} d\boldsymbol{\alpha} \right]^{\frac{1}{p}} \right\} \\ &= O\left[ (\lambda+1)^{\sigma-\frac{1}{p}} \times \left\{ \int\_{0}^{\frac{\pi}{\lambda+1}} \left( \frac{\xi(\boldsymbol{\alpha})}{\boldsymbol{\alpha}^{\beta+1-\sigma}} \right)^{q} d\boldsymbol{\alpha} \right\}^{\frac{1}{p}} \right] \\ &= O\left[ (\lambda+1)^{\sigma-\frac{1}{p}} (\lambda+1)^{\beta+\frac{1}{p}-\sigma} \xi\left( \frac{\pi}{\lambda+1} \right) \right] \\ &= O\left[ (\lambda+1)^{\theta} \xi\left( \frac{1}{\lambda+1} \right) \right] \end{split} \tag{27}$$

in view of condition (22) and *<sup>p</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>q</sup>*�<sup>1</sup> <sup>¼</sup> 1 and Remark 1.

$$\begin{split} I\_{2} &= O\left(\frac{1}{\lambda+1}\right) \int\_{\frac{\pi}{\lambda+1}}^{\frac{1}{\mu}} \frac{|\eta(x,a)|}{a^{2}} \, da \\ &= O\left\{\frac{1}{\lambda+1} \int\_{\frac{\pi}{\lambda+1}}^{\pi} \left(\frac{a\eta - \eta|\eta(x,a)|\sin\frac{\theta}{\lambda}(\frac{a}{2})}{\xi(a)}\right)^{p} \, da\right\}^{\frac{1}{p}} \times \left\{\int\_{\frac{\pi}{\lambda+1}}^{\pi} \left(\frac{a^{p-1}\xi(a)}{a^{p-\eta+1+\theta}}\right)^{q} \, da\right\}^{\frac{1}{q}} \\ &= O\left[\left(\lambda+1\right)^{-1+\eta-\frac{1}{p}} \xi\left(\frac{\pi}{\lambda+1}\right) \left(\frac{\lambda+1}{\pi}\right) \left(\int\_{\frac{\pi}{\lambda+1}}^{\pi} a^{-(\beta+1-\eta)q} \, da\right)^{\frac{1}{q}}\right] \\ &= O\left[\left(\lambda+1\right)^{\eta-\frac{1}{p}} \xi\left(\frac{\pi}{\lambda+1}\right) \left(\lambda+1\right)^{\beta+1-\eta-\frac{1}{q}}\right] \\ &= O\left[\left(\lambda+1\right)^{\theta} \xi\left(\frac{1}{\lambda+1}\right)\right] \end{split} \tag{28}$$

$$\left| t\_{\lambda}^{C^{\delta}T}(\tilde{\zeta}, \varkappa) - \tilde{\zeta}(\varkappa) \right| = O\left[ (\lambda + 1)^{\beta} \xi \left( \frac{1}{\lambda + 1} \right) \right].$$

$$\left\| t\_{\lambda}^{C^{\delta}T}(\tilde{\zeta}, \varkappa) - \tilde{\zeta}(\varkappa) \right\|\_{p} = \mathcal{O}\left[ (\lambda + \mathbf{1})^{\beta} \xi \left( \frac{\mathbf{1}}{\lambda + \mathbf{1}} \right) \right]$$

$$\|\mathfrak{t}\_{\lambda}^{\mathcal{C}^{\mathcal{T}}}(\tilde{\zeta};\mathfrak{x}) - \tilde{\zeta}(\mathfrak{x})\|\_{1} = \mathcal{O}\left[ (\lambda + \mathbf{1})^{\beta} \xi \left( \frac{\mathbf{1}}{\lambda + \mathbf{1}} \right) \right],$$

