Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 4s (2025) 

 

118 
https://internationalpubls.com 

Degree of Approximation of the Conjugate of Functions Belonging to 

LIP (𝜶, 𝒓) −Class by (𝑪, 𝟏)(𝑬, 𝒒)(𝑬, 𝒒) Means of Conjugate Fourier 

Series  

 

Rupesh Kumar Mishra 𝟏 and Shambhu Kumar Mishra  𝟐 

 1Research Scholar, Department of Mathematics, Patliputra University, Patna, Bihar, India 

 2Professor, Department of Mathematics, Patliputra University, Patna, Bihar, India 

E-mail:  1rupeshmishra043@gmail.com, 2shambhumishra5@gmail.com 

 

Article History: 

Received: 16-09-2024 

Revised: 21-11-2024 

Accepted: 29-11-2024 

Abstract:  

This research paper is related to the degree of approximation of the conjugate of 

2𝜋 −periodic function belonging to the Lip(𝛼, 𝑟)(0 < 𝛼 ≤ 1, 𝑟 ≥ 1)- class by using 

(𝐶, 1)(𝐸, 𝑞)(𝐸, 𝑞) means of the conjugate Fourier series. Our result may be useful for the 

coming researchers in the future. 

Keywords: Lip(𝛼, 𝑟) − class, conjugate Fourier series, (𝐶, 1)(𝐸, 𝑞)(𝐸, 𝑞) means. 

 

1.  Introduction 

 Let ∑∞
𝑛=0 𝑢𝑛 be a given infinite series and the sequence {𝑠𝑛} its nth partial sum.The sequence -to- 

sequence transform  

 𝐶𝑛
1 =

1

𝑛+1
∑𝑛

𝑘=0 𝑠𝑘,    𝑛 = 0,1,2, . .. (1) 

 defines the Cesàro means of order one of {𝑠𝑛}. If 𝑙𝑖𝑚𝑛→∞𝐶𝑛
1 = 𝑠, the series ∑∞

𝑛=0 𝑢𝑛 is said to be 

(𝐶, 1) summable to s. 

The sequence-to-sequence transform  

 𝐸𝑛
𝑞 =

1

(1+𝑞)𝑛
∑𝑛

𝑘=0 (
𝑛
𝑘

) 𝑞𝑛−𝑘𝑠𝑘, 𝑞 > 0, 𝑛 = 0,1,2, . .. (2) 

 defines the Euler mean of order 𝑞 > 0 of {𝑠𝑛}.  

 𝐶𝑛
1𝐸𝑛

𝑞𝐸𝑛
𝑞 =

1

𝑛+1
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣𝑠𝑣 (3) 

The series ∑∞
𝑛=0 𝑢𝑛 is said to be (𝐶, 1)(𝐸, 𝑞)(𝐸, 𝑞) summable to s, if im 𝑛→∞𝐶𝑛

1𝐸𝑛
𝑞𝐸𝑛

𝑞 = 𝑠. 

For a 2𝜋 periodic signal which is integrable in the sense of Lebesgue over (−𝜋, 𝜋). 

The conjugate of Fourier series is defined by  

 ∑∞
𝑘=1 (𝑏𝑘𝑐𝑜𝑠𝑘𝑥 − 𝑎𝑘𝑐𝑜𝑠𝑘𝑥) (4) 

               and nth partial sum is defined by  

 �̃�𝑛(𝑓; 𝑥) = ∑∞
𝑘=1 (𝑏𝑘𝑐𝑜𝑠𝑘𝑥 − 𝑎𝑘𝑐𝑜𝑠𝑘𝑥) (5) 



Communications on Applied Nonlinear Analysis 

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 The conjugate of f denoted by f̃ is defined by  

