EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 5, 2020, 1306-1324
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Special Issue Dedicated to
Professor Hari M. Srivastava

On the Occasion of his 80th Birthday

Further approximations on Durrmeyer modification of
Szász-Mirakjan operators

Rishikesh Yadav1,∗, Ramakanta Meher1, Vishnu Narayan Mishra2

1 Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute
of Technology Surat, Surat-395 007 (Gujarat), India
2 Department of Mathematics, Indira Gandhi National Tribal University, Lalpur,
Amarkantak-484 887, Anuppur, Madhya Pradesh, India

Abstract. The main purpose of this paper is to determine the approximations of Durrmeyer
modification of Szász-Mirakjan operators, defined by Mishra et al. (Boll. Unione Mat. Ital.
(2016) 8(4):297-305). We estimate the order of approximation of the operators for the functions
belonging to the different spaces. Here, the rate of convergence of the said operators is established
by means of the function with derivative of the bounded variation. At last, the graphical analysis
is discussed to support the approximation results of the operators.

2020 Mathematics Subject Classifications: 41A25, 41A35, 41A36.

Key Words and Phrases: Szász-Mirakjan operators, rate of convergence, Peetres K-functional,
function of bounded variation.

1. Introduction

In 1944, Mirakjan [8] and 1950, Szász [16] introduced operators on unbounded interval
[0,∞), known as Szász-Mirakjan operators defined by

Sn(g;x) =

∞∑
j=0

sn,j(x)g

(
j

n

)
, (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3728

Email addresses: rishikesh2506@gmail.com (R. Yadav),
meher ramakanta@yahoo.com (R. Meher), vishnunarayanmishra@gmail.com (V. N. Mishra)

https://www.ejpam.com 1306 c© 2020 EJPAM All rights reserved.



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1307

where sn,j = e−nx (nx)j

j! , g ∈ C2[0,∞) = {g ∈ C[0,∞) : lim
x→∞

f(x)
1+x2

exists and finite}, x ≥ 0

and for all n ∈ N.

An integral modification of the above operators (1) can be seen in [2] to estimate
the approximation results for the integrable function. The important properties including
global results, local results, simultaneous approximation, convergence properties, etc. have
been studied with the above operators and their modifications in various studies (see
[1, 3, 11–13]). One of them, an interesting modification was the Durrmeyer modification
of the Szász-Mirakjan operators and is written as:

Dn(g;x) =
∞∑
j=0

sn,j(x)

∞∫
0

sn,j(t)g(t)dt, (2)

seen in [7]. Also, another modification into Stancu variant appeared in [9] of the above
operators (2) and related properties like density, direct results as well as Voronovskaya
type theorem are studied. Many approximation results are also discussed in [14, 20].

A natural generalization is carried out for the above operators (2) in [10] by Mishra et
al. for the study of simultaneous approximation, like

B∗n(g;x) = un

∞∑
j=0

sun,j(x)

∞∫
0

sun,j(t)g(t)dt, (3)

where sun,j(x) = e−unx (unx)j

j! by considering the sequence un is strictly increasing of posi-
tive real number as well as un →∞ as n→∞ with u1 = 1.

Our main motive is to study the approximation properties of the proposed operators
(3) for the functions from different spaces. The important properties of the above proposed
operators (3) are studied by authors which can also be applied to the operators defined
by (2).

In order to study the operators (3), we divide the paper into sections. Section second
contains preliminary results, which are used to prove the main theorems. Section third
deals with the approximation properties of the operators for the function belongs to the
different spaces of functions classes. In section fourth, the rate of convergence of the
operators is estimated for the functions with derivative of bounded variation. At last, we
present the graphical and numerical representation for the operators in order to show the
convergence of the operators.

2. Preliminary

This section contains the basic properties of the defined operators (3). In order to
prove approximations properties, we need basic lemmas.

Lemma 1. For all x ≥ 0 and n ∈ N, we have

B∗n(1;x) = 1



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1308

B∗n(t;x) =
1

un
+ x

B∗n(t2;x) =
2 + 4xun + x2u2n

u2n

B∗n(t3;x) =
6 + 18xun + 9x2u2n + x3u3n

u3n
.

Proof. We can easily proof the above parts of the lemma, so we omit the proof.

