EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 3, 2020, 567-578
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Approximation of a function in Hölder class using
double Karamata (Kλ,µ) method

H. K. Nigam1, Md Hadish1,∗

1 Department of Mathematics, Cental University of South Bihar, Gaya, Bihar, India

Abstract. In this paper, we establish a new theorem on the best approximation of a function of
two variables belonging to Hölder class by double Karamata (Kλ,µ) means of its double Fourier
series.

2020 Mathematics Subject Classifications: 40C10, 40G05, 40G10, 42A10, 42A24, 40C05,
41A25, 42B05

Key Words and Phrases: Hölder class, double Karamata (Kλ,µ), double Fourier series, stirling
numbers, error approximation.

1. Introduction

Kλ-method was first introduced by Karamata [7]. Lotosky [10] re-introduced the spe-
cial case λ = 1. Only after the study of Agnew [1], an intensive study of these and
similar cases took place. Vuĉkoviĉ [19] applied this method for summability of Fourier
series. Kathal[8] extended the result of Vuĉkoviĉ [19]. The approximation of a 2π-periodic
function f(x) in different Lipschitz classes using Cesàro, Nörlund and Kλ summability
methods of Fourier series and conjugate Fourier series has been studied by the researchers
[2, 5, 6, 12, 14–16].
The approximation of a 2π-periodic function f(x) in Hölder metric by different summa-
bility transforms of Fourier series has been studied by the researchers like [4, 11, 13]. The
approximation of a function f(x, y) (2π-periodic with respect to the variables x, y) of their
Fourier series has been studied by [17, 18]. Lal [9] studied the approximation of a function
in Lipschitz class by matrix means of its double Fourier series. But nothing seems to have
done to obtain the best approximation of the function f(x, y), a 2π-periodic with respect
to the variable x, y, of its double Fourier series. Thus, in this paper, we obtain the best
approximation of the function h(ζ,Θ) in Hölder class by Kλ,µ method of its double Fourier
series.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i3.3663

Email addresses: hknigam@cusb.ac.in (H. K. Nigam), hadish@cusb.ac.in (Md Hadish)

https://www.ejpam.com 567 c© 2020 EJPAM All rights reserved.



H. K. Nigam, Md Hadish / Eur. J. Pure Appl. Math, 13 (3) (2020), 567-578 568

2. Preliminaries

Under usual assumptions, Fourier series of h(t) is defined as

h(t) ∼ 1

2
a0 +

∞∑
ρ=1

(aρ cos ρt+ bρ sin ρt). (1)

Under usual assumptions, double Fourier series of h(ζ,Θ) is given by

h(ζ,Θ) ∼
∞∑
ν=0

∞∑
ρ=0

αν,ρ[aν,ρ cos νζ cos ρΘ+bν,ρ sin νζ cos ρΘ+cν,ρ cos νζ sin ρΘ+dν,ρ sin νζ sin ρΘ],

(2)
where

αν,ρ =


1
4 for ν = ρ = 0
1
2 for ν > 0, ρ = 0 and ν = 0, ρ > 0

1 for ν > 0, ρ > 0.

(3)

and

aν,ρ =
1

π2

∫∫
S2

h(ζ,Θ) cos νζ cos ρΘ dζ dΘ (4)

with similar expressions for bν,ρ, cν,ρ and dν,ρ for ν = 0, 1, 2, . . . and ρ = 0, 1, 2, . . ., where
S2 denotes the fundamental square (−π, π;−π, π).
The partial sums of (2) can be denoted by

sν,ρ(h; ζ,Θ) =

ν∑
i=0

ρ∑
j=0

[aij cos iζ cos jΘ+bij sin iζ cos jΘ+cij cos iζ sin jΘ+dij sin iζ sin jΘ].