$$\begin{aligned} I\_1 &= O\left\{ \left| \int\_0^{\frac{\pi}{\ell+1}} \left( \frac{\alpha^{-\sigma} |\varphi(\mathbf{x}, \boldsymbol{\alpha})| \sin^{\beta} \left( \frac{\alpha \mathbf{v}}{2} \right)}{\xi(\boldsymbol{\alpha})} \right) d\boldsymbol{\alpha} \times \operatorname\*{ess\,sup}\_{0 < \alpha \le \frac{\pi}{\ell+1}} \left| \frac{\xi(\boldsymbol{\alpha})}{\alpha^{-\sigma+1} \sin^{\beta} \left( \frac{\alpha \mathbf{v}}{2} \right)} \right| \right\} \right\} \\ &= O\left( (\lambda + 1)^{\sigma-1} \right) \operatorname\*{ess\,sup}\_{0 < \alpha \le \frac{\pi}{\ell+1}} \left| \frac{\xi(\boldsymbol{\alpha})}{\alpha^{\beta-\sigma+1}} \right| \end{aligned}$$

$$\begin{aligned} \mathbf{I} &= O\left( (\lambda + \mathbf{1})^{\sigma - 1} \right) \left\{ \frac{\xi\left(\frac{\pi}{\lambda + \mathbf{1}}\right)}{\left(\frac{\pi}{\lambda + \mathbf{1}}\right)^{\beta - \sigma + 1}} \right\} \\\\ \mathbf{I} &= O\left( (\lambda + \mathbf{1})^{\beta} \xi\left(\frac{\mathbf{1}}{\lambda + \mathbf{1}}\right) \right) \end{aligned} \tag{29}$$

$$\begin{split} I\_{2} &= O\left(\frac{1}{\lambda+1}\right) \int\_{\frac{\pi}{\lambda+1}}^{\pi} \frac{|\psi(x,o)|}{o^{2}} \, do \\ &= O\left\{ \frac{1}{\lambda+1} \int\_{\frac{\pi}{\lambda+1}}^{\pi} \frac{o^{-\eta}|\psi(x,o)|\sin^{\beta}\left(\frac{o\eta}{2}\right)}{\xi(o)} \, doo \right\} \times \underset{\frac{\pi}{\lambda+1} \le o \le \pi}{\text{sup}} \left| \frac{\xi(o)}{o^{-\eta+\beta+2}} \right| \\ &= O\left[ (\lambda+1)^{\eta-2} \xi\left(\frac{\pi}{\lambda+1}\right) \left(\frac{(\lambda+1)^{2+\beta-\eta}}{\pi^{2+\beta-\eta}}\right) \right] \\ &= O\left[ (\lambda+1)^{\beta} \xi\left(\frac{\pi}{\lambda+1}\right) \right] \end{split} \tag{30} $$

$$|t\_{\lambda}^{\mathbb{C}^{\otimes T}}(\tilde{\boldsymbol{\xi}};\boldsymbol{x}) - \tilde{\boldsymbol{\xi}}(\boldsymbol{x})| = O\left[ (\lambda + \mathbf{1})^{\beta} \tilde{\boldsymbol{\xi}}\left(\frac{\mathbf{1}}{\lambda + \mathbf{1}}\right) \right] \tag{31}$$

$$\|\mathfrak{t}\_{\lambda}^{\mathcal{C}^{\mathcal{S}}}(\tilde{\zeta};\mathfrak{x}) - \tilde{\zeta}(\mathfrak{x})\|\_{1} = \mathcal{O}\left[ (\lambda + \mathbf{1})^{\beta} \xi \left( \frac{\mathbf{1}}{\lambda + \mathbf{1}} \right) \right]$$

$$\left\| t\_{\lambda}^{C^{\tilde{\lambda}}T}(\tilde{\zeta}, \varkappa) - \tilde{\zeta}(\varkappa) \right\|\_{p} = O\left[ \xi \left( \frac{1}{\lambda + 1} \right) \right],$$

$$\left\| t\_{\lambda}^{C^{\tilde{s}}T}(\tilde{\zeta}, \varkappa) - \tilde{\zeta}(\varkappa) \right\|\_{p} = O[(\lambda + 1)^{-a}],$$

*A Study on Approximation of a Conjugate Function Using Cesàro-Matrix Product Operator DOI: http://dx.doi.org/10.5772/intechopen.103015*

where, *C<sup>δ</sup> T* is as defined in (8).

