EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 4, 2019, 1508-1523
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Simultaneous Approximation of New Sequence of
Integral Type Operators with Parameter δ0

Ali Jassim Mohammad1, Hadeel Omar Muslim2,∗

1 Department of Mathematics, Faculty of Education for Pure Sciences, University of Basra,
Basra, Iraq.
2 Department of Thermal Mechanical Engineering, Faculty of Engineering Technology,
Southern Technical University, Basra, Iraq.

Abstract. In this paper, we define a new sequence of linear positive operators of integral type
Wn(f ;x) to approximate functions in the space Cα[0,∞), α > 0. First, we study the basic con-
vergence theorem in simultaneous approximation and then study Voronovskaja-type asymptotic
formula. Then, we estimate an error occurs by this approximation in the terms of the modulus
of continuity. Next, we give numerical examples to approximate two test functions in the space
Cα[0,∞) by the sequence Wn(f ;x). Finally, we compare the results with the classical sequence of
Szãsz operators Sn(f ;x) on the interval [a, b]. It turns out that, the sequence Wn(f ;x) gives better
results than the results of the sequence Sn(f ;x) for the two test functions using in the numerical
examples.

2010 Mathematics Subject Classifications: 41A10, 41A25,41A36

Key Words and Phrases: Linear positive operators, Simultaneous approximation, Voronovskaja-
type asymptotic formula, Modulus of continuity

1. Introduction

Bernstein in 1912, using a sequence known by his name, Bernstein sequence, which is
defined as:[1]

Bn(f ;x) =
n∑
k=0

bn,k(x)f

(
k

n

)
(1.1)

where, bn,k (x) =

(
n

k

)
xk(1− x)n−k and f ∈ C[0, 1].

Next, Voronovskaja in 1932 shown that the order of approximation is O(n−1). Also,
she showed that this order of approximation by Bernstein sequence cannot be improved
beyond O(n−1).[18]. Many papers interested in the classical sequences of Bernstein and
gave some modifications of them [5], [11]. In addition, the numerical application for this

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3547

http://www.ejpam.com 1508 c© 2019 EJPAM All rights reserved.



A. J. Mohammad, H. O. Muslim / Eur. J. Pure Appl. Math, 12 (4) (2019), 1508-1523 1509

sequence are very limited [3], [13].

Szãsz in 1950, generalized the Bernstein sequence to approximate the space of contin-
uous functions on the interval [0,∞) as [16]

Sn(f ;x) =
∞∑
k=0

qn,k(x)f

(
k

n

)
(1.2)

where qn,k(x) =
(nx)k

k!enx
, x ∈ [0,∞).

Several new modifications of Szãsz sequence were constructed and studied, here we refer to
[4, 14, 17, 20]. Also,many authors have discussed the approximation behavior of different
summation-integral type operators (see [6, 7, 10, 12] )

The sequence of integral type operators obviously appeared in the proof of Weierstrass
theorem (the fundamental theorem in approximation theory), these sequences variety via
the effort of some of the researchers who re-proved the Weierstrass theorem by using dif-
ferent sequences of integral type [8, 9, 19].

In the equation (1.1) Bernstein used the finite discrete sequence of a linear positive
operator to give another proof of Weierstrass theorem. The Bernstein sequence gives a
better result than the previous sequences in applications because it is simplest, finite and
discrete sequence [3], [13]. We believe that the same case occurs when we replaced Bern-
stein by Szãsz sequence so, we will use the classical Szãsz sequence to compare with the
numerical results of our sequence.

For x ≥ 0 is arbitrary but fixed, we define that
Cα[0,∞) =

{
f ∈ C[0,∞) : |f(t)| = O(eαt), for some α > 0

}
and the norm ‖f‖Cα [0,∞) =

supt∈[0,∞) |f(t)|e−αt.