Lemma 2. Consider the function g is integrable, continuous, bounded on given interval
[0,∞), then the central moments can be obtained as:

Ωn,m = un

∞∑
j=0

sun,j(x)

∞∫
0

sun,j(t)(t− x)mdt, (4)

where m = 0, 1, 2, . . .. So for m = 0, 1, we get the the central moments as follows:

Ωn,0 = 1,Ωn,1 =
1

un
, (5)

in general, we have

unΩn,m+1 = x
(
Ω′n,m + 2mΩn,m−1 + (1 +m)Ωn,m

)
, (6)

this lead us to

Ωn,m = O

(
u
−[m+1

2 ]
n

)
. (7)

Lemma 3. Let the function g be the continuous and bounded on [0,∞) endowed with
supremum norm ‖g(x)‖ = sup

x≥0
|g| then, we have

|B∗n(g;x)| ≤ ‖g‖. (8)

Remark 1. For second order central moment, it can be written as

Ωn,2 =
2(1 + unx)

u2n
=

2

un

(
x+

1

un

)
=

2

un
ζ2n(x), (9)

where ζ2n(x) =
(
x+ 1

un

)
.



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1309

3. Approximation properties

Consider CB[0,∞) be the space of all continuous and bounded function defined on
[0,∞), endowed with supremum norm ‖g‖ = sup

x≥0
|g(x)|, also let for any δ > 0

K2(g; δ) = inf
f∈E
{‖g − f‖+ δ‖f ′′‖} (10)

be the Peetre’s K-functional, where E = {f ∈ CB[0,∞) : f ′, f ′′ ∈ CB[0,∞)}. Also a
relation can be seen for which there exists a positive constant M such that:

K2(g; δ) ≤Mω2(g,
√
δ), δ > 0, (11)

where ω2(g,
√
δ) is second order modulus of smoothness for the function g ∈ CB[0,∞),

which is defined by:

ω2(g, δ) = sup{g(x+ h)− 2g(x) + g(x− h) : x, x± h ∈ [0,∞), 0 ≤ h ≤ δ}, (12)

also usual modulus of continuity can be defined for the function g ∈ CB[0,∞) as follows:

ω(g, δ) = {g(y)− g(x) : x, y ∈ [0,∞), |y − x| ≤ δ, δ > 0}. (13)

Theorem 1. Consider g ∈ CB[0,∞) and for all x ≥ 0 then there exists a positive constant
C such that

|B∗n(g;x)− g(x)| ≤ Cω2

(
g,

√
δn
2

)
+ ω (g, γn) , (14)

where δn = B̃∗n((t− x)2;x) + 1
u2n

and γn = B̃∗n((t− x);x).

Proof. Here, we consider the auxiliary operators as follows:

S̃∗n(g;x) = B∗n(g;x)− g
(

1

un
+ x

)
+ g(x). (15)

Let f ∈ E, x ≥ 0 then using Taylor’s formula, we get

f(t)− f(x) = (t− x)f ′(x) +

t∫
0

(t− v)f ′′(v)dv. (16)

Applying the operators B̃∗n on the both sides to the above expression, it yields:

B̃∗n(f ;x)− f(x) = f ′(x)B̃∗n(t− x;x) + B̃∗n

 t∫
x

(t− v)f ′′(v)dv





R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1310

= B̃∗n

 t∫
x

(t− v)f ′′(v)dv



= S∗n

 t∫
x

(t− v)f ′′(v)dv

−


(
1
un

+x
)∫

x

(
1

un
+ x− v

)
f ′′(v)dv

 .(17)

Here, the following inequalities are as:∣∣∣∣∣∣
t∫

x

(t− v)f ′′(v)dv

∣∣∣∣∣∣ ≤ (t− x)2‖f ′′‖ (18)

and ∣∣∣∣∣∣∣∣∣
(

1
un

+x
)∫

x

(
1

un
+ x− v

)
f ′′(v)dv

∣∣∣∣∣∣∣∣∣ ≤
1

u2n
‖f ′′‖. (19)

By considering the above inequalities (18, 19) and with the help of (17), we obtain

B̃∗n(f ;x)− f(x) =

{
B̃∗n((t− x)2;x) +

1

u2n

}
‖f ′′‖ (20)

= δn‖f ′′‖. (21)

Also, |S∗n(g;x)| ≤ ‖g‖. Using this property, we get

|S∗n(g;x))− g(x)| ≤ |B̃∗n(g − f ;x)− (g − f)(x)|+ |B̃∗n(f ;x)− f(x)|

+

∣∣∣∣g( 1

un
+ x

)
− g(x)

∣∣∣∣
≤ 4‖g − f‖+ |B̃∗n(f ;x)− f(x)|+

∣∣∣∣g( 1

un
+ x

)
− g(x)

∣∣∣∣ ,
using (20) and with the help of modulus of continuity, we obtain

|S∗n(g;x)− g(x)| ≤ 4‖g − f‖+ δn‖f ′′‖+ ω (g, γn) .