*Proof.* If we consider *<sup>β</sup>* <sup>¼</sup> 0 & *ξ ω*ð Þ¼ *ωα* in Theorem 4.1, we can obtain the proof. **Corollary 5.3** *The error estimate of* <sup>~</sup>*<sup>ζ</sup> in Lip<sup>α</sup>* ð Þ <sup>0</sup><*<sup>α</sup>* <sup>&</sup>lt;<sup>1</sup> *class by C<sup>δ</sup> T product means of its CFS is given by*

$$\left\| \left| t\_{\lambda}^{C^{\tilde{\varsigma}}T} (\tilde{\zeta}, \varkappa) - \tilde{\zeta}(\varkappa) \right| \right\|\_{p} = O[(\lambda + 1)^{-a}]$$

where, *C<sup>δ</sup> T* is as defined in (8).

*Proof.* If we take *<sup>β</sup>* <sup>¼</sup> 0 & *ξ ω*ð Þ¼ *<sup>ω</sup><sup>α</sup>* & *<sup>p</sup>* ! <sup>∞</sup> in Theorem 4.1, we can obtain the proof. For *α* ¼ 1, we can write an independent proof to obtain

$$\left\| t\_{\lambda}^{C^{\tilde{\lambda}}T}(\tilde{\zeta}, \varkappa) - \tilde{\zeta}(\varkappa) \right\|\_{\infty} = O\left[ \frac{\log \left( \lambda + \mathbf{1} \right)}{\lambda + \mathbf{1}} \right]$$

**Corollary 5.4** *The error estimate of* <sup>~</sup>*<sup>ζ</sup>* <sup>∈</sup> *W L<sup>p</sup>* ð Þ , *ξ ω*ð Þ *class by C<sup>δ</sup> H means*

$$t\_{\lambda}^{C^{\delta}H} = \sum\_{r=0}^{\lambda} \frac{\binom{\lambda - r + \delta - 1}{\delta - 1}}{\binom{\delta + \lambda}{\delta}} \left( \log \left( r + 1 \right) \right)^{-1} \sum\_{k=0}^{r} \frac{1}{\left( r - k + 1 \right)} s\_k,$$

of the CFS is given by

$$\|\mathfrak{t}\_{\lambda}^{\mathcal{C}^{\mathrm{SH}}}(\tilde{\zeta};\mathfrak{x}) - \tilde{\zeta}(\mathfrak{x})\|\_{p} = \mathcal{O}\left[ (\lambda + \mathbf{1})^{\beta} \xi \left( \frac{\mathbf{1}}{\lambda + \mathbf{1}} \right) \right]$$

provided *C<sup>δ</sup> T* defined in (8) and *ξ ω*ð Þ satisfies the conditions (21) to (24). **Corollary 5.5** *The error estimate of* <sup>~</sup>*<sup>ζ</sup>* <sup>∈</sup> *W L<sup>p</sup>* ð Þ , *ξ ω*ð Þ *class by C<sup>δ</sup> Np means*

$$t\_{\lambda}^{C^{\diamond}N\_p} = \sum\_{r=0}^{\lambda} \frac{\binom{\lambda - r + \delta - 1}{\delta - 1}}{\binom{\delta + \lambda}{\delta}} \frac{1}{P\_r} \sum\_{k=0}^r p\_{r-k} s\_k$$

of the CFS is given by

$$\|t\_{\lambda}^{\mathcal{C}^{\aleph\_{P}}}(\tilde{\zeta};\varkappa) - \tilde{\zeta}(\varkappa)\|\_{p} = O\left[ (\lambda + \mathbf{1})^{\beta} \xi \left(\frac{\mathbf{1}}{\lambda + \mathbf{1}}\right) \right]$$

