On the approximation of Kantorovich-type Szàsz-Charlier operators

Ümit Karabıyık; Adem Ayık

doi:10.1515/dema-2025-0144

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On the approximation of Kantorovich-type Szàsz-Charlier operators

Ümit Karabıyık and Adem Ayık

Published/Copyright: July 31, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

In this study, we introduce the Kantorovich-type modified Szàsz-Charlier operators and examine their approximation properties within the framework of fractional modeling and control theory. These operators are defined using the Korovkin-type theorem, and their local approximation properties are analyzed in detail. Additionally, the approximation rate of these operators is estimated using the modulus of continuity and functions in the Lipschitz class. To gain a deeper understanding of their approximation capabilities, we calculate their central moments. Furthermore, the convergence rates are determined based on the modulus of continuity, and a Voronovskaja-type asymptotic formula is provided for these operators. Our study also investigates the approximation speed in weighted spaces, which is crucial for addressing complex dynamical systems in science and engineering. The approximation speed in weighted spaces is evaluated using the weighted modulus of continuity and the Peetre-K function. Finally, illustrative graphics generated using Maple are presented to visualize the convergence of operators to specific functions.

Keywords: Kantorovich-type Szàsz-Charlier operators; the weighted modulus of continuity; local approximation; weighted space; Voronovskaja-type asymptotic formula

MSC 2010: 41A25; 41A36

1 Introduction

Approximation theory plays a fundamental role not only in mathematical analysis but also in various scientific disciplines that depend on mathematics. Its applications are broad, covering fields such as physics, computer-aided geometric design, and engineering, where it is particularly important for modeling and system control. The primary objective of approximation theory is to represent functions using simpler, more practical functions that facilitate a clearer, more efficient understanding of their behavior. In control theory and fractional modeling, these methods are essential for analyzing and optimizing complex dynamic systems.

Szàsz [1] defined the following operator. Such that these operators generalized Bernstein operators to the infinite interval

S n ( f ; x ) = e − n x ∑ k = 0 ∞ ( n x ) k k ! f k n , x ∈ [ 0 , ∞ ) .

Several researchers have extensively investigated this operator and its generalizations [2–9].

Recently, Varma and Taşdelen [10] introduced and studied the Szàsz-type operators and their Kantorovich variant which involves Charlier polynomials as follows:

(1) S n ( f ; x , a ) = e − 1 1 − 1 a ( a − 1 ) n x ∑ k = 0 ∞ C k ( a ) ( − ( a − 1 ) n x ) k ! f k n ; a > 1 ,

where C k ( a ) ( u ) are Charlier polynomials [11] that generate functions of the form

(2) e t 1 − t a u = ∑ k = 0 ∞ C k ( a ) ( u ) t k k ! , ∣ t ∣ < a ,

where C k ( a ) ( u ) = ∑ r = 0 k k r ( − u ) r 1 a r and ( m ) 0 = 1 , ( m ) j = m ( m + 1 ) … ( m + j − 1 ) , for j ≥ 1 .

Some approximation properties of this operator and its modifications were investigated in [12,13].

Now, for the approximation integrable function, we define the Kantorovich variant of the modified Szàsz-Charlier operator defined as

(3) S m ∗ ( f ; x , a ) = γ m e − 1 1 − 1 a ( a − 1 ) β m x ∑ j = 0 ∞ C j ( a ) ( − ( a − 1 ) β m x ) j ! ∫ j γ m j + 1 γ m f ( s ) d s ; a > 1 ,

( γ m ) , ( β m ) are unbounded and increasing sequences such that γ m ≥ 1 , β m ≥ 1 and

(4) lim m → ∞ 1 γ m = 0 , β m γ m = 1 + O 1 γ m .

Specially, when taken a ⟶ ∞ and x − ( 1 ∕ m ) instead of x , the operators S m ∗ reduce to the modified Szàsz operators given by Walczak [14]. If we take β m = γ m = m , S m ∗ reduces to the operators defined by Varma and Taşdelen [10].

In this article, we study the Kantorovich-type modified Szàsz-Charlier operators defined in (3), and we give approximation properties of our operators in the weighted space of continuous functions. Also, we estimate the rate of convergence using the weighted modulus of continuity and present the Voronovskaja-type asymptotic formula for these operators. In addition, using Maple, we will present illustrative graphics showing the rate of convergence of our new operators to certain functions.