Taking the infimum for all f ∈ E on the right hand side and by relation (11), we get

|S∗n(g;x)− g(x)| ≤ 4K2

(
g;

1

4
δn

)
+ ω (g, γn)

≤ Cω2

(
g,

√
δn
2

)
+ ω (g, γn) .



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1311

Thus, the proof is completed.

Now, we estimate the approximation of the defined operators (3), by new type of
Lipschitz maximal function with order s ∈ (0, 1], defined by Lenze [6] as

τs(g, x) = sup
x,t≥0

|g(t)− g(x)|
|t− x|s

, t 6= x. (22)

Using definition of Lipschitz maximal function, we have a theorem.

Theorem 2. For any g ∈ CB[0,∞) with s ∈ (0, 1] then one can obtain

|B∗n(g;x)− g(x)| ≤ τs(g, x) (Ωn,2)
s
2 .

Proof. By equation (22), we can write

|B∗n(g;x)− g(x)| ≤ τs(g, x)B∗n(|t− x|s;x).

Using, Hölder’s inequality with j = 2
s , l = 2

2−s , one can get

|B∗n(g;x)− g(x)| ≤ τs(g, x)
(
B∗n(g;x)((t− x)2;x)

) s
2 = τs(f, x) (Ωn,2)

s
2 .

Next theorem is based on modified Lipschitz type spaces [15] and this spaces is defined
by

Lipm1,m2

M (s) =

{
g ∈ CB[0,∞) : |g(l1)− g(l2)| ≤M

|l1 − l2|s(
l1 + l22m1 + l2m2

) s
2

, where l1, l2 ≥ 0

are variables, s ∈ (0, 1]

}

and m1,m2 are the fixed numbers and M > 0 is a constant.

Theorem 3. For g ∈ Lipm1,m2

M (s) and 0 < s ≤ 1, an inequality holds:

|B∗n(g;x)− g(x)| ≤ M

(
Ωn,2

x(xm1 +m2)

) s
2

, M > 0, x ∈ [0,∞).

Proof. We have s ∈ (0, 1] and in order to prove the above theorem, we discuss the
cases on s.

Case 1. if we consider s = 1 then for all t, x ≥ 0, we can observe that 1
t+x2m1+xm2)

≤
1

x(xm1+m2)
then

|B∗n(g;x)− g(x)| ≤ B∗n(|g(t)− g(x)|;x)



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1312

≤ MB∗n

(
|t− x|

(t+ x2m1 + xm2)
1
2

;x

)

≤ M

(x(xm1 +m2))
1
2

B∗n(|t− x|;x)

≤ M

(x(xa1 + a2))
1
2

(Ωn,2)
1
2

≤ M

(
Ωn,2

x(xm1 +m2)

) 1
2

.

Case 2. for s ∈ (0, 1) then using Hölder inequality with p = 2
s , q = 2

2−s , we get

|B∗n(g;x)− g(x)| ≤
(
B∗n(|g(t)− g(x)|

2
s ;x)

) s
2 ≤MB∗n

(
|t− x|2

(t+ x2m1 + xm2)
;x

) s
2

≤ MB∗n

(
|t− x|2

(x(xm1 +m2))
;x

) s
2

≤ M

(
Ωn,2

x(xm1 +m2)

) s
2

.

This complete the proof.

Theorem 4. For the function g which is continuous and bounded on [0,∞), the conver-
gence of the operators can be obtained as:

lim
n→∞

B∗n(g;x) = g(x), (23)

uniformly on any compact interval of [0,∞).

Proof. Using Bohman-Korovkin theorem, we can get our required result. Since
lim
n→∞

B∗n(1;x) → 1, lim
n→∞

B∗n(t;x) → x, lim
n→∞

B∗n(t2;x) → x2 and hence the proposed opera-

tors B∗n(g;x) converge uniformly to the function g(x) on any compact interval of [0,∞).