provided *C<sup>δ</sup> T* defined in (8) and *ξ ω*ð Þ satisfies the conditions (21) to (24). **Corollary 5.6** *The error estimate of* <sup>~</sup>*<sup>ζ</sup>* <sup>∈</sup>*W Lp* ð Þ , *ξ ω*ð Þ *class by C<sup>δ</sup> Npq means*

$$t\_{\lambda}^{C^{\delta}N\_{\mathbb{P}\mathbb{I}}} = \sum\_{r=0}^{\lambda} \frac{\binom{\lambda-r+\delta-1}{\delta-1}}{\binom{\delta+\lambda}{\delta}} \frac{1}{R\_r} \sum\_{k=0}^r p\_{r-k} q\_k s\_k,$$

of the CFS is given by

$$\|\mathfrak{t}\_{\lambda}^{\mathcal{C}^{\partial\mathbb{N}\_{\mathbb{P}^{\mathcal{I}}}}}(\tilde{\zeta};\mathfrak{x}) - \tilde{\zeta}(\mathfrak{x})\|\_{p} = O\left[ (\lambda+\mathbf{1})^{\beta}\xi\left(\frac{\mathbf{1}}{\lambda+\mathbf{1}}\right) \right],$$

provided *C<sup>δ</sup> T* defined in (8) and *ξ ω*ð Þ satisfies the conditions (21) to (24). **Corollary 5.7** *The error approximation of* <sup>~</sup>*<sup>ζ</sup>* <sup>∈</sup>*W Lp* ð Þ , *ξ ω*ð Þ *class by C<sup>δ</sup> Np means*

$$t\_{\lambda}^{\mathcal{C}\overline{N}\_{\mathbb{P}}} = \sum\_{r=0}^{\lambda} \frac{\binom{\lambda - r + \delta - 1}{\delta - 1}}{\binom{\delta + \lambda}{\delta}} \frac{1}{P\_r} \sum\_{k=0}^r p\_k s\_k,$$

of the CFS is given by

$$\|t\_{\lambda}^{C^{\beta}\overline{N}\_{p}}(\tilde{\zeta};\varkappa) - \tilde{\zeta}(\varkappa)\|\_{p} = O\left[ (\lambda+\mathbf{1})^{\beta}\xi\left(\frac{\mathbf{1}}{\lambda+\mathbf{1}}\right) \right],$$

provided *C<sup>δ</sup> T* defined in (8) and *ξ ω*ð Þ satisfies the conditions (21) to (24). **Corollary 5.8** *The error estimate of* <sup>~</sup>*<sup>ζ</sup>* <sup>∈</sup> *W Lp* ð Þ , *ξ ω*ð Þ *class by C<sup>δ</sup> Eq means*

$$t\_{\lambda}^{C^{\delta}E^{\delta}} = \sum\_{r=0}^{\delta} \frac{\binom{\delta-r+\delta-1}{\delta-1}}{\binom{\delta+\delta}{\delta}} \frac{1}{(1+q)^r} \sum\_{k=0}^r \binom{r}{k} q^{r-k} s\_k,$$

of the CFS is given by

$$\|\|t\_{\lambda}^{\mathcal{C}^{\rm SHP}}(\tilde{\zeta};\varkappa) - \tilde{\zeta}(\varkappa)\|\_{p} = \mathcal{O}\left[ (\lambda+1)^{\beta} \xi\left(\frac{1}{\lambda+1}\right) \right],$$

provided *C<sup>δ</sup> T* defined in (8) and *ξ ω*ð Þ satisfies the conditions (21) to (24). **Corollary 5.9** *The error estimate of <sup>ζ</sup>* <sup>∈</sup>*W Lp* ð Þ , *ξ ω*ð Þ *class by C<sup>δ</sup> E*<sup>1</sup> *means*