2 Approximation by Kantorovich-type-modified Szàsz-Charlier operators

First of all, in this section, we will examine the approximation properties of S m ∗ in a weighted space of continuous functions, which is the Kantorovich variant of the modified Szàsz-Charlier operator in the weighted space of continuous functions.

We consider the following class of functions that are defined in the interval [ 0 , ∞ ) .

Let B x 2 [ 0 , ∞ ) be the set of all functions f defined in [ 0 , ∞ ) satisfying the condition ∣ f ( x ) ∣ ≤ M f ( 1 + x 2 ) , where M f is a constant, depending on the function f . We will denote the subspace of all continuous functions of B x 2 [ 0 , ∞ ) by C x 2 [ 0 , ∞ ) . Provided that C x 2 ∗ [ 0 , ∞ ) is a subspace of all continuous functions for which lim ∣ x ∣ → ∞ f ( x ) 1 + x 2 is finite. The norm on C x 2 ∗ [ 0 , ∞ ) is defined as follows, as stated in [2],

‖ f ‖ x 2 = sup x ∈ [ 0 , ∞ ) ∣ f ( x ) ∣ 1 + x 2 .

Lemma 1

The following statements are valid

S m ∗ ( 1 ; x , a ) = 1 , S m ∗ ( e 1 ; x , a ) = β m x γ m + 3 2 γ m , S m ∗ ( e 2 ; x , a ) = β m 2 γ m 2 x 2 + β m γ m 2 4 + 1 a − 1 x + 10 3 γ m 2 , S m ∗ ( e 3 ; x , a ) = β m γ m 3 x 3 + β m 2 γ m 3 15 2 + 3 a − 1 x 2 + β m γ m 3 31 2 + 20 3 ( a − 1 ) + 2 ( a − 1 ) 2 x + 37 4 γ m 3 , S m ∗ ( e 4 ; x , a ) = β m γ m 4 x 4 + β m 3 γ m 4 12 + 6 a − 1 x 3 + β m 2 γ m 4 45 + 36 a − 1 + 11 ( a − 1 ) 2 x 2 + β m γ m 4 51 + 38 a − 1 + 22 ( a − 1 ) 2 + 6 ( a − 1 ) 3 x + 151 5 γ m 4 ,

where e i ( t ) = t i such that i = 0 , 1, 2, 3, 4.

Proof

We will now prove the validity of the moment identities stated in Lemma 1. Using the given definition in (2), we start by evaluating the zeroth moment:

∑ k = 0 ∞ C k ( a ) ( − ( a − 1 ) β m x ) k ! = e 1 − 1 a − ( a − 1 ) β m x .

This confirms S m ∗ ( 1 ; x , a ) = 1 .

Next, we compute the first moment by differentiating and applying the given formula:

∑ k = 0 ∞ k C k ( a ) ( − ( a − 1 ) β m x ) k ! = e 1 − 1 a − ( a − 1 ) β m x [ β m x + 1 ] .

Thus, we obtain

S m ∗ ( e 1 ; x , a ) = β m x γ m + 3 2 γ m .

Similarly, for the second moment, we proceed by considering the second derivative:

∑ k = 0 ∞ k 2 C k ( a ) ( − ( a − 1 ) β m x ) k ! = e 1 − 1 a − ( a − 1 ) β m x β m 2 x 2 + β m x 3 + 1 a − 1 .

From this, we derive

S m ∗ ( e 2 ; x , a ) = β m 2 γ m 2 x 2 + β m γ m 2 4 + 1 a − 1 x + 10 3 γ m 2 .

Continuing to the third moment, we use the corresponding expression

∑ k = 0 ∞ k 3 C k ( a ) ( − ( a − 1 ) β m x ) k ! = e 1 − 1 a − ( a − 1 ) β m x β m 3 x 3 + β m 2 x 2 6 + 3 a − 1 + 2 β m x 1 ( a − 1 ) 2 + 3 a − 1 + 5 + 5 .

By rearranging, we establish

S m ∗ ( e 3 ; x , a ) = β m γ m 3 x 3 + β m 2 γ m 3 15 2 + 3 a − 1 x 2 + β m γ m 3 31 2 + 20 3 ( a − 1 ) + 2 ( a − 1 ) 2 x + 37 4 γ m 3 .