4. Rate of convergence by means of the function with derivative of
bounded variation

This section consists the rate of convergence by means of the function with derivative
of bounded variation. Let DBV [0,∞) be the set of all class of function having derivative
of bounded variation on every compact interval of [0,∞). The following representation for
the function g ∈ DBV [0,∞), is as follows:

g(x) =

x∫
0

h(t)dt+ g(0), (24)



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1313

where h(t) is a function with derivative of bounded variation on any compact interval of
[0,∞).

For investigation of the convergence of the above operators (3) to the function with
derivative of bounded variation, we rewrite (3) as follows:

B∗n(g;x) =

∫ ∞
0

Yn(x, t)g(t)dt, (25)

where

Yn(x, t) = un

∞∑
j=0

sun,j(x) sun,j(t).

Such type of properties have been studied by researchers using various operators (see
[4, 5, 17–19]).

Lemma 4. For sufficiently large value of n and for all x ≥ 0, we have

(i) In(x, t) =
y∫
0

Yn(x, t)dt ≤ 2
(x−y)2un ζ

2
n(x), 0 ≤ y < x,

(ii) 1− In(x, t) =
∞∫
z
Yn(x, t)dt ≤ 2

(z−x)2un ζ
2
n(x), x ≤ z <∞.

Proof. Using the Lemma 2 and since the value of n is sufficiently large, so we have

In(x, t) =

y∫
0

Yn(x, t)dt ≤
y∫

0

(
(x− t)2

(x− y)2

)
Yn(x, t)dt

=
2

(x− y)2un
ζ2n(x).

Similarly, we can prove other inequality.

Theorem 5. Let g ∈ DBV [0,∞), then for all x ≥ 0, an upper bound of the operators to
the function can be as:

|B∗n(g;x)− g(x)| ≤ 1

2un
|g′(x+) + g′(x−)|+

√
1

2un
|g′(x+)− g′(x−)|ζn(x)

+
2ζ2n(x)

xun

[
√
un]∑

j=0

(
V t
x−x

j
g′x

)
+

x
√
un

(
V x
x− x√

un
g′x

)

+
x
√
un
V
x+ x√

un
x (g′x) +

2ζ2n(x)

xun

[
√
un]∑

j=0

V
x+x

j
x (g′x),



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1314

where

gx(t) =


g(t)− g(x−), 0 ≤ t < x,

0, t = x,

g(t)− g(x+), x < t <∞
(26)

be an auxiliary operator and V b
a g(x) denotes the total variation of the function g(x) on

[a, b].

Proof. Since, B∗n(1;x) = 1 and hence, one can write

B∗n(g;x)− g(x) =

∫ ∞
0

(g(t)− g(x))Yn(x, t)dt

=

∫ ∞
0

Yn(x, t)dt

t∫
x

g′(u)du.

Now, for g ∈ DBV [0,∞), we can write as:

g′(u) =
1

2
(g′(x+) + g′(x−)) + g′x(u) +

1

2
(g′(x+) + g′(x−)) (sgn(u− x))

+η(u)

(
g′(u)− 1

2
(g′(x+) + g′(x−))

)
,

where

η(u) =

{
1 u = x

0 u 6= x.
(27)

And then, one can show

∞∫
0

Yn(x, t)

t∫
x

(
η(u){g′(u)− 1

2
(g′(x+) + g′(x−))}du

)
dt = 0. (28)

Using (25), we can get

∞∫
0

Yn(x, t)

 s∫
x

1

2
(g′(x+) + g′(x−)) du

 dt =
1

2
(g′(x+) + g′(x−))

∞∫
0

Yn(x, t)(t− x) dt

=
1

2
(g′(x+) + g′(x−))Ωn,1. (29)

And



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1315

∣∣∣∣∣∣
∞∫
0

Yn(x, t)

1

2

t∫
x

(g′(x+)− g′(x−))sgn(u− x) du

 dt

∣∣∣∣∣∣ ≤ 1

2
|(g′(x+)− g′(x−))|

∞∫
0

Yn(x, t)|t− x| dt

≤ 1

2
|(g′(x+)− g′(x−))|

∞∫
0

|t− x|Yn(x, t)dt

≤ 1

2
|(g′(x+)− g′(x−)| (Ωn,2)

1
2 . (30)

Using (9), we get:

|B∗n(g;x)− g(x)| ≤ 1

2
|g′(x+) + g′(x−)|Ωn,1 +

1

2
|g′(x+)− g′(x−)|

√
2

un
ζn(x)

