131 
 

 American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  
http://asrjetsjournal.org/  

 

Approximation Theory on Summability of Fourier Series 

Sanjay Mukherjeea*, A J Khanb 

aResearch Scholor, MATS University, Raipur(C.G.), India 
bMATS University, Raipur(C.G.), India 

aEmail: sanjaymukherjeeruma@gmail.com 
bEmail: khanaj@matsuniversity.ac.in 

 

 

 

Abstract 

The results of Chandra to (e,c) means U.K.Shrivastava and S.K.Verma have proved the following theorem 

THEOREM : Let 𝑓𝑓 ∈ 𝐶𝐶2𝜋𝜋 ∩ 𝐿𝐿𝐿𝐿𝐿𝐿 ∝ ,0 <∝≤ 1. Then  

‖𝑡𝑡𝑛𝑛𝑐𝑐 − 𝑓𝑓‖ = 𝑜𝑜�𝑛𝑛−∝ 2� �,  

Where 𝑡𝑡𝑛𝑛𝑐𝑐(𝑓𝑓; 𝑥𝑥) is nth (e, c) means of fourier series of f at x.  

In this paper we obtain the Fourier series by (N,p,q)(E,1) which is the analogues  to  the (e , c) means given 

above .The theorem is as follows  

THEOREM: Let {𝐿𝐿𝑛𝑛}  and {𝑞𝑞𝑛𝑛}  be the positive monotonic, non increasing sequence of real numbers be 

summable (N,p,q)(E,1) to f(x) at the point t=x is  

𝑡𝑡𝑁𝑁
𝑝𝑝,𝑞𝑞,𝐸𝐸 − 𝑓𝑓(𝑥𝑥) = 𝑜𝑜(1)                    

Keywords: Fourier series; Borel means; Lebesgue series. 

 

----------------------------------------------------------------------- 

* Corresponding author.  

http://asrjetsjournal.org/


American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 54, No  1, pp 131-136 

132 
 

1. Introduction 

Let {𝐿𝐿𝑛𝑛} and {𝑞𝑞𝑛𝑛} be the sequences of constants, real or complex, such that 

𝑃𝑃𝑛𝑛 = 𝐿𝐿1 + 𝐿𝐿2 + 𝐿𝐿3 + ⋯𝐿𝐿𝑛𝑛 = �𝐿𝐿𝑟𝑟 → ∞, 𝑎𝑎𝑎𝑎 𝑛𝑛 → ∞,
𝑛𝑛

𝑟𝑟=0

 

𝑄𝑄𝑛𝑛 = 𝑞𝑞1 + 𝑞𝑞2 + 𝑞𝑞3 + ⋯𝑞𝑞𝑛𝑛 = ∑ 𝑞𝑞𝑟𝑟 → ∞, 𝑎𝑎𝑎𝑎 𝑛𝑛 → ∞𝑛𝑛
𝑟𝑟=0 ,                                                          (1.1) 

𝑅𝑅𝑛𝑛 = 𝐿𝐿0𝑞𝑞𝑛𝑛 + 𝐿𝐿1𝑞𝑞𝑛𝑛−1 + 𝐿𝐿3𝑞𝑞𝑛𝑛−2 + ⋯𝐿𝐿𝑛𝑛𝑞𝑞0 = �𝐿𝐿𝑟𝑟𝑞𝑞𝑛𝑛−𝑟𝑟 → ∞, 𝑎𝑎𝑎𝑎 𝑛𝑛 → ∞
𝑛𝑛

𝑟𝑟=0

 

Given two sequences {𝐿𝐿𝑛𝑛} and {𝑞𝑞𝑛𝑛} convolution (𝐿𝐿 ∗ 𝑞𝑞) is defined as 

𝑅𝑅𝑛𝑛 = (𝐿𝐿 ≠ 𝑞𝑞)𝑛𝑛 = ∑ 𝐿𝐿𝑛𝑛−𝑟𝑟𝑛𝑛
𝑟𝑟=0 𝑞𝑞𝑟𝑟                                                                                                       (1.2) 

Let ∑ 𝑢𝑢𝑛𝑛∞
𝑛𝑛=0    be an infinite series with the sequence of its nth partial sums{𝑎𝑎𝑛𝑛}. 