Finally, to derive the fourth moment, we extend the established pattern.

∑ k = 0 ∞ k 4 C k ( a ) ( − ( a − 1 ) β m x ) k ! = e 1 − 1 a − ( a − 1 ) β m x β m 4 x 4 + β m 3 x 3 10 + 6 a − 1 + β m 2 x 2 31 + 30 a − 1 + 11 ( a − 1 ) 2 + β m x 6 ( a − 1 ) 2 + 20 ( a − 1 ) 2 + 31 a − 1 + 67 + 15 .

From this, we conclude that

S m ∗ ( e 4 ; x , a ) = β m γ m 4 x 4 + β m 3 γ m 4 12 + 6 a − 1 x 3 + β m 2 γ m 4 45 + 36 a − 1 + 11 ( a − 1 ) 2 x 2 + β m γ m 4 51 + 38 a − 1 + 22 ( a − 1 ) 2 + 6 ( a − 1 ) 3 x + 151 5 γ m 4 .

Thus, all moment identities stated in Lemma 1 have been verified, which completes the proof.□

From Lemma 1 and by the linearity of S m ∗ , we have

(5) S m ∗ ( e 1 − x ; x , a ) = β m γ m − 1 x + 3 2 γ m ,

(6) S m ∗ ( ( e 1 − x ) 2 ; x , a ) = β m γ m − 1 2 x 2 + β m γ m 2 ( 4 + 1 1 − a ) − 3 γ m x + 10 3 γ m 2 .

Lemma 2

If f ∈ C x 2 [ 0 , ∞ ) and in case ρ ( x ) = 1 + x 2 be a weight function, then

‖ S m ∗ ( ρ ; x , a ) ‖ x 2 ≤ 1 + M ,

where M is a positive constant.

Proof

By Lemma 1,

S n ∗ ( ρ ; x , a ) = 1 + β m 2 γ m 2 x 2 + β m γ m 2 4 + 1 a − 1 x + 10 3 γ m 2 .

Then, we obtain

‖ S m ∗ ( ρ ; x , a ) ‖ x 2 = sup x ≥ 0 1 1 + x 2 + β m 2 x 2 γ m 2 ( x 2 + 1 ) + β m x γ m 2 ( x 2 + 1 ) 4 + 1 a − 1 + 10 3 γ m 2 ( x 2 + 1 ) ≤ 1 + β m 2 γ m 2 + β m γ m 2 4 + 1 a − 1 + 10 3 γ m 2 .

By (4), there exists a positive constant M such that

‖ S m ∗ ( ρ ; x , a ) ‖ x 2 ≤ 1 + M .

With this step, the lemma is proved.□

Based on Lemma 2, one can see that the operator S m ∗ defined (3) act from C x 2 [ 0 , ∞ ) to B x 2 [ 0 , ∞ ) .

Theorem 1

Let S m ∗ be a sequence of linear positive operators defined (3), then for each f ∈ C x 2 ∗ [ 0 , ∞ )

lim m → ∞ ‖ S m ∗ ( f ; x , a ) − f ( x ) ‖ x 2 = 0 .

Proof

Applying the weighted Korovkin theorem given by Gadjiev [15,16] is sufficient to verify the conditions

lim m → ∞ ‖ S m ∗ ( e v ; x , a ) − e v ( x ) ‖ x 2 = 0 , v = 0 , 1 , 2 .

From Lemma 1, we obtain

(7) ‖ S m ∗ ( 1 ; x , a ) − 1 ‖ x 2 = 0

and

‖ S m ∗ ( e 1 ; x , a ) − e 1 ( x ) ‖ x 2 = sup x ≥ 0 β m γ m − 1 x x 2 + 1 + 3 2 γ m ( x 2 + 1 ) ≤ β m γ m − 1 + 3 2 γ m .

Considering (4), we obtain

(8) lim m → ∞ ‖ S m ∗ ( e 1 ; x ) − e 1 ( x ) ‖ x 2 = 0 .