$$t\_{\lambda}^{C^{\delta}E^{4}} = \sum\_{r=0}^{\lambda} \frac{\binom{\lambda - r + \delta - 1}{\delta - 1}}{\binom{\delta + \lambda}{\delta}} \frac{1}{2^{r}} \sum\_{k=0}^{r} \binom{r}{k} s\_{k},$$

of the FS is given by

$$\|\mathfrak{t}\_{\lambda}^{C^{\text{sif}^1}}(\tilde{\zeta};\mathfrak{x}) - \tilde{h}(\mathfrak{x})\|\_{p} = \mathcal{O}\left[ (\lambda + \mathbf{1})^{\beta} \xi \left( \frac{\mathbf{1}}{\lambda + \mathbf{1}} \right) \right]$$

provided *C<sup>δ</sup> T* defined in (8) and *ξ ω*ð Þ 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*<sup>1</sup> *Np, C*<sup>1</sup> *Npq, C*<sup>1</sup> *N*~ *<sup>p</sup>*,*C*<sup>1</sup> *Eq and C*<sup>1</sup> *E*<sup>1</sup> *all things considered Remark 3.*

*A Study on Approximation of a Conjugate Function Using Cesàro-Matrix Product Operator DOI: http://dx.doi.org/10.5772/intechopen.103015*

## **6. Particular cases**

The following special cases of our theorems for *δ* ¼ 1 are.

**6.1.** If we take Remark 1ð Þ *iv* and *<sup>β</sup>* <sup>¼</sup> 0, *ξ ω*ð Þ¼ *<sup>ω</sup><sup>α</sup>*, 0 <sup>&</sup>lt;*α*<sup>≤</sup> 1 in our theorem, then the Theorem 2 of [8] become a special case of our theorem.

**6.2.** If *<sup>β</sup>* <sup>¼</sup> 0, *ξ ω*ð Þ¼ *ωα*, 0 <sup>&</sup>lt;*α*<sup>≤</sup> 1 & *<sup>p</sup>* ! <sup>∞</sup> in our theorem, then the Theorem 3.3 of [9] become a special case of our result.

**6.3.** If we consider Remark 2ð Þ *iv* 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 ð Þ *iv* then the Theorem 3.1 of [11] become a special case of our result.

**6.6.** If we consider Remark 2ð Þ *ii* 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 � <sup>4038</sup>P<sup>∞</sup> *<sup>j</sup>*¼<sup>1</sup> ð Þ �<sup>4038</sup> *<sup>j</sup>*�<sup>1</sup> is neither summable by matrix means(T) nor Cesáro means of order one *C*<sup>1</sup> � � but it summable by *C<sup>δ</sup> T* means for *δ* ¼ 1.

**Q. 7.2.** Prove that a function *f* is 2*π*-periodic and Lebesgue integrable then the error approximation of *f* in Lip*α* class by *C<sup>δ</sup> T* product means of its Fourier series is given by

$$E\_n(f) = \begin{cases} O[\left(n+1\right)^{-a}], & 0 \le a < 1\\ O\left[\left(n+1\right)^{-1}\{\log\left(n+1\right)\}\right], & a = 1, \end{cases}$$

where *C<sup>δ</sup> T* is as defined in (8) and provided (17) holds. {Hint: see [19]}.

**Q. 7.3.** Consider the matrix *T* ¼ ð Þ *an*,*<sup>k</sup>* as

$$a\_{n,k} = \begin{cases} \frac{2 \times 3^k}{3^{n+1} - 1}, & 0 \le k \le j \\ 0, & k > n \end{cases},$$

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 f g *a<sup>λ</sup>*,*<sup>k</sup>* , then prove that

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

## **Acknowledgements**

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

## **Author details**

Mohammed Hadish Government Inter College, Fatehpur, Uttar Pradesh, India

\*Address all correspondence to: hadish@cusb.ac.in

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*A Study on Approximation of a Conjugate Function Using Cesàro-Matrix Product Operator DOI: http://dx.doi.org/10.5772/intechopen.103015*