We can also write

sν,ρ(ζ,Θ) =
1

π2

∫∫
S2

h(ζ + σ,Θ + τ)
sin
(
ν + 1

2

)
σ sin

(
ρ+ 1

2

)
τ

4 sin
(
σ
2

)
. sin

(
τ
2

) dσ dτ.

The number
[
ν
p

]
for 0 ≤ p ≤ ν and ν ∈ N ∪ {0} is defined by

ν−1∏
l=0

(ζ + l) = ζ(ζ + 1) · · · (ζ + ν − 1) =
ν∑
p=0

[
ν

p

]
ζp =

Γ(ζ + ν)

Γζ
.

Let us also define for ρ ∈ N ∪ {0}, the number
[
ρ
q

]
for 0 ≤ q ≤ ρ, by

ρ−1∏
l=0

(Θ + l) = Θ(Θ + 1) · · · (Θ + ρ− 1) =

ρ∑
q=0

[
ρ

q

]
Θq =

Γ(Θ + ρ)

ΓΘ
.

The numbers
[
ν
p

]
and

[
ρ
q

]
are called the absolute values of Stirling numbers of first kind.



H. K. Nigam, Md Hadish / Eur. J. Pure Appl. Math, 13 (3) (2020), 567-578 569

Let {sν,ρ} be the sequence of partial sums of double infinite series
∑∞

ν=0

∑∞
ρ=0 aν,ρ and

write

sλ,µν,ρ =
Γλ

Γ(λ+ ν)
× Γµ

Γ(µ+ ρ)

ν∑
p=0

ρ∑
q=0

[
ν

p

][
ρ

q

]
λpµqsp,q (5)

to denote (ν, ρ)th Kλ,µ-means of order (λ, µ) > 0. If

sλ,µν,ρ → s as (ν, ρ)→∞, (6)

then the sequence sν,ρ or the series
∑∞

ν=0

∑∞
ρ=0 aν,ρ summable to s by double Karamata

(Kλ,µ) method of order (λ, µ) > 0.
Thus,

sλ,µν,ρ → s(Kλ,µ) as (ν, ρ)→∞.

The method Kλ,µ is regular for (λ, µ) > 0. The regularity of the Kλ,µ method is supposed
throughout the paper.
“Since h(ζ) is continuous and 2π-periodic function then the Hölder class for h(ζ) is defined
as

Hα = {h ∈ C2π : |h(ζ)− h(Θ)| ≤ K|ζ −Θ|} ,

where K is a positive constant. It can be verified that Hα is a Banach space with the
norm ‖.‖α defined by

‖h‖α = ‖h‖C + sup
ζ 6=Θ

∆αh(ζ,Θ), (7)

where

∆αh(ζ,Θ) =
|h(ζ)− h(Θ)|
|ζ −Θ|α

for ζ 6= Θ.

By convention ∆0h(ζ,Θ) = 0 and ‖h‖C = supζ∈[−π,π] |h(ζ)|. The metric induced by the
norm (7) on Hα is called the Hölder metric [13].”
Since h(ζ,Θ) is continuous and 2π-periodic function then the Hölder class for h(ζ,Θ) is
defined as

Hα,β =
{
h : |h(ζ,Θ; z, w) := |h(ζ,Θ)− h(z, w)| ≤ C1

(
|ζ − z|α + |Θ− w|β

)}
for some α, β > 0 and for all ζ,Θ, z, w. In above class of function, C1 is some positive
constant, which may depend on h, but not on ζ,Θ, t. Hα,β class of function is identical to
Lip(α, β) class of function.
Hα,β is a Banach space, whose norm ‖.‖α,β is defined by

‖h‖α,β = ‖h‖C + sup
ζ 6=z,
Θ 6=w

∆α,βh(ζ,Θ; z, w) (8)

i.e.