Again, by applying Lemma 1, we obtain the following expression/result:

‖ S m ∗ ( e 2 ; x , a ) − e 2 ( x ) ‖ x 2 = sup x ≥ 0 β m 2 γ m 2 − 1 x 2 x 2 + 1 + x β m ( x 2 + 1 ) γ m 2 4 + 1 a − 1 + 10 3 ( x 2 + 1 ) γ m 2 ≤ β m 2 γ m 2 − 1 + β m γ m 2 4 + 1 a − 1 + 10 3 γ m 2 .

If the limit of both sides of the above inequality is taken with m ⟶ ∞ and by (4), we obtain

(9) lim m → ∞ ‖ S m ∗ ( e 2 ; x , a ) − e 2 ( x ) ‖ x 2 = 0 .

From (7), (8), and (9), for v = 0 , 1, 2, we have lim m → ∞ ‖ S m ∗ ( e v ; x , a ) − e v ( x ) ‖ x 2 = 0 .□

Thus, the proof is completed.

2.1 Rate of convergence

The modulus of continuity ω ( δ ) does not tend to zero in infinite intervals. Thus, we use Ω ( f , δ ) (see [17])

Ω ( f , δ ) = sup ∣ h ∣ < δ , x ∈ R + ∣ f ( x + h ) − f ( x ) ∣ ( 1 + h 2 ) ( 1 + x 2 ) for each f ∈ C x 2 [ 0 , ∞ ) .

We will give some basic properties of Ω ( f , δ ) in the following lemma, as stated in [17].

Lemma 3

Let f ∈ C x 2 [ 0 , ∞ ) . Then,

Ω ( f , δ ) is a monotonically increasing function of δ such that δ ≥ 0 .
For every f ∈ C x 2 ∗ [ 0 , ∞ ) , lim δ → 0 Ω ( f , δ ) = 0 .
For each λ > 0 ,
(10) Ω ( f , λ δ ) ≤ 2 ( 1 + λ ) ( 1 + δ 2 ) Ω ( f , δ ) .
From inequality (10) and definition of Ω ( f , δ ) , we obtain
(11) ∣ f ( t ) − f ( x ) ∣ ≤ 2 ( 1 + x 2 ) ( 1 + ( t − x ) 2 ) 1 + ∣ t − x ∣ δ ( 1 + δ 2 ) Ω ( f , δ )
for every f ∈ C x 2 [ 0 , ∞ ) and x , t ∈ R + .

Theorem 2

Let f ∈ C x 2 ∗ [ 0 , ∞ ) and C 1 is a constant independent of m, then

sup x ≥ 0 ∣ S m ∗ ( f ; x , a ) − f ( x ) ∣ ( 1 + x 2 ) 3 ≤ C 1 Ω f ; 1 γ m .

Proof

By (11), we have

∣ f ( t ) − f ( x ) ∣ ≤ 2 1 + ∣ t − x ∣ δ ( 1 + δ 2 ) ( 1 + x 2 ) ( 1 + ( t − x ) 2 ) Ω ( f ; λ δ ) .

Thus,

∣ S m ∗ ( f ; x , a ) − f ( x ) ∣ ≤ γ m e − 1 1 − 1 a ( a − 1 ) β m x ∑ j = 0 ∞ C j ( a ) ( − ( a − 1 ) β m x ) j ! ∫ j γ m j + 1 γ m ∣ f ( t ) − f ( x ) ∣ d t ≤ 2 ( 1 + δ 2 ) ( 1 + x 2 ) Ω ( f ; δ ) γ m e − 1 1 − 1 a ( a − 1 ) β m x ∑ j = 0 ∞ C j ( a ) ( − ( a − 1 ) β m x ) j ! ∫ j γ m j + 1 γ m 1 + ∣ t − x ∣ δ ( 1 + ( t − x ) 2 ) d t = 2 ( 1 + δ 2 ) ( 1 + x 2 ) Ω ( f ; δ ) 1 + 1 δ γ m e − 1 1 − 1 a ( a − 1 ) β m x ∑ j = 0 ∞ C j ( a ) ( − ( a − 1 ) β m x ) j ! ∫ j γ m j + 1 γ m ( t − x ) 2 d t + 1 δ γ m e − 1 1 − 1 a ( a − 1 ) β m x ∑ j = 0 ∞ C j ( a ) ( − ( a − 1 ) β m x ) j ! ∫ j γ m j + 1 γ m ∣ t − x ∣ ( t − x ) 2 d t + 1 δ γ m e − 1 1 − 1 a ( a − 1 ) β m x ∑ j = 0 ∞ C j ( a ) ( − ( a − 1 ) β m x ) j ! ∫ j γ m j + 1 γ m ∣ t − x ∣ d t .