For f ∈ Cα[0,∞), we define and study the following sequence of linear positive opera-
tors:

Wn(f ;x) =

∫ x

0
Kn(t;x)f(t)dt (1.3)

Kn(t;x) is the Kernel of Wn(f ;x) which is define as:

Kn(t;x) =
ncosh(nt)

(δ0 + sinh(nx))
, x ∈ [0,∞), arbitrary but fixed, n ∈ N := 1, 2, 3, ... and δ0

positive parameter.
Firstly, we introduce some preliminary results for Wn(f ;x). Then, we study point-

wise convergence in simultaneous approximation and give a Voronovskaja-type asymp-
totic formula for the sequence Wn(f ;x). After that, we proceed to estimate an error
occurring by the approximation by this sequence in terms of the modulus of continuity.
Finally, we give numerical examples for our sequence to approximate two test functions
g1(t) = sin(10t)e−2t, g2(t) =

√
1− (t− 1)2 and evaluate the maximum errors occurring by



A. J. Mohammad, H. O. Muslim / Eur. J. Pure Appl. Math, 12 (4) (2019), 1508-1523 1510

this approximation, also, compare the results with the sequence of classical Szãsz operators
Sn(gi(t);x), i = 1, 2 on the intrval [a, b].

2. Preliminary Results

In this section, we give some preliminary results for the operators Wn(f ;x) which we
need in our study.

Lemma 2.1.
For x ∈ [0,∞) the following conditions hold:

(i) Wn(1;x) =
sinh(nx)

(δ0 + sinh(nx))
→ 1 as n→∞;

(ii) Wn(t;x) =
xsinh(nx)

(δ0 + sinh(nx))
− cosh(nx)

n (δ0 + sinh(nx))
+

1

n (δ0 + sinh(nx))
→ x as n→∞;

(iii) Wn(t2;x) =
x2sinh(nx)

(δ0 + sinh(nx))
− 2xcosh(nx)

n (δ0 + sinh(nx))
+

2sinh(nx)

n2 (δ0 + sinh(nx))
→ x2 as

n→∞.

Proof. By the direct computation, the proof of this lemma follows immediate.
In addition, from the above lemma and the Korovkin theorem [9], we have that:

lim
n→∞

Wn(f(t);x) = f(x). (2.1)

Further, if f is exists and is continuous on (a − η, b + η) ⊂ (0,∞), η > 0, the limit (2.1)
holds uniformly on [a, b].
Our next definition is the m− th order moment for Wn(f ;x).

Definition 2.1. For m ∈ N0 the m−th order moment Tn,m(x) for the operator Wn(f(t);x)
is define as:

Tn,m(x) = Wn((t− x)m;x) = (−1)mWn((x− t)m;x) =
n(−1)m

(δ0 + sinh(nx))

∫ x

0
cosh(nt)(x−t)mdt

(2.2)

the recurrence relations for Tn,m(x) are given in the next lamma.

Lemma 2.2.
For the function Tn,m(x), we have:

(i) Tn,0(x) =
sinh(nx)

(δ0 + sinh(nx))
;

(ii) Tn,1(x) =
1

n (δ0 + sinh(nx))
− cosh(nx)

n (δ0 + sinh(nx))
;



A. J. Mohammad, H. O. Muslim / Eur. J. Pure Appl. Math, 12 (4) (2019), 1508-1523 1511

(iii) Tn,2(x) =
2sinh(nx)

n2 (δ0 + sinh(nx))
− 2x

n (δ0 + sinh(nx))
.

Then, we have the following recurrence relation:

Tn,m(x) =
m(m− 1)

n2
Tn,m−2(x)− m(−1)m

n (δ0 + sinh(nx))
xm−1,m > 2. (2.3)

Further, we have:

(1) Tn,m(x) approximate a polynomial in x of degree < m, whenever n is sufficiently
large.

(2) for every x ∈ [0,∞), Tn,m(x) = O (n−m).

Proof. By direct computation and using lemma (2.1), we have (i),(ii),(iii). Now, we
prove (2.3), for x ∈ [0,∞), and all m > 2, we have:

Tn,m(x) =
n(−1)m

(δ0 + sinh(nx))

∫ x

0
cosh(nt)(x− t)mdt

=
n(−1)m

(δ0 + sinh(nx))

[
m

n

∫ x

0
sinh(nt)(x− t)m−1dt

]
=

n(−1)m

(δ0 + sinh(nx))

[
−m
n2
xm−1 +

m(m− 1)

n2

∫ x

0
cosh(nt)(x− t)m−2dt

]
=
m(m− 1)

n2
Tn,m−2(x)− m(−1)m

n (δ0 + sinh(nx))
xm−1.