+

∣∣∣∣∣∣
∞∫
0

Yn(x, t)

1

2

s∫
x

(g′x(u)) du

 dt

∣∣∣∣∣∣ . (31)

Here,

∞∫
0

Yn(x, t)

 s∫
x

(g′x(u)) du

 dt =

x∫
0

Yn(x, t)

 s∫
x

(g′x(u)) du

 dt+

∞∫
x

Yn(x, t)(x, t)

 t∫
x

(g′x(u)) du

 dt

= P1 + P2, (32)

where

P1 =

x∫
0

 t∫
x

(g′x(u)) du

 ∂

∂t
(In(x, t))dt

=

x∫
0

g′x(t)In(x, t)dt

=

y∫
0

g′x(t)In(x, t)dt+

x∫
y

g′x(t)In(x, t)dt. (33)

Here, we consider y = x− x√
un

then by the above equality, one can write

∣∣∣∣∣∣∣∣
x∫

x− x√
un

g′x(t)In(x, t)dt

∣∣∣∣∣∣∣∣ ≤
x∫

x− x√
un

|g′x(t)||In(x, t)|dt



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1316

≤
x∫

x− x√
un

|g′x(t)− g′x(x)|dt, g′x(x) = 0, (where |In(x, t)| ≤ 1)

≤
x∫

x− x√
un

V x
t g
′
xdt

≤ V x
x− x√

un
g′x

x∫
x− x√

un

dt

=
x
√
un

(
V x
x− x√

un
g′x

)
. (34)

Using Lemma 4 for solving second term by substituting t = x− x
u , we get

x− x√
un∫

x

|g′x(t)|In(x, t)dt ≤ 2ζ2n(x)

un

x− x√
un∫

x

|g′x(t)|
(x− t)2

dt

≤ 2ζ2n(x)

un

x− x√
un∫

x

V x
t g
′
x

1

(x− t)2
dt

=
2ζ2n(x)

xun

√
un∫

x

V s
x− x

u
g′xdu

≤ 2ζ2n(x)

xun

[
√
un]∑

j=0

(
V t
x−x

j
g′x

)
. (35)

Hence,

|P1| ≤
2ζ2n(x)

xun

[
√
un]∑

j=0

(
V t
x−x

j
g′x

)
+

x
√
un

(
V x
x− x√

un
g′x

)
. (36)

To solve P2, we reform P2 and integrating by parts, we have

|P2| =

∣∣∣∣∣
z∫
x

 t∫
x

g′x(u)du

 ∂

∂t
(1− In(x, t))dt+

∞∫
z

 t∫
x

g′x(u)du

 ∂

∂t
(1− In(x, t))dt

∣∣∣∣∣
≤

∣∣∣∣∣∣
z∫
x

 t∫
x

g′x(u)du

 ∂

∂t
(1− In(x, t))dt

∣∣∣∣∣∣+

∣∣∣∣∣∣
∞∫
z

 t∫
x

g′x(u)du

 ∂

∂t
(1− In(x, t))dt

∣∣∣∣∣∣



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1317

=

∣∣∣∣∣
 t∫
x

g′x(u)du(1− In(x, t))

z
x

−
z∫
x

g′x(t)(1− In(x, t))dt

+

 t∫
x

g′x(u)du(1− In(x, t))

∞
z

−
∞∫
z

g′x(t)(1− In(x, t))dt

∣∣∣∣∣
=

∣∣∣∣∣
z∫
x

g′x(u)du(1− In(x, z))−
z∫
x

g′x(t)(1− In(x, t))dt

−
z∫
x

g′x(u)du(1− In(x, z))−
∞∫
z

g′x(t)(1− In(x, t))dt

∣∣∣∣∣
=

∣∣∣∣∣
z∫
x

g′x(t)(1− In(x, t))dt+

∞∫
z

g′x(t)(1− In(x, t))dt

∣∣∣∣∣
≤

z∫
x

V t
x(g′x)dt+

2ζ2n(x)

un

∞∫
z

V t
x(g′x)

1

(t− x)2
dt

≤ x
√
un
V
x+ x√

un
x (g′x) +

2ζ2n(x)

un

∞∫
x+ x√

un

V t
x(g′x)

1

(t− x)2
dt.