We write 𝑡𝑡𝑛𝑛
𝑝𝑝,𝑞𝑞 = 1

𝑅𝑅𝑛𝑛
∑ 𝐿𝐿𝑛𝑛−𝑟𝑟𝑞𝑞𝑟𝑟𝑛𝑛
𝑟𝑟=0   (1.3) 

If 𝑅𝑅𝑛𝑛 ≠ 0, for all n, the generalized Norlund transform of the sequence {𝑎𝑎𝑛𝑛} is the sequence�𝑡𝑡𝑛𝑛
𝑝𝑝.𝑞𝑞�. 

If 𝑡𝑡𝑛𝑛
𝑝𝑝,𝑞𝑞 → 𝑆𝑆, 𝑎𝑎𝑎𝑎 𝑛𝑛 → ∞, then the series ∑ 𝑢𝑢𝑛𝑛∞

𝑛𝑛=0  or sequence {𝑎𝑎𝑛𝑛} is summable to S by  

𝑆𝑆𝑛𝑛 → 𝑆𝑆(𝑁𝑁, 𝐿𝐿, 𝑞𝑞)                              (1.4) 

The necessary and sufficient conditions for (N,p,q) method to be regular are  

∑ |𝐿𝐿𝑛𝑛−𝑟𝑟𝑞𝑞𝑟𝑟| = 𝑜𝑜(|𝑅𝑅𝑛𝑛|)𝑛𝑛
𝑟𝑟=0     (1.5) 

And 𝐿𝐿𝑛𝑛−𝑟𝑟 = 𝑜𝑜(|𝑅𝑅𝑛𝑛|), as 𝑛𝑛 → ∞ for every fixed 𝑘𝑘 ≥ 0, for which 𝑞𝑞𝑟𝑟 ≠ 0 

𝐸𝐸𝑛𝑛1 = 1
2𝑛𝑛
∑ �𝑛𝑛𝑟𝑟�
𝑛𝑛
𝑟𝑟=0 𝑎𝑎𝑟𝑟      (1.6) 

If 𝐸𝐸𝑛𝑛1 → 𝑎𝑎, 𝑎𝑎𝑎𝑎 𝑛𝑛 → ∞ , then the series ∑ 𝑢𝑢𝑛𝑛∞
𝑛𝑛=0  is said to be (E,1) summable to s (Hardy [1] ) : 

𝑡𝑡𝑛𝑛
𝑝𝑝,𝑞𝑞,𝐸𝐸 = 1

𝑅𝑅𝑛𝑛
∑ 𝐿𝐿𝑛𝑛−𝑟𝑟𝑞𝑞𝑟𝑟𝐸𝐸𝑟𝑟1𝑛𝑛
𝑟𝑟−0   

= 1
𝑅𝑅𝑛𝑛
∑ 𝐿𝐿𝑛𝑛−𝑟𝑟𝑞𝑞𝑟𝑟𝑛𝑛
𝑟𝑟=0

1
2𝑘𝑘
∑ �𝑘𝑘𝑟𝑟�𝑎𝑎𝑟𝑟
𝑛𝑛
𝑟𝑟=0              (1.7) 

If 𝑇𝑇𝑛𝑛
𝑝𝑝,𝑞𝑞,𝐸𝐸 → ∞,𝑎𝑎𝑎𝑎 𝑛𝑛 → ∞ , then we say that the series ∑ 𝑢𝑢𝑛𝑛∞

𝑛𝑛=0  or the sequence {𝑎𝑎𝑛𝑛}  is summable to S by 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 54, No  1, pp 131-136 

133 
 

(N,p,q)(E,1) summability method. 

2. Structure  

2. Degree of approximation by borel means and (E, Q) means were obtained by Chandra [4] and [5] respectively 

.Extending the results of Chandra to (e,c) means U.K.Shrivastava and S.K.Verma[9] have proved the following 

theorem 

THEOREM : Let 𝑓𝑓 ∈ 𝐶𝐶2𝜋𝜋 ∩ 𝐿𝐿𝐿𝐿𝐿𝐿 ∝ ,0 <∝≤ 1. Then  

‖𝑡𝑡𝑛𝑛𝑐𝑐 − 𝑓𝑓‖ = 𝑜𝑜�𝑛𝑛−∝ 2� �,  

Where 𝑡𝑡𝑛𝑛𝑐𝑐(𝑓𝑓; 𝑥𝑥) is nth (e,c) means of fourier series of f at x. (2.1) 

Our theorem fourier series by (N,p,q)(E,1) is the analogues to  the (e,c) means theorem, which is as follows 

