Ratio Mathematica Volume 42, 2022

On the approximation of conjugate of
functions belonging to the generalized

Lipschitz class by Euler-matrix product
summability method of conjugate series of

Fourier series

Jitendra Kumar Kushwaha*
Krishna Kumar †

Abstract

In this paper, a new theorem on the approximation of conjugate of
functions belonging to the generalized Lipschitz class by Euler-Matrix
product summability method of conjugate series of Fourier series has
been obtained. Sometimes a series is not summable by any individual
summability method. But it becomes summable by taking product
summability means of given series. So working in this direction we
have used Euler-Matrix product summability method. Since Lipα
Lip(α,p) classes are the particular cases of generalized Lipschitz
class. Therefore, many of the known results may become particu-
lar cases of our result. On the bases of above facts we can say that our
result may be useful for the coming researchers in future.
Keywords: generalized Lipschitz class, conjugate series of Fourier
series, product summability method, Euler mean, matrix mean.
2022 AMS subject classifications: 42B05, 42B08. 1

*Department of Math and Stat, DDU Gorakhpur University; kjitendrakumar@yahoo.com.
†Department of Math and Stat, DDU Gorakhpur University; kkmaths1986@gmail.com.

1Received on May 17th, 2022. Accepted on June 27th, 2022. Published on June 30th, 2022. doi:
10.23755/rm.v41i0.788. ISSN: 1592-7415. eISSN: 2282-8214. ©Jitendra Kumar Kushwaha and
Krishna Kumar.

271



J.K.Kushwaha, K.Kumar

1 Introduction
The degree of approximation of functions belonging to various classes by us-

ing Cesáro, Nörlund and generalized Nörlund summability methods has been ob-
tained by a number of researchers like Chandra [1], Holland [3], Lal et al ([5],[6])
and Kushwaha [4], Qureshi [7]. Later on Tiwary et al [9] has discussed the de-
gree of approximation of functions by using (E,q)A product summability means
of Fourier series. No work seems to have been done so for to find the degree of
approximation of conjugate of functions belonging to generalized Lipschitz class
by using Euler-Matrix product summability means. Now, in this paper, we are
presenting a new theorem on the degree of approximation of conjugate functions
belonging to the generalized Lipschitz class by Euler-Matrix product summability
method. This new result may become the generalization of many of the known
results.

2 Definitions
In this section,we have given following definitions:

Definition 2.1. A function f ∈ Lipα if

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

Definition 2.2. A function f ∈ Lip(α,p) if(
2π∫
0

|f(x + t) − f(x − t)|p dx
)1/p

= O(|tα|),0 ≤ α < 1,p ≥ 1.

Given a positive increasing function ξ(t) and integer p ≥ 1,

f ∈ Lip(ξ(t),p) if
 2π∫

0

|f(x + t) − f(x − t)|p dx


1/p = O (ξ(t)) .

If ξ(t) = tα, then Lip(ξ(t),p) coincides to Lip(α,p).

272



On the approximation conjugate of functions belonging to.......

Definition 2.3. L∞-norm of a function f : R → R is defined by

∥f∥∞ = sup{|f(x)|/f : R → R} .

Lp-norm is defined by

∥f∥p =


 2π∫

0

|f(x)|p

1/p ,p ≥ 1.

The degree of approximation of a function f : R → R by a trigonometric
polynomial tn(Zygmund [11]) is defined by

∥tn − f∥∞ = sup{|tn − f| : x ∈ R}or∥tn − f∥p = min∥tn − f∥.

Let f be 2π periodic and integrable over (−π,π) in Lebesgue sense and f ∈
Lip(ξ(t),p) . Let its Fourier series be given by

f(t) =
1

2
a0 +

∞∑
n=1

(an cosnx + bn sinnx)

=
1

2
a0 +

∞∑
n=1

An(x). (1)

The conjugate series of the Fourier series (1) is given by

∞∑
n=1

(an sinnx − bn cosnx) = −
∞∑
n=1

Bn(x). (2)

If f is Lebesgue integrable, then

f(x) = −
1

2π

π∫
0

ψ(t) cot(t/2)dt = −
1

2π
lim
ϵ→0

π∫
0

ψ(t) cot(t/2)dt

exists for almost all x (Zygmund [11]).

273



J.K.Kushwaha, K.Kumar

Let T = (an,k) be an infinite lower triangular matrix satisfying Töeplitz (p.
131) condition of regularity i.e. an,k → 1 as n → ∞, an,k = 0 for k > n and
n∑

k=0

|an,k| ≤ M a finite constant. Let
∞∑
n=0

un be an infinite series whose kth partial

sums is sk =
k∑

n=0

un. The sequence to sequence transformation tn =
n∑

k=0

aa,ksk

defines the sequence {tn} of lower triangular matrix summability means of se-
quence {sn} generated by the sequence of coefficients (an,k). The series

∞∑
n=0

un is

said to summable to sum s by lower triangular matrix method if lim
n→∞

tn exists and

is equal to s (Zygmund; p.74) and we write tn → s(T), as n → ∞.