Therefore,(2.3) satisfied.
The consequence (1) can be proved easily by using (2.3) and the induction on m, so the
details are omitted.

Lemma 2.3. For m > 1, we have:

Wn(tm;x) =

(
sinh(nx)

(δ0 + sinh(nx))

)
xm −

(
mcosh(nx)

n (δ0 + sinh(nx))

)
xm−1 +O

(
n−2

)
.

Clearly, we have limn→∞Wn(tm;x) = xm.
Proof.

Wn(tm;x) =
n

(δ0 + sinh(nx))

∫ x

0
cosh(nt)tmdt

=
n

(δ0 + sinh(nx))

[(
1

n
sinh(nx)

)
xm − m

n

∫ x

0
sinh(nt)tm−1dt

]
=

(
sinh(nx)

(δ0 + sinh(nx))

)
xm −

(
mcosh(nx)

n (δ0 + sinh(nx))

)
xm−1



A. J. Mohammad, H. O. Muslim / Eur. J. Pure Appl. Math, 12 (4) (2019), 1508-1523 1512

+

(
m(m− 1)

n (δ0 + sinh(nx))

)∫ x

0
cosh(nt)tm−2dt

=

(
sinh(nx)

(δ0 + sinh(nx))

)
xm −

(
mcosh(nx)

n (δ0 + sinh(nx))

)
xm−1 +O

(
n−2

)
.

Lemma 2.4. Let δ and α be any two positive real numbers and [a, b] ⊂ (0,∞). Then for
λ > 0, we have:

sup
x∈[a,b]

∣∣∣∣∫
x−t>δ

Kn(t;x)eαtdt

∣∣∣∣ = O
(
n−λ

)
.

Making use of Taylor’s expansion, Schwartz inequality and lemma 2.2(2), the proof of this
lemma easily follows.

Lemma 2.5. [15]

(1) Let r be a nonnegative integer and assume f and g are r-times differentiable functions
of x then:
dr

dxr
(fg) =

∑r
l=0

(
r
l

) dr−l
dxr−l

(f)
dl

dxl
(g);

(2) Let g(x) be a real or complex valued function that is r-times differentiable then:
dr

dxr

(
1

g(x)

)
=
∑r

l=0(−1)l
(
r+1
l+1

) 1

(g(x))l+1
(f)

dr

dxr
(g(x))l.

Lemma 2.6. For r ∈ N, we have:

(i) limn→∞
sinh(nx)

(δ0 + sinh(nx))
= 1;

(ii) limn→∞
cosh(nx)

(δ0 + sinh(nx))
= 1;

(iii) limn→∞
dr

dxr

(
sinh(nx)

(δ0 + sinh(nx))

)
= 0;

(iv) limn→∞
dr

dxr

(
cosh(nx)

(δ0 + sinh(nx))

)
= 0;

(v) limn→∞
dr

dxr

(
1

(δ0 + sinh(nx))

)
= 0;

(vi) limn→∞
dr

dxr

(
xsinh(nx)

(δ0 + sinh(nx))

)
= 0, r > 1.



A. J. Mohammad, H. O. Muslim / Eur. J. Pure Appl. Math, 12 (4) (2019), 1508-1523 1513

Lemma 2.7. [2]
For the function F (x) given by:

F (x) =

∫ b(x)

a(x)
f(x, y)dy

Then the chain rule of differentiation of the function F (x) gives:

F
′
(x) = b

′
(x)f(x, b(x))− a′(x)f(x, a(x)) +

∫ b(x)

a(x)

∂

∂x
f(x, y)dy

3. The main results

First,we prove that:

Wn
(r)(f(t);x)→ f (r)(x), as n→∞, r ∈ N.

Theorem 3.1. Suppose that r ∈ N, f ∈ Cα[0,∞) and f (r+1)(x) exists at a point x ∈
(0,∞), then:

lim
n→∞

Wn
(r)(f(t);x)→ f (r)(x). (3.1)

Further, if f (r+1)(x) exists and is continuous on (a − η, b + η) ⊂ (0,∞), η > 0, the limit
(3.1) holds uniformly on [a, b].