‖h‖α,β = ‖h‖C + sup
ζ 6=z,
Θ6=w

|f(ζ,Θ)− f(z, w)|
|ζ − z|α + |Θ− w|β

for ζ 6= z,Θ 6= w



H. K. Nigam, Md Hadish / Eur. J. Pure Appl. Math, 13 (3) (2020), 567-578 570

where ∆α,βh(ζ,Θ; z, w) = |h(ζ,Θ)−h(z,w)|
|ζ−z|α+|Θ−w|β (ζ 6= z,Θ 6= w).

By convention ∆0,0h(ζ,Θ; z, w) = 0 and

‖h‖C = sup
(ζ,Θ)∈S2

|h(ζ,Θ)|. (9)

“The η-order error of approximation of a function h ∈ C2π is defined by

Eη(h) = inf
tη
‖h− tη‖,

where tη is a trigonometric polynomial of degree η (Bernstein [3])”.
“If Eη(h) → 0 as η → ∞, then Eη(h) is said to be the best approximation of h ([20])”.
We write

φ(t) = h(ζ + t) + h(ζ − t)− 2h(ζ).

Φ(t) =

∫ t

0
|φ(σ)| dσ.

Φ(σ, τ) = Φ(ζ,Θ;σ, τ)

=
1

4
[h(ζ + σ,Θ + τ) + h(ζ + σ,Θ− τ) + h(ζ − σ,Θ + τ)

+ h(ζ − σ,Θ− τ)− 4h(ζ,Θ)]

where Ψ(σ, τ) := Ψ(z, w;σ, τ).

Since h(ζ,Θ) ∈ Hα,β, then |F (σ, τ)| = O(|ζ − z|α + |Θ− w|β).

Kλ
ν (σ) =

∑ν
p=0

[
ν
p

]
λp sin

(
p+ 1

2

)
σ

Γ(λ+ ν) sin
(
σ
2

) .

Kµ
ρ (τ) =

∑ρ
q=0

[
ρ
q

]
µq sin

(
q + 1

2

)
τ

Γ(µ+ ρ) sin
(
τ
2

) .

3. Main Theorem

Theorem 1. The best approximation of a 2π-periodic function h(ζ,Θ) of two variables ζ
and Θ and Lebesgue integrable over S2(−π, π;−π, π) in Hα,β, 0 < α, β ≤ 1 class by double
Karamata (Kλ,µ) method of its double Fourier series is given by

‖sλ,µν,ρ (ζ,Θ)− h(ζ,Θ)‖α,β

= O

[
M N ΓλΓµ

(ν + 1)(ρ+ 1)

(
1

(ν + 1)α
+

1

(ρ+ 1)β
+ 1

)]
+O

[
M ΓλΓµ

(ν + 1)Γ(µ+ ρ)

(
1 +

lnπ(ρ+ 1)

(ν + 1)α
+ lnπ(ρ+ 1)

)]



H. K. Nigam, Md Hadish / Eur. J. Pure Appl. Math, 13 (3) (2020), 567-578 571

+O

[
N ΓλΓµ

(ρ+ 1)Γ(λ+ ν)

(
1 +

lnπ(ρ+ 1)

(ρ+ 1)β
+ lnπ(ν + 1)

)]
+O

[
ΓλΓµ

Γ(λ+ ν)Γ(µ+ ρ)

(
ln
(
(ν + 1)(ρ+ 1)π2

)
+ {lnπ(ν + 1)}{lnπ(ρ+ 1)}

)]
where M = λ ln(ν + 1) + 1 and N = µ ln(ρ+ 1) + 1.

4. Lemmas

Lemma 1. “([19]). Let λ > 0 and 0 < t < π
2 then

ImΓ(λeit + ρ)

Γ(λ cos t+ ρ) sin
(
t
2

) =
| sin(λ ln(ρ+ 1). sin t)|

sin
(
t
2

) +O(1)

as ρ→∞ uniformly in t”.

Lemma 2. “([12]). For 0 < σ < 1
ν+1 ,

Kλ
ν (σ) = O [λ ln(ν + 1)] +O(1)

and for 0 < τ < 1
ρ+1 ,

Kµ
ρ (τ) = O [µ ln(ρ+ 1)] +O(1)

”.