The (E,q) means of {sn} is defined by -

Wn =
1

(1 + q)n

n∑
k=0

(
n

k

)
qn−ksk.

The (E,q) transform of Matrix transform A of sn is defined by

ηn =
1

(1 + q)n

n∑
k=0

(
n

k

)
qn−ktk

=
1

(1 + q)n

n∑
k=0

(
n

k

)
qn−k

{
k∑

ν=0

aν,ksk

}
.

If ηn → ∞ as n → ∞, then the series
∞∑
n=0

un is said to be (E,q)A-summable

to sum s.

We use the following notations :

(i) ψ(x,t) = f(x + t) − f(x − t).

(ii) Mn(t) =
1

2π(1+q)n

n∑
k=0

(
n
k

)
qn−k

{
k∑

ν=0

aν,k
cos(ν+1/2)t

sin(t/2)

}
.

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On the approximation conjugate of functions belonging to.......

3 Lemmas:-
For the proof of our theorem, we have required following lemmas:

Lemma 3.1. Mn(t) = O
(
1
t

)
for 0 ≤ t ≤ (n + 1)−1.

Proof. For 0 ≤ t ≤ (n + 1)−1, we have

∣∣Mn(t)∣∣ = 1
2π(1 + q)n

∣∣∣∣∣
[

n∑
k=0

(
n

k

)
qn−k

{
k∑

ν=0

aν,k
cos(ν + 1/2)t

sin(t/2)

}]∣∣∣∣∣
≤

1

2π(1 + q)n

∣∣∣∣∣
n∑

k=0

(
n

k

)
qn−k

{
k∑

ν=0

aν,k
| cos(ν + 1/2)t|

(t/π)

}]

≤
1

2π(1 + q)n

∣∣∣∣∣
n∑

k=0

(
n

k

)
qn−k

{
k∑

ν=0

aν,k

}]

= O

(
1

t

)
.

Lemma 3.2. Mn(t) = O
(

An,τ
t

)
for (n + 1)−1 ≤ t ≤ π.

Proof. For (n + 1)−1 ≤ t ≤ π, we have by Jordan’s Lemma, sin(t/2) ≥
(t/π), then

∣∣Mn(t)∣∣ = 1
2π(1 + q)n

∣∣∣∣∣
[

n∑
k=0

(
n

k

)
qn−k

{
k∑

ν=0

aν,k
cos(ν + 1/2)t

sin(t/2)

}]∣∣∣∣∣
≤

1

2π(1 + q)n

∣∣∣∣∣
[

n∑
k=0

(
n

k

)
qn−k

{
k∑

ν=0

aν,k
cos(ν + 1/2)t

(t/π)

}]∣∣∣∣∣
=

1

2t(1 + q)n

∣∣∣∣∣
[

n∑
k=0

(
n

k

)
qn−k

{
k∑

ν=0

aν,k cos(ν + 1/2)t

}]∣∣∣∣∣
=

1

2t(1 + q)n

∣∣∣∣∣
[

n∑
k=0

(
n

k

)
qn−k

{
O
(ak,k−τ−1

t

)
+ Ak,τ

}]∣∣∣∣∣
= O

(
An,τ
t

)
.

275



J.K.Kushwaha, K.Kumar

4 Theorem
In this section, we have proved the theorem:

Theorem 4.1. Let f be a 2π-periodic, Lebesgue integrable function belonging to
Lip(ξ(t),p) class and T = (am,n) be an infinite lower triangular matrix. Then
the degree of approximation of conjugate function by (E,q)A-summability means
of its conjugate series of Fourier series is given by

∥η − f(x)∥p = O
(
(n + 1)1/pξ

(
1

n + 1

))
,

provided ξ(t) satisfies following conditions:




1/(n+1)∫
0

(
t|ψ(t)|
ξ(t)

)p


1/p

= O

(
1

n + 1

)
, (3)

and 


π∫
1/(n+1)

(
t−δ|ψ(t)|
ξ(t)

)q


1/q

= O
(
(n + 1)δ

)
, (4)

where δ ia an arbitrary number such that q(1 − δ) − 1 > 0, p−1 + q−1 = 1 such
that 1 ≤ p < ∞, conditions (3) and (4) holds uniformly in x.

Proof. The kth partial sums of conjugate series of Fourier series (2) is given
by

sk(f;x) = −
1

2π

π∫
0

cot(t/2)ψ(t)dt +
1

2π

2π∫
0

cos(k + 1/2)t

sin(t/2)
ψ(t)dt.

sk(f;x) = −
1

2π

π∫
0

cot(t/2)ψ(t)dt =
1

2π

2π∫
0

cos(k + 1/2)t

sin(t/2)
ψ(t)dt.