Proof. By using Taylor’s expansion of f, when ξ lies between t and x, we get:

f(t) =
∑r

i=0

f (i)(x)

i!
(t− x)i +

f (r+1)(ξ)

(r + 1)!
(t− x)r+1,

Operating by the sequence W
(r)
n , we get:

Wn
(r)(f(t);x) =

r∑
i=0

f (i)(x)

i!
(−1)iWn

(r)((x− t)i;x) +
f (r+1)(ξ)

(r + 1)!
(−1)r+1Wn

(r)((x− t)r+1;x)

:= Σ1 + Σ2

Σ1 :=

r∑
i=0

f (i)(x)

i!
(−1)iWn

(r)((x− t)i;x)

=

r∑
i=0

f (i)(x)

i!
(−1)i

i∑
j=0

(
i

j

)
x(i−j)(−1)jWn

(r)(tj ;x)

=
f (r)(x)

r!
Wn

(r)(tr;x)

When j < r then Wn
(r)(tj ;x) −→ 0 as n −→∞.



A. J. Mohammad, H. O. Muslim / Eur. J. Pure Appl. Math, 12 (4) (2019), 1508-1523 1514

Using lemma 2.3, lemma 2.5, and lemma 2.6 we get:

Σ1 =
f (r)(x)

r!
{r!
(

sinh(nx)

(δ0 + sinh(nx))

)
+ rr!x

ncosh(nx)

(δ0 + sinh(nx))

(
1− sinh(nx)

(δ0 + sinh(nx))

)
+ ...+ xr

dr

dxr

(
sinh(nx)

(δ0 + sinh(nx))

)
− rr!

(
sinh(nx)

(δ0 + sinh(nx))
− cosh2(nx)

(δ0 + sinh(nx))2

)
− ...− r

n
xr−1

dr

dxr

(
cosh(nx)

(δ0 + sinh(nx))

)
} −→ f (r)(x) as n −→∞.

Σ2 =
f (r+1)(ξ)

(r + 1)!
(−1)r+1W (r)

n

(
(x− t)r+1 ;x

)
=
f (r+1)(ξ)

(r + 1)!
(−1)r+1 d

r

dxr

{
n

(δ0 + sinh(nx))

∫ x

0
cosh(nt) (x− t)r+1 dt

}
.

Using lemma 2.5, lemma 2.6 and lemma 2.7, we obtain:

Σ2 =
f (r+1)(ξ)

(r + 1)!
(−1)r+1

r∑
l=0

(
r

l

)
dr−1

dxr−1

(
n

(δ0 + sinh(nx))

)
dl

dxl

(∫ x

0
cosh(nt) (x− t)r+1 dt

)

=
f (r+1)(ξ)

(r + 1)!
(−1)r+1 d

r

dxr

(
n

(δ0 + sinh(nx))

)∫ x

0
cosh(nt) (x− t)r+1 dt

+
f (r+1)(ξ)

(r + 1)!
(−1)r+1

r∑
l=1

(
r

l

)
dr−1

dxr−1

(
n

(δ0 + sinh(nx))

)
dl

dxl

(∫ x

0
cosh(nt) (x− t)r+1 dt

)
= I1 + I2.

I1 ≤

∣∣∣∣∣f (r+1)(ξ)

(r + 1)!

∣∣∣∣∣
r∑
l=0

(
r + 1

l + 1

) ∣∣∣∣∣ n

(δ0 + sinh(nx))l+1

dr

dxr
(δ0 + sinh(nx))l

∣∣∣∣∣
∫ x

0
cosh(nt)

∣∣∣(−(x− t))r+1
∣∣∣ dt.

By Schwartz inequality, we have:

|I1| ≤

∣∣∣∣∣f (r+1)(ξ)

(r + 1)!

∣∣∣∣∣
r∑
l=0

(
r + 1

l + 1

) ∣∣∣∣∣ 1

(δ0 + sinh(nx))l
dr

dxr
(δ0 + sinh(nx))l

∣∣∣∣∣
×
(

n

(δ0 + sinh(nx))

∫ x

0
cosh(nx)dt

)1/2( n

(δ0 + sinh(nx))

∫ x

0
cosh(nt) (−(x− t))2(r+1) dt

)1/2

= M1n
rO(1)O(n−(r+1))

= M1o(1).