Lemma 3. For 1
ν+1 ≤ σ ≤ π

Kλ
ν (σ) = O

[
1

σ Γλ

]
.

Proof. Using sin σ
2 ≥

σ
π and | sin(ρσ)| ≤ 1

|Kλ
ν (σ)| =

∣∣∣∣∣
∑ν

p=0

[
ν
p

]
λp sin

(
p+ 1

2

)
σ

Γ(λ+ ν) sin
(
σ
2

) ∣∣∣∣∣
≤ 1

Γ(λ+ ν)

ν∑
p=0

[
ν

p

]
λp

1

sin
(
σ
2

)
= O

[
1

σ Γλ

]
.

Lemma 4. For 1
ρ+1 ≤ τ ≤ π

Kµ
ρ (τ) = O

[
1

τ Γµ

]
.

Proof. This can be proved along the same lines of Lemma 4.3.



H. K. Nigam, Md Hadish / Eur. J. Pure Appl. Math, 13 (3) (2020), 567-578 572

5. Proof of the Main Theorem

Let sν,ρ(ζ,Θ) denote the (ν, ρ)th partial sum of the series (2), we have

sν,ρ(ζ,Θ)− h(ζ,Θ) =
1

π2

∫ π

0

∫ π

0
Φ(σ, τ)

sin
(
ν + 1

2

)
σ

sin
(
σ
2

) .
sin
(
ρ+ 1

2

)
τ

sin
(
τ
2

) dσ dτ.

Denoting Kλ,µ means of {sν,ρ(ζ,Θ)} by sλ,µν,ρ (ζ,Θ), we get

sλ,µν,ρ (ζ,Θ)− h(ζ,Θ)

=
Γλ

Γ(λ+ ν)
.

Γµ

Γ(µ+ ρ)

ν∑
p=0

ρ∑
q=0

[
ν

p

][
ρ

q

]
λpµq (sp,q(ζ,Θ)− h(ζ,Θ))

=
ΓλΓµ

π2

∫ π

0

∫ π

0
Φ(σ, τ)

ν∑
p=0

[
ν

p

]
λp sin

(
p+ 1

2

)
σ

Γ(λ+ ν) sin
(
σ
2

) ρ∑
q=0

[
ρ

q

]
µq sin

(
q + 1

2

)
τ

Γ(µ+ ρ) sin
(
τ
2

) dσ dτ
=

ΓλΓµ

π2

∫ π

0

∫ π

0
Φ(σ, τ)Kλ

ν (σ)Kµ
ρ (τ) dσ dτ

= Iν,ρ(ζ,Θ) (say). (10)

Let us estimate

sup
ζ 6=z,
Θ 6=w

|Iν,ρ(ζ,Θ)− Iν,ρ(z, w)|
|ζ − z|α + |Θ− w|β

= O(1). (11)

Now,

|Iν,ρ(ζ,Θ)− Iν,ρ(z, w)|

=
ΓλΓµ

π2

∣∣∣∣∫ π

0

∫ π

0
F (σ, τ)Kλ

ν (σ)Kµ
ρ (τ) dσ dτ

∣∣∣∣
≤ ΓλΓµ

π2

(∫ 1
ν+1

0

∫ 1
ρ+1

0
+

∫ 1
ν+1

0

∫ π

1
ρ+1

+

∫ π

1
ν+1

∫ 1
ρ+1

0
+

∫ π

1
ν+1

∫ π

1
ρ+1

)
|F (σ, τ)Kλ

ν (σ)Kµ
ρ (τ)| dσ dτ

= O

[
ΓλΓµ

π2
(J1 + J2 + J3 + J4)

]
. (12)