Therefore making (A-transform) of sk(f;x), we get

tn − f(x) =
1

2π

π∫
0

ψ(t)
n∑

k=0

an,k
cos(ν + 1/2)t

sin(t/2)
dt.

276



On the approximation conjugate of functions belonging to.......

Now, making (E,q)A-transform of sk(f;x), we get

ηn − f(x) =
1

2π(1 + q)n

×
π∫

0

ψ(t)
n∑

k=0

(
n

k

)
qn−k

{
k∑

ν=0

an,k
cos(ν + 1/2)t

sin(t/2)

}
dt

=

π∫
0

ψ(t)Mn(t)dt

=




1/(n+1)∫
0

+

π∫
1/(n+1)


ψ(t)Mn(t)dt

= I1 + I2. (5)

Clearly,

|ψ(x + t) − ψ(x − t)| ≤ |f(u + x + t) − f(x + t)| + |f(u − x − t) − f(x − t)|.

Now,let
ψ(x,t) = ψ(x + t) − ψ(x − t)

ψ1(u,x,t) = f(u + x + t) − f(x + t)

ψ2(u,x,t) = f(u − x − t) − f(x − t)

Hence, by Minkowski’s inequality


2π∫
0

|ψ(x,t)|pdx




1/p

≤




2π∫
0

|ψ1(u,x,t)|pdx




1/p

+




2π∫
0

|ψ2(u,x,t)|pdx




1/p

= O (ξ(t)) . (6)

277



J.K.Kushwaha, K.Kumar

Then
f ∈ Lip(ξ(t),p) ⇒ ψ ∈ Lip(ξ(t),p) .

Using the Hölder’s inequality, ψ(t) ∈ Lip(ξ(t),p), condition (3), sint ≥ (2π/t),
Lemma 1, and second mean value theorem for integrals, we have

|I1| ≤


 1/(n+1)∫

0

(
tψ(t)

ξ(t)

)p
dt




1/p 
 1/(n+1)∫

0

(
ξ(t)|Mn(t)|

t

)q
dt




1/q

= O
(
(n + 1)−1

)  1/(n+1)∫
0

(
ξ(t)|Mn(t)|

t

)q
dt




1/q

= O
(
(n + 1)−1

)  1/(n+1)∫
0

(
ξ(t)

t2

)q
dt




1/q

= O

(
(n + 1)1/pξ

(
1

n + 1

))
,p−1 + q−1 = 1. (7)

Using the Hölder’s inequality, Lemma 2, | sint| ≤ 1, sint ≥ (2π/t), and con-
dition (4), we have

|I2| ≤


 π∫
1/(n+1)

(
t−δψ(t)

ξ(t)

)p
dt




1/p  π∫
1/(n+1)

(
ξ(t)|Mn(t)|

t−δ

)q
dt




1/q

≤ O
(
(n + 1)δ

)  π∫
1/(n+1)

(
ξ(t)|Mn(t)|

t−δ

)q
dt




1/q

≤ O
(
(n + 1)δ

)  π∫
1/(n+1)

(
ξ(t)

t−δ
O

(
An,τ
t

))q
dt




1/q

278



On the approximation conjugate of functions belonging to.......

= O
(
(n + 1)δ

)  π∫
1/(n+1)

((
ξ(t)

t1−δ
An,τ

))q
dt




1/q

= O
(
(n + 1)δ

) 
(n+1)∫
1/π

(
ξ(1/y)

yδ−1
An,[y]

)q
dy

y2




1/q

= O

(
(n + 1)δξ

(
1

n + 1

)) 
(n+1)∫
1/π

dy

yδq−q+2




1/q

=

{
(n + 1)δξ

(
1

n + 1

) (
O(n + 1)−q(δ−1)−1

)1/q}
= O

(
(n + 1)1/pξ

(
1

n + 1

))
,p−1 + q−1 = 1. (8)

Collecting equations from (5) to (8), we get

∥η − f(x)∥p = O
(
(n + 1)1/pξ

(
1

n + 1

))
,1 ≤ p < ∞.

5 Corollaries:-
Corolary 5.1. If ξ(t) = tα then the degree of approximation of a function f(x),
conjugate of f ∈ Lip(α,p), 1

p
< α < 1 by (E,q)A means is given by

∥η − f(x)∥p = O
(
(n + 1)−α+1/p

)
,1 ≤ p < ∞.

Corolary 5.2. If p → ∞ in case 1, then for 0 < α < 1, the degree of approx-
imation of a function f(x), conjugate of f ∈ Lipα by (E,q)A means is given
by

∥η − f(x)∥p = O ((n + 1)−α) .

6 Acknowledgement
Auther is highley thankful to Professeor Shyam lal, Department of Math-

ematics, Institute of Science, Banaras Hindu University, Varanasi, India for his
encouragement and support to this work.

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J.K.Kushwaha, K.Kumar

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