Now, using lemma 2.5 and lemma 2.7, we obtain:

|I2| ≤

∣∣∣∣∣f (r+1)(ξ)

(r + 1)!

∣∣∣∣∣
r∑
l=1

(
r

l

) r−1∑
l1=0

(
r − l + 1

l1 + 1

) ∣∣∣∣∣ n

(δ0 + sinh(nx))l1+1

dr−1

dxr−1
(δ0 + sinh(nx))l1

∣∣∣∣∣



A. J. Mohammad, H. O. Muslim / Eur. J. Pure Appl. Math, 12 (4) (2019), 1508-1523 1515

×
∫ x

0
cosh(nt)

∣∣∣∣ dldxl (−(x− t))r+1

∣∣∣∣ dt
= M2{[

r2(r − 1)

2

n

(δ0 + sinh(nx))2
dr−1

dxr−1
(δ0 + sinh(nx))

+ ...+ r
n

(δ0 + sinh(nx))r
dr−1

dxr−1
(δ0 + sinh(nx))r−1]

×
(∫ x

0
cosh(nt) |−(r + 1) (−(x− t))r| dt

)
+ [

r(r − 1)2(r − 2)

4

n

(δ0 + sinh(nx))2
dr−2

dxr−2
(δ0 + sinh(nx))

+ ...+
r(r − 1)

2

n

(δ0 + sinh(nx))r
dr−2

dxr−2
(δ0 + sinh(nx))r−1]

×
(∫ x

0
cosh(nt)

∣∣∣r(r + 1) (−(x− t))r−1
∣∣∣ dt)

+
n

(δ0 + sinh(nx))

(∫ x

0
cosh(nt) |(−1)r(r + 1)! (−(x− t))| dt

)
}.

By Schwarz inequality, we have:
I2 = M2O(n−1)
Hence,
I2 = o(1).
Now, it follows I1 → 0 as n → ∞. also I2 → 0 as n → ∞. Hence Σ2 = o(1). The
uniformity assertion follows easily from the fact that δ(ε) in the above proof can be chosen
to be independent of x ∈ [a, b] and all the other estimates hold uniformly on [a, b].

Our next results is a Voronovskaja-type asymptotic formula for the operatorsWn
(r)(f(t);x), r ∈

N.

Theorem 3.2.
Suppose that r ∈ N, f ∈ Cα[0,∞) and f (r+3)(x) exists at a point x ∈ (0,∞), then:

lim
n→∞

n
(
Wn

(r)(f(t);x)− f (r)(x)
)

= −f (r+1)(x) (3.2)

Further, if f (r+3)(x) exists and is continuous on (a − η, b + η) ⊂ (0,∞), η > 0, the limit
(3.2) holds uniformly on [a, b].

Proof. By using Taylor’s expansion of f, when ξ lies between t and x, we get:

f(t) =
r+2∑
i=0

f (i)(x)

i!
(t− x)i +

f (r+)(ξ)

(r + 3)!
(t− x)r+3

Operating by the sequence W
(r)
n , we get:



A. J. Mohammad, H. O. Muslim / Eur. J. Pure Appl. Math, 12 (4) (2019), 1508-1523 1516

Wn
(r)(f(t);x) =

r+2∑
i=0

f (i)(x)

i!
(−1)iWn

(r)((x− t)i;x) +
f (r+3)(ξ)

(r + 3)!
(−1)r+3Wn

(r)((x− t)r+3;x)

:= Σ1 + Σ2

Using the same technique of theorem 3.1, we get:
Σ2 → 0 as n→ ∞.

When i < r then from lemma 2.3, we have
Wn

(r)((t− x)i;x) −→ 0 as n −→∞.
Using lemma 2.3, lemma 2.5, and lemma 2.6, we get:

Σ1 =
f (r)(x)

r!
{r!
(

sinh(nx)

(δ0 + sinh(nx))

)
+ rr!x

ncosh(nx)

(δ0 + sinh(nx))

(
1− sinh(nx)

(δ0 + sinh(nx))

)
+ ...+ xr

dr

dxr

(
sinh(nx)

(δ0 + sinh(nx))

)
− rr!