Using the fact that |F (σ, τ)| = O(|ζ − z|α + |Θ − w|β) and Lemma 4.2 for 0 < σ < 1
ν+1 ,

we obtain

J1 =

∫ 1
ν+1

0

∫ 1
ρ+1

0
|F (σ, τ)Kλ

ν (σ)Kµ
ρ (τ)| dσ dτ

= [O{λ ln(ν + 1)}+O(1)] [O{µ ln(ρ+ 1)}+O(1)]

∫ 1
ν+1

0

∫ 1
ρ+1

0
|F (σ, τ)| dσ dτ

= O(1)[{λ ln(ν + 1)}+ 1][{µ ln(ρ+ 1)}+ 1][(|ζ − z|α + |Θ− w|β)]

∫ 1
ν+1

0

∫ 1
ρ+1

0
dσ dτ



H. K. Nigam, Md Hadish / Eur. J. Pure Appl. Math, 13 (3) (2020), 567-578 573

= O(1)

{
[{λ ln(ν + 1)}+ 1]

ν + 1
× [{µ ln(ρ+ 1)}+ 1]

ρ+ 1

}
(|ζ − z|α + |Θ− w|β). (13)

For 0 < α, β ≤ 1, by using Lemmas 4.2 for 0 < σ < 1
ν+1 , 4.4 and the fact that |F (σ, τ)| =

O(|ζ − z|α + |Θ− w|β), we get

J2 =

∫ 1
ν+1

0

∫ π

1
ρ+1

|F (σ, τ)Kλ
ν (σ)Kµ

ρ (τ)| dσ dτ

= O{λ log(ν + 1) + 1}
∫ 1

ν+1

0

∫ π

1
ρ+1

|F (σ, τ)||Kµ
ρ (τ)| dσ dτ

= O(1)
{λ log(ν + 1) + 1}

ν + 1

∫ π

1
ρ+1

|F (σ, τ)||Kµ
ρ (τ)| dτ

= O

{
{λ ln(ν + 1)}+ 1

ν + 1

}
(|ζ − z|α + |Θ− w|β)

∫ π

1
ρ+1

1

τΓµ
dτ

= O

{
{λ ln(ν + 1)}+ 1

(ν + 1)Γµ

}
(|ζ − z|α + |Θ− w|β) lnπ(ρ+ 1). (14)

Similarly by changing the order of integration in J3 and using Lemmas 4.2 for 0 < τ < 1
ρ+1

and 4.3, we obtain

J3 = O

{
{µ ln(ρ+ 1)}+ 1

(ρ+ 1)Γλ

}
(|ζ − z|α + |Θ− w|β) ln((ν + 1)π). (15)

Now, using Lemmas 4.3 and 4.4, we get

J4 =

∫ π

1
ν+1

∫ π

1
ρ+1

|F (σ, τ)Kλ
ν (σ)Kµ

ρ (τ)| dσ dτ

=

∫ π

1
ν+1

Kλ
ν (σ)

(∫ π

1
ρ+1

1

τΓµ
dτ

)
|F (σ, τ)| dσ

= O

{
1

Γµ

}∫ π

1
ν+1

Kλ
ν (σ) ln((ρ+ 1)π)|F (σ, τ)| dσ

= O

{
ln((ρ+ 1)π). ln((ν + 1)π)

ΓλΓµ
(|ζ − z|α + |Θ− w|β)

}
. (16)

Combining (12) to (16), we obtain

|Iν,ρ(ζ,Θ)− Iν,ρ(z, w)|
(|ζ − z|α + |Θ− w|β)

= O

(
[{λ ln(ν + 1)}+ 1][{µ ln(ρ+ 1)}+ 1]ΓλΓµ

(ν + 1)(ρ+ 1)
+

[{λ ln(ν + 1)}+ 1]Γλ lnπ(ρ+ 1)

ν + 1

)



H. K. Nigam, Md Hadish / Eur. J. Pure Appl. Math, 13 (3) (2020), 567-578 574

+O

(
[{µ ln(ρ+ 1)}+ 1]Γµ lnπ(ν + 1)