## **References**

[1] Rhaodes BE. On the degree of approximation of functions belonging to Lipschitz class by Hausdorff means of its Fourier series. Tamkang Journal of Mathematics. 2003;**34**(3):245-247 MR2001920 (2004g: 41023). Zentralblatt für Mathematik und ihre Grenzgebiete 1039.42001

[2] Nigam HK, Sharma K. A study on degree of approximation by Karamata summability method. J. Inequal. Appl. 2011;**85**(1):1-28

[3] Kushwaha JK. On the approximation of generalized Lipschitz function by Euler means of conjugate series of Fourier series. Scientific World Journal. 2013;**2013**:508026

[4] Mittal ML, Singh U, Mishra VN. approximation of functions (signals) belonging to *W Lp* ð Þ , *<sup>ξ</sup>*ð Þ*<sup>t</sup>* -class by means of conjugate Fourier series using Norlund operators. Varahmihir J. Math. Sci. India. 2006;**6**(1):383-392

[5] Nigam HK. A study on approximation of conjugate of functions belonging to Lipschitz class and generalized Lipschitz class by product summability means of conjugate series of Fourier series. Thai Journal of Mathematics. 2012;**10**(2): 275-287

[6] Nigam HK, Hadish M. Best approximation of functions in generalized Hölder class. Journal of Inequalities and Applications. 2018;**1**:1-15

[7] Nigam HK, Hadish M. A study on approximation of a conjugate function using Cesàro-Matrix product operator. Nonlinear Functional Analysis and Applications. 2020;**25**(4):697-713

[8] Sonkar S, Singh U. Degree of approximation of the conjugate of signals (functions) belonging to Lip ð Þ *α*,*r* -class by ð Þ *C*, 1 ð Þ *E*, *q* means of conjugate trigonometric Fourier series. Journal of inequalities and applications. 2012;**2012**: 278

[9] Deger U. On approximation to the functions in the *W Lp*, *<sup>ξ</sup>*ð Þ*<sup>t</sup>* class by a new matrix mean. Novi Sad J. Math. 2016;**46**(1):1-14

[10] Singh U, Srivastava SK. Trigonometric approximation of functions belonging to certain Lipschitz classes by *C*<sup>1</sup> *T* operator. Asian-European J. Math. 2014;**7**(4):1450064

[11] Mishra VN, Khan HH, Khatri K, Mishra LN. Degree of approximation of conjugate signals (functions) belonging to the generalized weighted *W Lr* ð Þ , *ξ*ð Þ*t* ,*r*≥1 class by ð Þ *C*, 1 ð Þ *E*, *q* means of conjugate Trigonometric Fourier series. Bulletin of Mathematical Analysis and Applications. 2013;**5**(4):40-53

[12] Mishra VN, Khan HH, Khatri K, Mishra LN. Approximation of functions belonging to the generalized Lipschitz class by *C*<sup>1</sup> *Np* summability method of conjugate series of Fourier series. Mathematicki vesnik. 2014;**66**(2):155-164

[13] Zygmund A. Trigonometric Series. 3rd rev. ed. Cambridge: Cambridge Univ. Press; 2002

[14] Bernshtein SN. On the best approximation of continuous functions by polynomials of given degree, in: Collected Works [in Russian], Izd. Nauk SSSR, Moscow. 1952;**1**:11-1047

[15] Hardy GH. Divergent Series. Oxford University Press; 1949

[16] Riesz M. Sur l'equivalence decertaines method de sommation, proceeding of London. Mathematical Society. 1924;**22**:412-419

[17] Borwein D. On products of sequences. J. London Math. Soc. 1958;**33**: 352-357

[18] Töeplitz O. Uberallagemeine lineara Mittelbil dunger. P.M.F. 1913;**22**:113-119

[19] Nigam HK, Hadish M. On approximation of function in generalized Zygmund class using *Cη<sup>T</sup>* operator. Journal of Mathematical Inequalities. 2020;**14**(1):273-289

## **Chapter 6**