If the Cauchy-Schwarz inequality is applied to the last term above, the following result is obtained:

(12) ∣ S m ∗ ( ( f ; x , a ) − f ( x ) ) ∣ ≤ 2 ( 1 + δ 2 ) ( 1 + x 2 ) Ω ( f ; δ ) 1 + S m ∗ ( ( t − x ) 2 ; x , a ) 1 δ S m ∗ ( ( t − x ) 2 ; x , a ) + 1 δ . S m ∗ ( ( t − x ) 2 ; x , a ) S m ∗ ( ( t − x ) 4 ; x , a ) .

By linearity of S m ∗ , we have, following estimates:

(13) S m ∗ ( ( e 1 − x ) 2 ; x , a ) = O 1 γ m ( x 2 + x + 1 ) ,

(14) S m ∗ ( ( e 1 − x ) 4 ; x , a ) = O 1 γ m ( x 4 + x 3 + x 2 + x + 1 ) .

Substituting the above inequality into (12), we obtain the following:

∣ S n ∗ ( ( f ; x , a ) − f ( x ) ) ∣ ≤ 2 ( 1 + δ 2 ) ( 1 + x 2 ) Ω ( f ; δ ) × 1 + O 1 γ m ( x 2 + x + 1 ) + 1 δ O 1 γ m ( x 2 + x + 1 ) + 1 δ O 1 γ m ( x 2 + x + 1 ) O 1 γ m ( x 4 + x 3 + x 2 + x + 1 ) .

If, choose δ = 1 γ m , we obtain the desired result.□

Now, we prove a local approximation theorem related to operators S m ∗ . If assumed that C B [ 0 , ∞ ) denote the space of all real value bounded and continuous functions f on [ 0 , ∞ ) with the norm

‖ f ‖ = sup x ∈ [ 0 , ∞ ) ∣ f ( x ) ∣ .

For all δ > 0 , the Peetre’s K − functional is given by

K 2 ( f , δ ) = inf h ∈ C B 2 [ 0 , ∞ ) { ‖ f − h ‖ + δ ‖ h ″ ‖ } ,

where C B 2 [ 0 , ∞ ) = { h ∈ C B [ 0 , ∞ ) : h ′ , h ″ ∈ C B [ 0 , ∞ ) } . By Lorentz [18], ∃ C > 0 such that

K 2 ( f , δ ) ⩽ C ω 2 ( f , δ ) .

The second-order modulus of continuity of f ∈ C B [ 0 , ∞ ) is denoted by

ω 2 ( f , δ ) = sup 0 < p < δ sup x ∈ [ 0 , ∞ ) ∣ f ( x + 2 p ) − 2 f ( x + p ) + f ( x ) ∣ .

Also, let ω ( f , δ ) be denoted as the usual modulus of continuity of f ∈ C B [ 0 , ∞ ) .

We now define the new operator constructed using S m ∗

S m ∗ ˜ ( f ; x , a ) = S m ∗ ( f ; x , a ) − f β m x γ m + 3 2 γ m + f ( x ) ,

where f ∈ C B [ 0 , ∞ ) , x ≥ 0 .

Lemma 4

Let h ∈ C B 2 [ 0 , ∞ ) . Then, for all x ≥ 0 , we obtain ∣ S m ∗ ˜ ( f ; x ) − h ( x ) ∣ ≤ ϕ m ( x ) ‖ h ″ ‖ where

ϕ m ( x ) = 2 β m γ m − 1 2 x 2 + β m γ m 2 ( 7 + 1 1 − a ) − 6 γ m x + 67 12 γ m 2 .