 𝑓(𝑥) = −
1

2𝜋
𝑙𝑖𝑚𝜉→0 ∫

𝜋

𝜉
𝜓(𝑡)𝑐𝑜𝑠 (

𝑡

2
) 𝑑𝑡, 

            where 𝜓(𝑡) = 𝑓(𝑥 + 𝑡) − 𝑓(𝑥 − 𝑡)  

A function 𝑓 ∈ Lip𝛼, if 

|𝑓(𝑥 + 𝑡) − 𝑓(𝑥 + 𝑡)| = 𝑂(|𝑡|𝛼)   𝑓𝑜𝑟    0 < 𝛼 ≤ 1. 

and 𝑓 ∈ 𝐿𝑖𝑝(𝛼, 𝑟) if  

 (∫
2𝜋

0
|𝑓(𝑥)|𝑟)

1

𝑟
= 𝑂(𝑡𝛼),   0 < 𝛼 ≤ 1, 𝑟 ≥ 1. 

𝐿𝑝- norm is defined by  

 𝑓𝑝 = (∫
2𝜋

0
|𝑓(𝑥)|𝑝)

1

𝑝
, 𝑝 ≥ 1. 

L∞-norm of a function 𝑓: 𝑅 → 𝑅 is defined by 𝑓∞ 

𝑓∞ = 𝑠𝑢𝑝{|𝑓(𝑥)|/𝑓: 𝑅 → 𝑅} 

The degree of approximation of function 𝑓: 𝑅 → 𝑅 by a trigonometric polynomial 𝑡𝑛[1] is defined by  

𝑡𝑛 − 𝑓∞ = 𝑠𝑢𝑝{|𝑡𝑛 − 𝑓|: 𝑥 ∈ 𝑅}𝑜𝑟𝑡𝑛 − 𝑓𝑝 = 𝑚𝑖𝑛𝑡𝑛 − 𝑓. 

This method of approximation is called trigonometric Fourier approximation. 

We also write 

𝐶𝑛
1𝐸𝑛

𝑞𝐸𝑛
𝑞 =

1

𝑛 + 1
∑

𝑛

𝑘=0

1

(1 + 𝑞)𝑘
∑

𝑘

𝑢=0

(
𝑘
𝑢

)
𝑞𝑘−𝑢

(1 + 𝑞)𝑢
∑

𝑢

𝑣=0

(
𝑢
𝑣

) 𝑞𝑢−𝑣
𝑐𝑜𝑠 (𝑣 +

1
2

) 𝑡

𝑠𝑖𝑛 (
𝑡
2

)
 

and 𝜏 = [
1

𝑡
], the integral part of 

1

𝑡
.  

2.  Known theorem 

Various investigators such as Dhakal[2], Lal and Singh[8], Mittal et al. [6,7], Qureshi[4,5] Sonker and 

Singh[9] have studied the degree of approximation in various function spaces such as Lip 𝛼 , Lip(𝛼, 𝑟), 

Lip(𝜉(𝑡), 𝑟) and weighted (𝐿𝑟 , 𝜉(𝑡)) by using triangular matrix summability and product summability 

(C,1)(E,1), (N,𝑝𝑛)(E,1).  Sonker and Singh[9] have determined the degree of approximation of the 

conjugate of signals (functions) belonging to Lip(𝛼, 𝑟)-class by(𝐶, 1)(𝐸, 𝑞) means of conjugate 

trigonometric Fourier series. Sonker and Singh have proved the following:  

Theorem 1 

 [9] Let 𝑓(𝑥) be a 2𝜋-periodic, Lebesgue integrable function and belonging to the Lip(𝛼, 𝑟)- class with 

𝑟 ≥ 1 and 𝛼𝑟 ≥ 1. Then the degree of approximation of f̃(x), the conjugate of 𝑓(𝑥) by (𝐶, 1)(𝐸, 𝑞) 

means of its conjugate Fourier series is given by  



Communications on Applied Nonlinear Analysis 

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 𝐶𝑛
1𝐸𝑛

𝑞 − 𝑓𝑟 = 𝑂 (𝑛
1

𝑟
−𝛼) , 𝑛 = 0,1,2, . . . . ., (6) 

Main theorem 

 The objective of this paper is to establish the following theorem.  