(
sinh(nx)

(δ0 + sinh(nx))
− cosh2(nx)

(δ0 + sinh(nx))2

)
− ...− r

n
xr−1

dr

dxr

(
cosh(nx)

(δ0 + sinh(nx))

)
} − f r(x)

+
f (r+1)(x)

(r + 1)!
{−r(r + 1)!

2
x2

ncosh(nx)

(δ0 + sinh(nx))

(
1− sinh(nx)

(δ0 + sinh(nx))

)
− ...− rxr+1 d

r

dxr

(
sinh(nx)

(δ0 + sinh(nx))

)
− (r + 1)!

n

cosh(nx)

(δ0 + sinh(nx))

+ ..+
(r + 1)(r − 1)

n
xr

dr

dxr

(
cosh(nx)

(δ0 + sinh(nx))

)
}

+
f (r+2)(x)

(r + 2)!
{r(r + 2)!

6
x3

ncosh(nx)

(δ0 + sinh(nx))

(
1− sinh(nx)

(δ0 + sinh(nx))

)
+ ...+

r(r + 1)

2
xr+2 d

r

dxr

(
sinh(nx)

(δ0 + sinh(nx))

)
− ...− r(r − 1)(r + 2)

2n
xr+1

× dr

dxr

(
cosh(nx)

(δ0 + sinh(nx))

)
}.

Therefore,
limn→∞ n

(
Wn

(r)(f(t);x)− f (r)(x)
)

= −f (r+1)(x)
The uniformity assertion follows easily from the fact that δ(ε) in the above proof can be
chosen to be independent of x ∈ [a, b] and all the other estimates hold uniformly on [a, b].

Finally, we give an estimate of the degree of approximation by Wn
(r)(f ;x).

Theorem 3.3. Let f ∈ Cα[0,∞), α > 0 for some h > 0 and r 6 q 6 r+ 2. If f (q+1) exists
and is continuous on (a− η, b+ η) ⊂ (0,∞), η > 0, then for sufficiently large n,



A. J. Mohammad, H. O. Muslim / Eur. J. Pure Appl. Math, 12 (4) (2019), 1508-1523 1517

∥∥∥Wn
(r)(f(t);x)− f (r)(x)

∥∥∥
C[a,b]

6 C1n
−1

q∑
i=r

∥∥∥f (i)∥∥∥
C[a,b]

+ C2n
−1/2ωf (q+1)

×
(
n−1/2; (a− η, b+ η)

)
+O(n−2),

where, C1, C2 are constants independent of f and n.
Proof. By our hypothesis

f(t) =

q∑
i=0

f (i)(x)

i!
(t− x)i +

f (q+1)(ξ)− f (q+1)(x)

(q + 1)!
(t− x)q+1χ(t) + h(t, x)(1− χ(t)),

where, ξ lies between t, x and χ(t) is the characteristic function of the interval (a−η, b+η).
For t ∈ (a− η, b+ η) and x ∈ (0,∞) we get:

f(t) =

q∑
i=0

f (i)(x)

i!
(t− x)i +

f (q+1)(ξ)− f (q+1)(x)

(q + 1)!
(t− x)q+1.

For t ∈ [0,∞)\(a− η, b+ η) and x ∈ [a, b], we define

h(t, x) = f(t)−
q∑
i=0

f (i)(x)

i!
(t− x)i.

∂r

∂xr
h(t, x) =

q∑
i=0

f (i)(x)

i!
(t− x)i

Now,

Wn
(r)(f(t);x)− f (r)(x) =

[
q∑
i=0

f (i)(x)

i!
(−1)iWn

(r)((x− t)i;x)− f (r)(x)

]

+ (−1)q+1Wn
(r)

(
f (q+1)(ξ)− f (q+1)(x)

(q + 1)!
(x− t)q+1χ(t);x)

)
+Wn

(r) (h(t, x)(1− χ(t));x))

= Σ1 + Σ2 + Σ3.