On substituting t = x
(

1 + 1
β

)
, we obtain

|P2| ≤
x
√
un
V
x+ x√

un
x (g′x) +

2ζ2n(x)

xun

√
un∫

0

V
x+ x

β
x (g′x)dβ

≤ x
√
un
V
x+ x√

un
x (g′x) +

2ζ2n(x)

xun

[
√
un]∑

j=0

√
j+1∫
j

V
x+x

j
x (g′x)dβ

=
x
√
un
V
x+ x√

un
x (g′x) +

2ζ2n(x)

xun

[
√
un]∑

j=0

V
x+x

j
x (g′x).

Using the value of P1, P2 in (32), we obtain

∞∫
0

Yn(x, t)

 s∫
x

(g′x(u)) du

 dt =
2ζ2n(x)

xun

[
√
un]∑

j=0

(
V t
x−x

j
g′x

)
+

x
√
un

(
V x
x− x√

un
g′x

)

+
x
√
un
V
x+ x√

un
x (g′x) +

2ζ2n(x)

xun

[
√
un]∑

j=0

V
x+x

j
x (g′x). (37)



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1318

Put the above value from (37) in (31), we obtain required result

|B∗n(g;x)− g(x)| ≤ 1

2un
|g′(x+) + g′(x−)|+

√
1

2un
|g′(x+)− g′(x−)|ζn(x)

+
2ζ2n(x)

xun

[
√
un]∑

j=0

(
V t
x−x

j
g′x

)
+

x
√
un

(
V x
x− x√

un
g′x

)

+
x
√
un
V
x+ x√

un
x (g′x) +

2ζ2n(x)

xun

[
√
un]∑

j=0

V
x+x

j
x (g′x).

5. Graphical and numerical analysis of the operators

In this section, we study the graphical representation and numerical analysis of the
operators to the function.

Example 1. Let the function g : [0, 2.5] → [0,∞) such that g(x) = −x3e−5x(blue)
for all x ∈ [0, 2.5]. Choosing un = n = 15, 35, 50 and then corresponding operators are
S∗15(g;x), S∗35(g;x), S∗50(g;x) represent green, red and black colors respectively in the given
Figure 1. One can observe that as the value of n is increased, the error of the operators
to the function is going to be least. We can say that the approach of the operators to the
function is good for the large value of n.

But for the same function, if we move towards the truncation type error, we can observe
by Figure 2, the approximation is not better throughout the interval [0, 2.5]. Here we
consider the un = n = 15, 35, 50 and j = 15, 35, 50, using these values, the truncation is
determined. So one can observe that at a some stage, its going good but not at all.

g

15

35

50

0.0 0.5 1.0 1.5 2.0 2.5

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

Figure 1: The convergence of the operators S∗n(g;x) to the function g(x)(blue).

Now, we determine the convergence of the operators to the function by considering the
different sequences for the operators and then we see that the variation of the convergence
to the function is changed.



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1319

g

15

35

50

0.0 0.5 1.0 1.5 2.0 2.5

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

Figure 2: The convergence of the operators S∗n(g;x) to the function g(x)(blue).

Example 2. Let the function g(x) = x2e2x(black), for all 0 ≤ x ≤ 2.5. Here, we consider
un = n and choosing the value of n = 10, 50, 100, 200, 250, 500, 1000, for which the opera-
tors’s curve is red for the all values of n. Then, we can observe the error estimations by
Figure 3 as well Table 1 at different points of x, which is going to be better as the value
of n is increased.

Figure 3: The convergence of the operators S∗n(g;x) to the function g(x).

Table 1: Convergence estimations of the operators B∗n(g;x) to the function g(x)

x ↓, un = n→ at n=10 at n=50 at n=100 at n=200 at n=250 at n=500 at n=1000

0.1 0.202522 0.0156053 0.0069326 0.00326665 0.00258244 0.00126086 0.000622967

0.5 3.82396 0.325365 0.148479 0.0710035 0.0563036 0.0276615 0.0137104

0.9 27.2622 2.13631 0.969982 0.462837 0.366865 0.180094 0.0892291

1.0 42.1618 3.22439 1.46137 0.696735 0.552174 0.270979 0.134238

1.5 310.724 20.8491 9.3538 4.43876 3.51461 1.72172 0.852162

2.0 1888.96 110.236 48.9145 23.0939 18.2677 8.93151 4.4164

2.5 10237.6 516.742 226.689 106.464 84.1292 41.0503 20.2783



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1320

Example 3. Let for the same function g(x) = x2e2x(black), for all 0 ≤ x ≤ 2.5. Here, we

consider un = n
3
2 and choosing the value of n = 10, 50, 100, 200, 250, 500, 1000, the curves

of the operators (3) represent green color for all values of n
3
2 for the operators (3). Hence,

we can observe the error estimations by Figure 4 as well Table 2 at the different points of
x.