Theorem 2 

 Let 𝑓(𝑥) be a 2𝜋-periodic, Lebesgue integrable function and belonging to the Lip(𝛼, 𝑟)- class with 

𝑟 ≥ 1 and 𝛼𝑟 ≥ 1. Then the degree of approximation of f̃(x), the conjugate of 𝑓(𝑥) by 

(𝐶, 1)(𝐸, 𝑞)(𝐸, 𝑞) means of its conjugate Fourier series is given by  

 𝐶𝑛
1𝐸𝑛

𝑞
𝐸𝑛

𝑞
− 𝑓𝑟 = 𝑂 (𝑛

1

𝑟
−𝛼) , 𝑛 = 0,1,2, . . . . ., (7) 

 provided  

 (∫
𝜋

𝑛+1

0
(|𝜓(𝑡)|/𝑡𝛼)𝑟𝑑𝑡)

1

𝑟

= 𝑂 (
1

𝑛+1
), (8) 

 (∫
𝜋
𝜋

𝑛+1

(𝑡−𝛿|𝜓(𝑡)|/𝑡𝛼)
𝑟

𝑑𝑡)

1

𝑟
= 𝑂((𝑛 + 1)𝛿), (9) 

 Where 𝛿 is an arbitrary number such that (𝛼 + 𝛿)𝑠 < −1 and 1/𝑠 = 1 − 1/𝑟 for 𝑟 > 1.  

4.  Lemmas 

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

4.1  Lemma 

 |𝐾𝑛(𝑡)| = 𝑂 (
1

𝑡
) + 𝑂((𝑛 + 1)𝑡)  for 0 ≤ 𝑡 ≤

𝜋

𝑛+1
≤

𝜋

𝑣+1
  

Proof.  

 |𝐾𝑛(𝑡)| =
1

2𝜋(𝑛+1)
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣
𝑐𝑜𝑠(𝑣+

1

2
)𝑡

𝑠𝑖𝑛(
𝑡

2
)

 

 =
1

2𝜋(𝑛+1)
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣
𝑐𝑜𝑠(𝑣+1−

1

2
)𝑡

𝑠𝑖𝑛(
𝑡

2
)

 

 ≤
1

(𝑛+1)
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣
𝑐𝑜𝑠(𝑣+1)𝑡𝑐𝑜𝑠(

𝑡

2
)+𝑠𝑖𝑛(𝑣+1)𝑡𝑠𝑖𝑛(

𝑡

2
)

𝑠𝑖𝑛(
𝑡

2
)

 

 =
1

(𝑛+1)
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣 [𝑂 (
1

𝑡
) + 𝑂(𝑠𝑖𝑛(𝑣 + 1)𝑡)] 

 = [
1

(𝑛+1)𝑡
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣] 

 + [
1

(𝑛+1)
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣(𝑣 + 1)𝑡] 

 = 𝑂 [
1

(𝑛+1)𝑡
(𝑛 + 1)] + 𝑂 [

1

(𝑛+1)
(𝑛 + 1)(𝑛 + 1)𝑡] 



Communications on Applied Nonlinear Analysis 

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 = 𝑂 (
1

𝑡
) + 𝑂((𝑛 + 1)𝑡),  

In view of sin(𝑣 + 1)𝑡 ≤ (𝑣 + 1)𝑡 for 0 ≤ 𝑡 ≤
𝜋

𝑣+1
  and (𝑠𝑖𝑛 (

𝑡

2
))

−1

< 
𝜋

𝑡
  for 

0<t≤ 𝜋  [3, p.247]. 

4.2  Lemma 

 |𝐾𝑛(𝑡)| = 𝑂 (
1

𝑡
) + 𝑂(1) for 

𝜋

𝑣+1
≤ 𝑡 ≤ 𝜋.  