THEOREM: Let {𝐿𝐿𝑛𝑛}  and {𝑞𝑞𝑛𝑛}  be the positive monotonic ,non increasing sequence of real numbers be 

summable (N,p,q)(E,1) to f(x) at the point t=x is  

𝑡𝑡𝑁𝑁
𝑝𝑝,𝑞𝑞,𝐸𝐸 − 𝑓𝑓(𝑥𝑥) = 𝑜𝑜(1)                    

Proof of the above theorem required some lemmas 

3. Lemmas 

Lemma  3.1- For 0 ≤ 𝑡𝑡 ≤ 1
𝑛𝑛
   |𝐾𝐾𝑛𝑛(𝑡𝑡)| = 𝑜𝑜(𝑛𝑛) 

Lemma 3.2- If  {𝐿𝐿𝑛𝑛} and {𝑞𝑞𝑛𝑛} are non negative and non increasing, then for 0 ≤ 𝑎𝑎 ≤ 𝑏𝑏 < ∞, 0 ≤ 𝑡𝑡 ≤ 𝜋𝜋, and any 

n we have 1
2𝜋𝜋𝑅𝑅𝑛𝑛

�∑ 𝐿𝐿𝑛𝑛−𝑟𝑟𝑞𝑞𝑟𝑟
𝑐𝑐𝑐𝑐𝑐𝑐𝑟𝑟�𝑡𝑡 2� �𝑐𝑐𝑠𝑠𝑛𝑛(𝑟𝑟+1)�𝑡𝑡 2� �

𝑐𝑐𝑠𝑠𝑛𝑛�𝑡𝑡 2� �
𝑏𝑏
𝑟𝑟=𝑎𝑎 � = 𝑜𝑜 � 𝑅𝑅𝑘𝑘

𝑡𝑡𝑅𝑅𝑛𝑛
� 

4. Proof of Theorem  

Let f(t) be a periodic function with period 2𝜋𝜋 and integrable in the same sense of Lebesgue over the interval 

�–𝜋𝜋,𝜋𝜋� 

Let its Fourier series be given by  

𝑓𝑓(𝑡𝑡)~ 1
2
𝑎𝑎0 + ∑ (𝑎𝑎𝑛𝑛𝑐𝑐𝑜𝑜𝑎𝑎𝑛𝑛𝑡𝑡 + 𝑏𝑏𝑛𝑛𝑎𝑎𝐿𝐿𝑛𝑛𝑛𝑛𝑡𝑡)∞

𝑛𝑛=1   (4.1) 

Following Zygmund [3] , the nth sum 𝑎𝑎𝑛𝑛(𝑥𝑥) of the series  at t=x is given by  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 54, No  1, pp 131-136 

134 
 

𝑎𝑎𝑛𝑛(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) + 1
2𝜋𝜋 ∫ ∅𝑥𝑥(𝑡𝑡)𝜋𝜋

0
𝑐𝑐𝑠𝑠𝑛𝑛(𝑛𝑛+1)𝑡𝑡
𝑐𝑐𝑠𝑠𝑛𝑛�𝑡𝑡 2� �

𝑑𝑑𝑡𝑡  (4.2) 

So the (E,1) mean of the series at t=x is given by 

𝐸𝐸𝑛𝑛1(𝑥𝑥) =
1

2𝑛𝑛
��

𝑛𝑛
𝑟𝑟
�

𝑛𝑛

𝑟𝑟=0

𝑎𝑎𝑟𝑟(𝑥𝑥) 

= 𝑓𝑓(𝑥𝑥) +
1

2𝑛𝑛+1𝜋𝜋
�

∅𝑥𝑥(𝑡𝑡)
𝑎𝑎𝐿𝐿𝑛𝑛�𝑡𝑡 2� �

𝜋𝜋

𝑟𝑟=0
���

𝑛𝑛
𝑟𝑟
�

𝑛𝑛

𝑟𝑟=0

𝑎𝑎𝐿𝐿𝑛𝑛 �𝑟𝑟 +
1
2
� 𝑡𝑡� 𝑑𝑑𝑡𝑡 

= 𝑓𝑓(𝑥𝑥) +
1

2𝑛𝑛+1𝜋𝜋
�

∅𝑥𝑥(𝑡𝑡)
𝑎𝑎𝐿𝐿𝑛𝑛(𝑡𝑡/2) 𝐼𝐼𝐼𝐼

𝜋𝜋

0

�𝑒𝑒𝑠𝑠𝑡𝑡/2(1 + 𝑒𝑒𝑠𝑠𝑡𝑡)𝑛𝑛�𝑑𝑑𝑡𝑡 

= 𝑓𝑓(𝑥𝑥) + 1
2𝑛𝑛+1𝜋𝜋 ∫

∅𝑥𝑥(𝑡𝑡)
𝑐𝑐𝑠𝑠𝑛𝑛(𝑡𝑡/2)