ρ+ 1
+

ln(ρ+ 1)π) ln((ν + 1)π)

ΓλΓµ

)
. (17)

Now, from (10), we have

|Iν,ρ(ζ,Θ)|

=
ΓλΓµ

π2

∣∣∣∣∫ π

0

∫ π

0
Φ(σ, τ)Kλ

ν (σ)Kµ
ρ (τ) dσ dτ

∣∣∣∣
≤ΓλΓµ

π2

[∫ 1
ν+1

0

∫ 1
ρ+1

0
+

∫ ν+1

0

∫ π

1
ρ+1

+

∫ π

1
ν+1

∫ 1
ρ+1

0
+

∫ π

1
ν+1

∫ π

1
ρ+1

|Φ(σ, τ)||Kλ
ν (σ)||Kµ

ρ (τ)| dσ dτ

]

=
ΓλΓµ

4π2
(I1 + I2 + I3 + I4). (18)

Now,

I1 =

∫ 1
ν+1

0

∫ 1
ρ+1

0
|Φ(σ, τ)||Kλ

ν (σ)||Kµ
ρ (τ)| dσ dτ

I1 = O

[
(λ log(ν + 1) + 1)(µ log(ρ+ 1) + 1)

∫ 1
ν+1

0

∫ 1
ρ+1

0
(σα + τβ) dσ dτ

]

= O

[
MN

∫ 1
ν+1

0
σα

{∫ 1
ρ+1

0
dτ

}
dσ

]
+O

[
MN

∫ 1
ρ+1

0
τβ

{∫ 1
m+1

0
dσ

}
dτ

]
where M = {λ ln(ν + 1)}+ 1 and N = {µ ln(ρ+ 1)}+ 1

=

[
MN

ρ+ 1

∫ 1
ν+1

0
σα dσ

]
+

[
MN

ν + 1

∫ 1
ρ+1

0
τβ dτ

]

= O(1)
MN

(ν + 1)(ρ+ 1)

[
1

(ν + 1)α
+

1

(ρ+ 1)β

]
. (19)

I2 =

∫ 1
ν+1

0

∫ π

1
ρ+1

|Φ(σ, τ)||Kλ
ν (σ)||Kµ

ρ (τ)| dσ dτ

= O

[
{λ log(ν + 1) + 1}

∫ 1
ν+1

0

∫ π

1
ρ+1

1

τΓµ
(σα + τβ) dσ dτ

]

= O

[
Γµ

∫ 1
ν+1

0

∫ π

1
ρ+1

1

τ
σα dσ dτ

]
+O

[
M

Γµ

∫ 1
ν+1

0

∫ π

1
ρ+1

1

τ
τβ dσ dτ

]

= O(1)
M

(ν + 1)Γµ

(
ln(ρ+ 1)π

(ν + 1)α
+ 1

)
. (20)

Similarly,

I3 = O(1)
N

(ρ+ 1)Γλ

(
ln(ν + 1)π

(ρ+ 1)β
+ 1

)
(21)



H. K. Nigam, Md Hadish / Eur. J. Pure Appl. Math, 13 (3) (2020), 567-578 575

and

I4 =

∫ π

1
ν+1

∫ π

1
ρ+1

|Φ(σ, τ)||Kλ
ν (σ)||Kµ

ρ (τ)| dσ dτ

= O

[∫ π

1
ν+1

∫ π

1
ρ+1

1

στΓλΓµ
[σα + τβ] dσ dτ

]

= O

[
1

ΓλΓµ

∫ π

1
ν+1

σα

σ

{∫ π

1
ρ+1

dτ

τ

}
dσ

]
+O

[
1

ΓλΓµ

∫ π

1
ν+1

1

σ

{∫ π

1
ρ+1

τβ

τ
dτ

}
dσ

]
.