Proof.  

 |𝐾𝑛(𝑡)| ≤
1

2𝜋(𝑛+1)
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣
𝑐𝑜𝑠(𝑣+

1

2
)𝑡

𝑠𝑖𝑛(
𝑡

2
)

 

 =
1

2𝜋(𝑛+1)
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣
𝑐𝑜𝑠(𝑣+1−

1

2
)𝑡

𝑠𝑖𝑛(
𝑡

2
)

 

 ≤
1

(𝑛+1)
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣
𝑐𝑜𝑠(𝑣+1)𝑡𝑐𝑜𝑠(

𝑡

2
)+𝑠𝑖𝑛(𝑣+1)𝑡𝑠𝑖𝑛(

𝑡

2
)

𝑠𝑖𝑛(
𝑡

2
)

 

 =
1

(𝑛+1)
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣 [𝑂 (
1

𝑡
) + 𝑂(1)] 

 = [
1

(𝑛+1)𝑡
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣] 

 + [
1

(𝑛+1)
∑𝑛

𝑘=0

1

(1+𝑞)𝑘
∑𝑘

𝑢=0 (
𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣] 

 = 𝑂 [
1

(𝑛+1)𝑡
(𝑛 + 1)] + [

1

(𝑛+1)
(𝑛 + 1)] 

 = 𝑂 (
1

𝑡
) + 𝑂(1), 

In view of |𝑠𝑖𝑛(𝑣 + 1)𝑡| ≤ 1 and (𝑠𝑖𝑛 (
𝑡

2
))

−1

≤  
𝜋

𝑡
    for 0 < 𝑡 ≤ 𝜋  [3, p.247] 

5.  Proof of main Theorem 

 The integral representation of �̃�𝑛(𝑓; 𝑥) is given by  

 �̃�𝑛(𝑓; 𝑥) = −
1

𝜋
∫

𝜋

0
𝜓(𝑡)

𝑐𝑜𝑠(
𝑡

2
)−𝑐𝑜𝑠(𝑛+

1

2
)𝑡

2𝑠𝑖𝑛(
𝑡

2
)

𝑑𝑡. 

Therefore, we have  

 �̃�𝑛(𝑓; 𝑥) − 𝑓(𝑥) =
1

2𝜋
∫

𝜋

0
𝜓(𝑡)

𝑐𝑜𝑠(𝑛+
1

2
)𝑡

𝑠𝑖𝑛(
𝑡

2
)

𝑑𝑡. 

Now, denoting (𝐶, 1)(𝐸, 𝑞)(𝐸, 𝑞) transform of �̃�𝑛(𝑓; 𝑥) by 𝐶𝑛
1𝐸𝑛

𝑞𝐸𝑛
𝑞 , we write  



Communications on Applied Nonlinear Analysis 

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 𝐶𝑛
1𝐸𝑛

𝑞𝐸𝑛
𝑞 − 𝑓 =

1

2𝜋(𝑛+1)
[∑𝑛

𝑘=0

1

(1+𝑞)𝑘 ∫
𝜋

0

𝜓(𝑡)

𝑠𝑖𝑛
𝑡

2

∑𝑘
𝑢=0 (

𝑘
𝑢

)
𝑞𝑘−𝑢

(1+𝑞)𝑢
∑𝑢

𝑣=0 (
𝑢
𝑣

) 𝑞𝑢−𝑣𝑐𝑜𝑠 (𝑣 +

1

2
) 𝑡] (10) 

 = [∫
𝜋

𝑛+1

0
+ ∫

𝜋
𝜋

𝑛+1

] 𝜓(𝑡)𝐾𝑛(𝑡)𝑑𝑡 = 𝐼1 + 𝐼2, 𝑠𝑎𝑦. (11) 