Using lemma 2.3, we get:

Σ1 =

q∑
i=0

f (i)(x)

i!
(−1)i

i∑
j=0

(
i

j

)
xi−j(−1)jWn

(r)(tj ;x)− f (r)(x)

=

q∑
i=r

f (i)(x)

i!
(−1)i

i∑
j=0

(
i

j

)
xi−j(−1)j

dr

dxr
[

(
sinh(nx)

(δ0 + sinh(nx))

)
xj



A. J. Mohammad, H. O. Muslim / Eur. J. Pure Appl. Math, 12 (4) (2019), 1508-1523 1518

− j

n

(
cosh(nx)

(δ0 + sinh(nx))

)
xj−1 +O(n−2)] − f r(x)

Consequently,

‖Σ1‖C[a,b] 6 C1n
−1
[∑q

i=r

∥∥f (i)∥∥
C[a,b]

]
+O(n−2), uniformly on [a, b].

To estimate Σ2 we proceed as follows:

|Σ2| 6Wn
(r)

(
|f (q+1)(ξ)− f (q+1)(x)|

(q + 1)!
| − (x− t)|q+1χ(t);x

)

6
ωf (q+1) (δ; (a− η, b+ η))

(q + 1)!
Wn

(r)

(
δ + | − (x− t)|

δ
| − (x− t)|q+1χ(t);x

)
6
ωf (q+1) (δ; (a− η, b+ η))

(q + 1)!

dr

dxr
[

n

(δ0 + sinh(nx))

∫ x

0
cosh(nt)(| − (x− t)|q+1

+ δ−1| − (x− t)|q+2)dt] δ > 0.

Now, for k = 0, 1, 2, .... and using lemma 2.5, lemma 2.7,, we have:∣∣∣∣ drdxr
[

n

(δ0 + sinh(nx))

∫ x

0
cosh(nt) (−(x− t))k+1 dt

]∣∣∣∣ (3.3)

=

r∑
l=0

(
r

l

) ∣∣∣∣ dr−ldxr−l

(
n

δ0 + sinh(nx)

)∣∣∣∣ ∣∣∣∣ dldxl
(∫ x

0
cosh(nx) (−(x− t))k+1 dt

)∣∣∣∣
=

∣∣∣∣ drdxr
(

n

δ0 + sinh(nx)

)∣∣∣∣ ∫ x

0
cosh(nt) |−(x− t)|k+1 dt

+

r∑
l=1

(
r

l

) ∣∣∣∣ dr−ldxr−l

(
n

δ0 + sinh(nx)

)∣∣∣∣ ∣∣∣∣ dldxl
(∫ x

0
cosh(nx) |−(x− t)|k+1 dt

)∣∣∣∣
:= J1 + J2

Using the same technique in I1, I2(theorem 3.1 ), we get:
J1 = O(nr−(k+1)), uniformly on [a, b]
and J2 = O(n−1). Choosing δ = n−1/2 and applying 3.3, we are led to:

‖Σ2‖C[a,b] 6
ωf (q+1)

(
n−1/2; (a− η, b+ η)

)
(q + 1)!

[
O
(
n(r−(q+1))

)
+ n1/2O

(
n(r−(q+2))

)
+O(n−1)

]
6 C2n

−(r−(q+1))ωf (q+1)

(
n−1/2; (a− η, b+ η)

)
.

Since t ∈ [0,∞)\(a − η, b + η), we can choose δ > 0 in such a way that x − t > δ for all
x ∈ [a, b]. Thus,

|Σ3| 6
∣∣∣∣ drdxr

[
n

(δ0 + sinh(nx))

∫ x

0
cosh(nt) (h(t, x)(1− χ(t))) dt

]∣∣∣∣



A. J. Mohammad, H. O. Muslim / Eur. J. Pure Appl. Math, 12 (4) (2019), 1508-1523 1519

6

∣∣∣∣ drdxr
[

n

(δ0 + sinh(nx))

∫
x−t≥δ

cosh(nt)h(t, x)dt

]∣∣∣∣
=

∣∣∣∣ drdxr
(

n

(δ0 + sinh(nx))

)∣∣∣∣ ∫
x−t≥δ

cosh(nt)h(t, x)dt

+
r∑
l=1

(
r

l

) ∣∣∣∣ dr−ldxr−l

(
n

(δ0 + sinh(nx))

)∣∣∣∣ ∣∣∣∣ dldxl
(∫

x−t≥δ
cosh(nt)h(t, x)dt

)∣∣∣∣
= J3 + J4.