𝐼𝐼𝐼𝐼𝜋𝜋
0 �𝑒𝑒𝑠𝑠𝑡𝑡/2(1 + 𝑐𝑐𝑜𝑜𝑎𝑎𝑡𝑡 + 𝐿𝐿𝑎𝑎𝐿𝐿𝑛𝑛𝑡𝑡)𝑛𝑛�𝑑𝑑𝑡𝑡          (4.3) 

= 𝑓𝑓(𝑥𝑥) +
1

2𝑛𝑛+1𝜋𝜋
�

∅𝑥𝑥(𝑡𝑡)
𝑎𝑎𝐿𝐿𝑛𝑛(𝑡𝑡/2) 𝐼𝐼𝐼𝐼

𝜋𝜋

0

�𝑒𝑒𝑠𝑠𝑡𝑡/22𝑛𝑛𝑐𝑐𝑜𝑜𝑎𝑎𝑛𝑛 �
𝑡𝑡
2
� �𝑐𝑐𝑜𝑜𝑎𝑎

𝑡𝑡
2

+ 𝐿𝐿𝑎𝑎𝐿𝐿𝑛𝑛
𝑡𝑡
2
�
𝑛𝑛
� 𝑑𝑑𝑡𝑡 

= 𝑓𝑓(𝑥𝑥) +
1

2𝑛𝑛+1𝜋𝜋
�

∅𝑥𝑥(𝑡𝑡)
𝑎𝑎𝐿𝐿𝑛𝑛(𝑡𝑡/2) 𝐼𝐼𝐼𝐼

𝜋𝜋

0

�𝑒𝑒𝑠𝑠𝑡𝑡/22𝑛𝑛𝑐𝑐𝑜𝑜𝑎𝑎𝑛𝑛 �
𝑡𝑡
2
� �𝑐𝑐𝑜𝑜𝑎𝑎

𝑛𝑛𝑡𝑡
2

+ 𝐿𝐿𝑎𝑎𝐿𝐿𝑛𝑛
𝑛𝑛𝑡𝑡
2
�� 𝑑𝑑𝑡𝑡 

= 𝑓𝑓(𝑥𝑥) +
1

2𝜋𝜋
�∅𝑥𝑥(𝑡𝑡)
𝜋𝜋

0

𝑐𝑐𝑜𝑜𝑎𝑎𝑛𝑛(𝑡𝑡/2)𝑎𝑎𝐿𝐿𝑛𝑛(𝑛𝑛 + 1)(𝑡𝑡/2)

sin (𝑡𝑡
2
)

𝑑𝑑𝑡𝑡 

Therefore  

𝑡𝑡𝑛𝑛
𝑝𝑝,𝑞𝑞,𝐸𝐸(𝑥𝑥) − 𝑓𝑓(𝑥𝑥) = �� + �+�

𝜋𝜋

𝛿𝛿

𝛿𝛿

1/𝑛𝑛

1/𝑛𝑛

0

� 𝐾𝐾𝑛𝑛(𝑡𝑡)∅𝑥𝑥(𝑡𝑡)𝑑𝑑𝑡𝑡 

= 𝐼𝐼1 + 𝐼𝐼2 + 𝐼𝐼3 (say) (4.4) 

We have  

|𝐼𝐼1| ≤ � |𝐾𝐾𝑛𝑛(𝑡𝑡)|

1/𝑛𝑛

0

|∅𝑥𝑥(𝑡𝑡)|𝑑𝑑𝑡𝑡 

= 𝑂𝑂(𝑛𝑛)∫ |∅𝑥𝑥(𝑡𝑡)|1/𝑛𝑛
0 𝑑𝑑𝑡𝑡 ( 𝑢𝑢𝑎𝑎𝐿𝐿𝑛𝑛𝑢𝑢 𝐿𝐿𝑒𝑒𝐼𝐼𝐼𝐼𝑎𝑎 3.1)  (4.5) 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 54, No  1, pp 131-136 