 Using Lemma 4.1, Hölder’s inequality, condition (8) and Minkwiski’s inequality, we have 

 |𝐼1| = ∫
𝜋

𝑛+1

0
|𝜓(𝑡)||𝐾𝑛(𝑡)|𝑑𝑡 

 ≤ [∫
𝜋

𝑛+1

0
(|𝜓(𝑡)/𝑡𝛼)𝑟]

1

𝑟

[𝑙𝑖𝑚𝜖→0 ∫
𝜋

𝑛+1

𝜖
(𝑡𝛼|𝐾𝑛(𝑡)|)𝑠𝑑𝑡]

1

𝑠

 

 = 𝑂((𝑛 + 1)−1) [𝑙𝑖𝑚𝜖→0 ∫
𝜋

𝑛+1

𝜖
(𝑡𝛼−1 + (𝑛 + 1)𝑡𝛼+1)𝑠𝑑𝑡]

1

𝑠

 

 = 𝑂((𝑛 + 1)−1) [(𝑙𝑖𝑚𝜖→0 ∫
𝜋

𝑛+1

𝜖
𝑡(𝛼−1)𝑠𝑑𝑡)

1

𝑠

+ (𝑙𝑖𝑚𝜖→0 ∫
𝜋

𝑛+1

𝜖
(𝑛 + 1)𝑡(𝛼+1)𝑠𝑑𝑡)

1

𝑠

] 

 = 𝑂((𝑛 + 1)−1)[(𝑛 + 1)−𝛼+1−1/𝑠 + (𝑛 + 1)(𝑛 + 1)−𝛼−1−1/𝑠] 

 = 𝑂((𝑛 + 1)−1)[(𝑛 + 1)−𝛼+1/𝑟 + (𝑛 + 1)(𝑛 + 1)−𝛼−1−1+1/𝑟] 

 = 𝑂 [(𝑛 + 1)−𝛼+
1

𝑟
−1 + (𝑛 + 1)−𝛼−2+

1

𝑟] 

 = 𝑂 ((𝑛 + 1)−𝛼−1+
1

𝑟)                                                                               (12) 

 Now, we consider  

 |𝐼2| = ∫
𝜋

𝑛+1
𝜋

𝑛+1

|𝜓(𝑡)||𝐾𝑛(𝑡)|𝑑𝑡. 

 Using Lemma 4.2, condition (9) and Minkowiski’s inequality, we have  

 |𝐼2| ≤ [∫
𝜋
𝜋

𝑛+1

(
𝑡−𝛿|𝜓(𝑡)|

𝑡𝛼 )
𝑟

𝑑𝑡]

1

𝑟

[∫
𝜋
𝜋

𝑛+1

(
𝑡𝛼|𝐾𝑛(𝑡)|

𝑡−𝛿 )
𝑠

]

1

𝑠
 

 = 𝑂((𝑛 + 1)𝛿) [∫
𝜋
𝜋

𝑛+1

(
𝑡𝛼

𝑡−𝛿
(𝑂 (

1

𝑡
) + 𝑂(1)))

𝑠

𝑑𝑡]

1

𝑠

 

 = 𝑂((𝑛 + 1)𝛿) [∫
𝜋
𝜋

𝑛+1

(𝑡𝛼+𝛿−1 + 𝑡𝛼+𝛿)
𝑠
𝑑𝑡]

1

𝑠
 

 = 𝑂((𝑛 + 1)𝛿) [(∫
𝜋
𝜋

𝑛+1

𝑡(𝛼+𝛿−1)𝑠𝑑𝑡)

1

𝑠
+ (∫

𝜋
𝜋

𝑛+1

𝑡(𝛼+𝛿)𝑠𝑑𝑡)

1

𝑠
] 



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 = 𝑂((𝑛 + 1)𝛿) [(𝑛 + 1)(−𝛼−𝛿+1)−
1

𝑠 + (𝑛 + 1)(−𝛼−𝛿)−
1

𝑠]    (1 + (𝛼 + 𝛿)𝑠 ≤ 0) 