For x− t > δ, we can find a constant C > 0 such that |h(t, x)| 6 Ceαt and

∣∣∣∣ ∂r∂xr h(t, x)

∣∣∣∣ 6
Ceαt

Using lemma 2.4, we get |Σ3| = O(n−λ), λ > 0 uniformly on [a, b].
Combining the estimates of Σ1,Σ2,Σ3 the required results are immediate.

4. Numerical Examples

In this section, we give some numerical examples for the sequence of linear posi-
tive operators Wn(.;x), by using two test functions g1(t) = sin(10t)e−2t and g2(t) =√

1− (t− 1)2 , t ∈ [0, 2]. we calculate the maximum error and compare the results of the
sequence Wn(.;x), with the results of classical Szãsz sequence Sn(.;x) in the interval [0, 2]
and we describe the results by figures (1 − 24) for some n = 30, 60, 100 and the positive
parameter δ0 = 0.01, 0.1, 1. respectvely.

Definition 4.1. Given a sequence Mn, here (Mn = WnorSn), and let f be a function,
the error function E(x) occurring by approximate the function f by the sequence Mn is
defined as E(x) = |Mn(f ;x) − f(x)|. Also, the maximum error of the function E(x) is
denote and define as MaxE = maxx∈[0,2] |E(x)|.

Example 4.1. For n = 30, 60, 100 and δ0 = 0.01, 0.1, 1. respectively the sequences Wn(g1;x)
and Sn(g1;x) converge to the test function g1(x) = sin(10x)e−2x, with maximum error(MaxE)
given in the following figures (1− 12)

(a) MaxE := 0.1844077862 (b) MaxE := 0.1201880456 (c) MaxE := 0.0806800064



A. J. Mohammad, H. O. Muslim / Eur. J. Pure Appl. Math, 12 (4) (2019), 1508-1523 1520

(a) MaxE := 0.192609996 (b) MaxE := 0.1229776897 (c) MaxE := 0.08243795854

(a) MaxE := 0.2624406460 (b) MaxE := 0.1553141469 (c) MaxE := 0.0984378898

(a) MaxE := 0.2259276806 (b) MaxE := 0.1323992412 (c) MaxE := 0.085016286

Example 4.2. For n = 30, 60, 100 and δ0 = 0.01, 0.1, 1. respectively the sequences Wn(g2;x)
and Sn(g2;x) converge to the test function g2(t) =

√
1− (t− 1)2, with maximum error

(MaxE) given in the following figures (13− 24)

(a) MaxE := 0.1319652402 (b) MaxE := 0.0791847229 (c) MaxE := 0.0528549386



REFERENCES 1521

(a) MaxE := 0.1319652402 (b) MaxE := 0.0791847229 (c) MaxE := 0.0528549386

(a) MaxE := 0.1605861444 (b) MaxE := 0.1142832076 (c) MaxE := 0.08801539766

(a) MaxE := 0.2753815754 (b) MaxE := 0.2368816505 (c) MaxE := 0.2109249528

5. Conclusions

In this section, we gave some numerical examples for our sequences Wn(.;x), in cases
n = 30, 60, 100 and the positive parameter δ0 = 0.01, 0.1, 1. to approximate two test
functions g1(t) = sin(10t)e−2t and g2(t) =

√
1− (t− 1)2 , in the space Cα[0,∞) and

compared the results of the sequence Wn(.;x), with the results of classical Szãsz sequence
Sn(.;x) in the interval [0, 2]. it turns out that: If i = 1, the sequence Wn(gi(t);x) gives
better results than the result of the Szãsz sequence Sn(gi;x) for all value of n and δ0 =
0.01, 0.1, except δ0 = 1 the sequence of Szãsz sequence give little better results than the
sequence Wn(.;x). When i = 2, the sequence Wn(gi(t);x) gives better results than the
Szãsz sequence Sn(gi;x) for all value of n and δ0. Hence, we recommend to use the sequence
Wn(.;x) instead of the sequence Sn(.;x) in the application.

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