135 
 

= 𝑜𝑜 �
1

𝛼𝛼(𝑛𝑛)� 

= 𝑜𝑜(1) 𝑎𝑎𝑎𝑎 𝑛𝑛 → ∞   (4.6) 

Now  

|𝐼𝐼2| ≤ �|𝐾𝐾𝑛𝑛(𝑡𝑡)|
𝛿𝛿

1/𝑛𝑛

|∅𝑥𝑥(𝑥𝑥)|𝑑𝑑𝑡𝑡 (𝑤𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒 0 < 𝛿𝛿 < 1) 

= �𝑜𝑜 �
𝑅𝑅(1/𝑡𝑡)
𝑡𝑡𝑅𝑅(𝑛𝑛)

� |∅𝑥𝑥(𝑡𝑡)|
𝛿𝛿

1/𝑛𝑛

𝑑𝑑𝑡𝑡 (𝑢𝑢𝑎𝑎𝐿𝐿𝑛𝑛𝑢𝑢 𝐿𝐿𝑒𝑒𝐼𝐼𝐼𝐼𝑎𝑎 3.2) 

= 𝑜𝑜 �
1

𝑅𝑅(𝑛𝑛)
� � �

𝑅𝑅(1/𝑡𝑡)
𝑡𝑡

� |∅𝑥𝑥(𝑡𝑡)|
𝛿𝛿

1/𝑛𝑛

𝑑𝑑𝑡𝑡 

= 𝑜𝑜 �
1

𝑅𝑅(𝑛𝑛)
� ��

𝑅𝑅(1/𝑡𝑡)
𝑡𝑡

∅𝑥𝑥(𝑡𝑡)�
1/𝑛𝑛

𝛿𝛿

− � 𝑑𝑑 �
𝑅𝑅(1/𝑡𝑡)

𝑡𝑡
�

𝛿𝛿

1/𝑛𝑛

∅𝑥𝑥(𝑡𝑡)� 

= 𝑜𝑜 �
1

𝑅𝑅(𝑛𝑛)� + 𝑜𝑜 �
1

𝛼𝛼(𝑛𝑛)� + 𝑜𝑜 �
1

𝑅𝑅(𝑛𝑛)� � � ∅𝑥𝑥(𝑡𝑡)
𝛿𝛿

1/𝑛𝑛

�𝑑𝑑 �
𝑅𝑅(1/𝑡𝑡)𝛼𝛼(1/𝑡𝑡)

𝑡𝑡𝛼𝛼(1/𝑡𝑡)
��� 

= 𝑜𝑜 �
1

𝑅𝑅(𝑛𝑛)� + 𝑜𝑜 �
1

𝛼𝛼(𝑛𝑛)� + 𝑜𝑜(1) 

= 𝑜𝑜(1), 𝑎𝑎𝑎𝑎 𝑛𝑛 → ∞  (4.7) 

Now  

𝐼𝐼3 = �|𝐾𝐾𝑛𝑛(𝑡𝑡)||∅𝑥𝑥(𝑡𝑡)|
𝜋𝜋

𝛿𝛿

𝑑𝑑𝑡𝑡 

By Riemann-Lebesgue theorem and regularity of the method of summability we have 

𝐼𝐼3 = 𝑜𝑜(1), 𝑎𝑎𝑎𝑎 𝑛𝑛 → ∞  (4.8) 

Combining (4.6),(4.7) and (4.8) we get  

𝑡𝑡𝑁𝑁
𝑝𝑝,𝑞𝑞,𝐸𝐸 − 𝑓𝑓(𝑥𝑥) = 𝑜𝑜(1)                   

This completes the proof of the theorem. 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 54, No  1, pp 131-136 

136 
 

5. Conclusion  

We conclude that the above theorem which is proved in (e,c) means can be proved by (N,p,q)(E,1) means. 

Acknowledgements                                        

History of all great works into witness that no great work was ever done without either active or passive support 

of a person ‘surrounding and one’s close quarters. Thus is it not hard to conclude how active assistance from 

senior could positively impact the execution of the project. I am highly thankful to our faculty head A.J. Khan 

Sir for his active guidance throughout the completion of the paper. Last but not the least, I would also want to 

extend my appreciation to those who could not be mentioned here but have well played their role in inspire me 

behind the certain. 

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