= O

[
1

ΓλΓµ
ln(ρ+ 1)π

]
+O

[
1

ΓλΓµ
ln(ν + 1)π

]
= O

[
1

ΓλΓµ
ln(ν + 1)(ρ+ 1)π2

]
(22)

Combining (18) to (22), we get

Iν,ρ = O

[
ΓλΓµ

π2

{
MN

(ν + 1)(ρ+ 1)

(
1

(ν + 1)α
+

1

(ρ+ 1)β

)
+

M

(ν + 1)Γµ

(
1 +

ln(ρ+ 1)π

(ν + 1)α

)}]
+O

[
ΓλΓµ

π2

{
N

(ν + 1)Γλ

(
1 +

ln(ν + 1)π

(ρ+ 1)β

)
+

1

ΓλΓµ
log(ν + 1)(ρ+ 1)π2

}]
. (23)

By (17) and (23), we obtain

‖Iν,ρ‖α,β = ‖sλ,µν,ρ (ζ,Θ)− h(ζ,Θ)‖α,β = O

[
MNΓλΓµ

(ν + 1)(ρ+ 1)

(
1

(ν + 1)α
+

1

(ρ+ 1)β
+ 1

)]
+O

[
MΓλΓµ

(ν + 1)Γ(µ+ ρ)

(
1 +

log((ρ+ 1)π)

(ν + 1)α
+ log((ρ+ 1)π)

)]
+

[
NΓλΓµ

(ν + 1)Γ(λ+ ν)

(
1 +

log((ρ+ 1)π)

(ρ+ 1)β
+ log((ν + 1)π)

)]
+O

[
ΓλΓµ

Γ(λ+ ν)Γ(µ+ ρ)

(
log
(
(ν + 1)(ρ+ 1)π2

)
+ {log((ν + 1)π)}{log((ρ+ 1)π)

)
}
]

where M = {λ ln(ν + 1)}+ 1 and N = {µ ln(ρ+ 1)}+ 1.

6. Verification

We calculate error by putting some values of ν, ρ, α, β, λ, µ.

6.1. ν = ρ = 10, α = β = 0.5, λ = 2, µ = 3

error E1 ∼ 1.25827



H. K. Nigam, Md Hadish / Eur. J. Pure Appl. Math, 13 (3) (2020), 567-578 576

6.2. ν = ρ = 20, α = β = 0.5, λ = 2, µ = 3

error E2 ∼ 0.46798

6.3. ν = ρ = 50, α = β = 0.5, λ = 2, µ = 3

error E3 ∼ 0.111632

6.4. ν = ρ = 500, α = β = 0.5, λ = 2, µ = 3

error E4 ∼ 0.0011456

6.5. ν = ρ = 1000, α = β = 0.5, λ = 2, µ = 3

error E5 ∼ 6.83193× 10−4.

6.6. ν = ρ = 10000, α = β = 0.5, λ = 2, µ = 3

error E6 ∼ 1.13411× 10−5.

6.7. ν = ρ = 100000, α = β = 0.5, λ = 2, µ = 3

error E7 ∼ 1.1191× 10−7.

6.8. ν = ρ = 1010, α = β = 0.5, λ = 2, µ = 3

error E8 ∼ 5.36301× 10−15.

7. Conclusion

From above verification, we observed that error estimation approaches to zero rapidly
as ν, ρ increase infinitely. Thus, we arrive at the best approximation of the function.

Acknowledgements

First author expresses his gratitude towards his mother for her blessings. The first au-
thor also expresses his gratitude towards his father in heaven, whose soul is always guiding
and encouraging him. Second author is thankful to the University Grants Commission,
India for providing Senior Research fellowship(SRF) to carry out the present work as a
part of Ph.D, degree. The Second author also expresses his gratitude towards his parents
for blessings. Both the authors are also grateful to the Hon’ble vice-chancellor, Central
University of South Bihar, for motivation to this work.



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