Figure 4: The convergence of the operators S∗n(g;x) to the function g(x).

Table 2: Convergence estimations of the operators B∗n(g;x) to the function g(x).

x ↓, un = n
3
2 → at n=10 at n=50 at n=100 at n=200 at n=250 at n=500

0.1 0.0282979 0.0018008 0.000622967 0.000218562 0.000156203 0.0000551185

0.5 0.574288 0.0394044 0.0137104 0.0048201 0.00344596 0.0012166

0.9 3.79761 0.256632 0.0892291 0.0313623 0.0224205 0.00791509

1.0 5.74555 0.386191 0.134238 0.0471774 0.033726 0.011906

1.5 37.6466 2.45554 0.852162 0.299321 0.213959 0.0755213

2.0 201.92 12.7484 4.4164 1.55031 1.10808 0.391058

2.5 960.667 58.6418 20.2783 7.11386 5.08411 1.79398

x ↓, un = n
3
2 → at n=1000

0.1 0.0000194739

0.5 0.000429916

0.9 0.00279694

1.0 0.00420715

1.5 0.0266852

2.0 0.138172

2.5 0.633827

Example 4. Further for the function g(x) = x2e2x(black), for all 0 ≤ x ≤ 2.5, one can
see the error estimations of the operators (3). Here, we consider un = n2 and choosing
the value of n = 10, 50, 100, 200, 250, 500, 1000, the curves of the operators (3) represent
Magenta color for all values of n2 of the operators. Hence, we can observe the error
estimations by Figure 5 as well Table 3 at different points of x. By observing, we can see,
the function’s curve almost overlapped by the curves of the operators.



R. Yadav, R. Meher, V. N. Mishra / Eur. J. Pure Appl. Math, 13 (5) (2020), 1306-1324 1321

Figure 5: The convergence of the operators S∗n(g;x) to the function g(x).

Table 3: Convergence estimations of the operators B∗n(g;x) to the function g(x)

un = n2 → at n=10 at n=50 at n=100 at n=200 at n=250 at n=500

x=0.1 0.0069326 0.000247412 0.0000616321 0.0000153943 9.85127×10−6 2.462477×10−6

x=0.5 0.148479 0.00545553 0.00136032 0.000339859 0.000217493 0.0000543675

x=0.9 0.969982 0.0354973 0.00885019 0.00221104 0.00141495 0.000353699

x=1.0 1.46137 0.053398 0.0133126 0.00332584 0.00212836 0.000532032

x=1.5 9.3538 0.338802 0.0844444 0.0210951 0.0134997 0.00337451

x=2.0 48.9145 1.75487 0.437267 0.109226 0.069898 0.0174722

x=2.5 226.689 8.0529 2.00598 0.501045 0.320634 0.080147

un = n2 → at n=1000

x=0.1 6.15594×10−7

x=0.5 0.0000135915

x=0.9 0.0000884224

x=1.0 0.000133004

x=1.5 0.000843601

x=2.0 0.0043679

x=2.5 0.020036



REFERENCES 1322

Example 5. At the same time for the same function g(x) = x2e2x, 0 ≤ x ≤ 2.5, we can
observe by the given Figure 6 that the accuracy of the convergence for the operators (3)
is better when un = n2 is taken rather than when we choose the sequences un = n and
un = n

3
2 for the same operators (3).

Figure 6: The convergence of the operators S∗n(g;x) to the function g(x).

Remark: After observing by all the Figures (1)-(6) and Tables (1)-(3), we can conclude
that the better approximation can be obtained by choosing the appropriate sequence for
the operators (3) and in addition, will get good approximation by the operators (3) for
the large value of n of the positive and real sequence.

Conclusion: The approximation properties have been determined for the functions
belonging to different spaces and moreover the rate of the convergence of the operators
has been discussed. To validate the approximation results, the graphical representation
and numerical analysis have been studied.

Acknowledgements

The authors would like to express their deep gratitude to the anonymous learned
referees for their keen reading, valuable suggestions, and constructive comments, which
resulted in the subsequent improvement of this research article. All the authors read
and approved the final version of the manuscript. The authors declare that there are no
conflicts of interest to this work.

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