 = 𝑂 [(𝑛 + 1)−𝛼+1−
1

𝑠 + (𝑛 + 1)−𝛼−
1

𝑠] 

 = 𝑂 [(𝑛 + 1)−𝛼+
1

𝑟 + (𝑛 + 1)−𝛼−1+
1

𝑟] 

 = 𝑂 [(𝑛 + 1)−𝛼+
1

𝑟(1 + (𝑛 + 1)−1)] 

 = 𝑂 ((𝑛 + 1)−𝛼+
1

𝑟)                                                                        (13) 

 Combining (12) and (13), we have  

 |𝐶𝑛
1𝐸𝑛

𝑞𝐸𝑛
𝑞 − 𝑓| = 𝑂 ((𝑛 + 1)−𝛼+

1

𝑟). 

Hence,  

 𝐶𝑛
1𝐸𝑛

𝑞𝐸𝑛
𝑞 − 𝑓𝑟 = (∫

2𝜋

0
|𝐶𝑛

1𝐸𝑛
𝑞𝐸𝑛

𝑞 − 𝑓(𝑥)|𝑟𝑑𝑥)

1

𝑟
= 𝑂 (𝑛−𝛼+

1

𝑟). 

This completes the proof of theorem 2.  

6  Corollaries 

6.1  Corollary  

 If one (𝐸, 𝑞) = 1, then (𝐶, 1)(𝐸, 𝑞)(𝐸, 𝑞) means reduces to (𝐶, 1)(𝐸, 𝑞) means. 

Hence, Theorem 2 reduces to theorem 1.  

6.2  Corollary 

 When 𝑞 = 1 then (𝐶, 1)(𝐸, 𝑞)(𝐸, 𝑞) means reduces to (𝐶, 1)(𝐸, 1)(𝐸, 1) means.  

6.3  Corollary 

 If (𝐶, 1) = 1, then (𝐶, 1)(𝐸, 𝑞)(𝐸, 𝑞) means reduces to (𝐸, 𝑞)(𝐸, 𝑞) means.  

7.  Conclusion 

The result established here is more general form than some earlier existing results in the sense that, 

one (𝐸, 𝑞) = 1 our proposed mean reduces to (𝐶, 1)(𝐸, 𝑞) Mean. 

8. Acknowledgments 

 The first author expresses his gratitude towards his parents for their blessings. The second author 

expresses his gratitude towards his parents for their blessing. Both authors are also grateful to the 

Honorable Vice-Chancellor, Patliputra University, Patna, Bihar, India for the inspiration for this work. 

9. Refrences 

[1]   A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, 3rd rev. ed., 2002.  

[2]  B.P. Dhakal: Approximation of the conjugate of a function belonging to the W𝐿𝑝, 𝜉(𝑡) class by (𝑁, 𝑝𝑛)(𝐸, 1) means 

of the the conjugate series of the Fourier series. J. Sci. Eng. Technol. 5(II), 30-36 (2009) . 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 4s (2025) 

 

124 
https://internationalpubls.com 

[3]   G.Bachman,L. Narici, E. Beckenstein: Fourier and Wavelet Analysis. Springer, New York (2000). 

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by means of conjugate Fourier series using linear operators. Indian J. Math. 47, 217-229 (2005) . 

[7]  M.L. Mittal, B.E. Rhoades, V.N. Mishra : Approximation of signals (functions) belonging to the weighted 

W(𝐿𝑃, 𝜉(𝑡))-class by linear operators. Int. J. Math. Math. Sci. 2006, article I.D. 53538 (2006).  

[8]   S.Lal, P.N.Singh : Degree of approximation of conjugate of Lip(𝛼, 𝑝) function by (𝐶, 1)(𝐸, 1) means of conjugate 

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[9]  S.Sonker, U. Singh: Degree of approximation of the conjugate of signals (functions) belonging to -class by means of 

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