Proof

One can easily verify that S m ∗ ˜ ( e 1 − x ; x , a ) = 0 . Let h ∈ C B 2 [ 0 , ∞ ) . Using Taylor’s expansion of h

h ( t ) − h ( x ) = ( t − x ) h ′ ( x ) + ∫ x t ( t − u ) h ″ ( u ) d u

where t ∈ [ 0 , ∞ ) . When S m ∗ ˜ is applied to both sides of the above equation, we obtain

S m ∗ ˜ ( h ; x , a ) − h ( x ) = h ′ ( x ) S m ∗ ˜ ( t − x ; x , a ) + S m ∗ ˜ ∫ x t ( t − u ) h ″ ( u ) d u ; x , a = S m ∗ ˜ ∫ x t ( t − u ) h ″ ( u ) d u ; x , a

= S m ∗ ∫ x t ( t − u ) h ″ ( u ) d u ; x , a − ∫ x β m x γ m + 3 2 γ m β m x γ m + 3 2 γ m − u h ″ ( u ) d u

knowing that

(15) ∣ S m ∗ ˜ ( h ; x , a ) − h ( x ) ∣ ≤ S m ∗ ∫ x t ( t − u ) h ″ ( u ) d u ; x , a + ∫ x β m x γ m + 3 2 γ m β m x γ m + 3 2 γ m − u h ″ ( u ) d u .

Since

∫ x t ( t − u ) h ″ ( u ) d u ⩽ ( t − x ) 2 ‖ h ″ ‖ ,

we obtain

∫ x β m x γ m + 3 2 γ m β m x γ m + 3 2 γ m − u h ″ ( u ) d u ≤ β m − γ m γ m x + 3 2 γ m 2 ‖ h ″ ‖ .

By (15), we have the following result:

∣ S m ∗ ˜ ( h ; x , a ) − h ( x ) ∣ ≤ S m ∗ ( ( t − x ) 2 ; x , a ) + β m − γ m γ m x + 3 2 γ m 2 ‖ h ″ ‖ .

By (6), one can see that the following holds:

□ ∣ S m ∗ ˜ ( h ; x , a ) − h ( x ) ∣ ≤ 2 β m γ m − 1 2 x 2 + β m γ m 2 ( 7 + 1 1 − a ) − 6 γ m x + 67 12 γ m 2 ‖ h ″ ‖ = ϕ m ( x ) ‖ h ″ ‖ .

Theorem 3

Let us suppose that f ∈ C B [ 0 , ∞ ) . For every x ≥ 0 , there exists a constant C > 0 such that

∣ S m ∗ ( f ; x , a ) − f ( x ) ∣ ≤ C ω 2 ( f , ϕ m ( x ) ) + ω f ; β m − γ m γ m x + 3 2 γ m .

Proof

For f ∈ C B [ 0 , ∞ ) , h ∈ C B 2 [ 0 , ∞ ) , by the definition of S m ∗ ˜ , we obtain

∣ S m ∗ ( f ; x , a ) − f ( x ) ∣ ≤ ∣ S m ∗ ˜ ( f − h ; x , a ) ∣ + ∣ ( f − h ) ( x ) ∣ + ∣ S m ∗ ˜ ( h ; x , a ) − h ( x ) ∣ + f β m x γ m + 3 2 γ m − f ( x )

and

∣ S m ∗ ˜ ( f ; x , a ) ∣ ≤ ∥ f ∥ S m ∗ ( 1 ; x , a ) + 2 ∥ f ∥ = 3 ∥ f ∥ .

Thus, we obtain

∣ S m ∗ ( f ; x , a ) − f ( x ) ∣ ≤ 4 ∥ f − h ∥ + ∣ S m ∗ ˜ ( h ; x , a ) − h ( x ) ∣ + ω f ; β m − γ m γ m x + 3 2 γ m

and using (2.1), it is easily obtained

∣ S m ∗ ( f ; x , a ) − f ( x ) ∣ ≤ 4 ( ∥ f − h ∥ + ϕ n ( x ) h ′ ′ ) + ω f ; β m − γ m γ m x + 3 2 γ m .

From here, we obtain the desired result by taking the infimum of all h ∈ C B 2 [ 0 , ∞ ) on the right-hand side of the last inequality.□

Theorem 4

Let f be a bounded function on [ 0 , ∞ ) . f has a second derivative at a point x ∈ [ 0 , ∞ ) , then

lim m → ∞ γ m ( S m ∗ ( f ; x , a ) − f ( x ) ) = 3 2 f ′ ( x ) + 1 2 1 + 1 1 − a x f ″ ( x ) .

Proof

From the Taylor expansion of f , we obtain

f ( t ) = f ( x ) + f ′ ( x ) ( t − x ) + 1 2 f ″ ( x ) ( t − x ) 2 + ε ( t , x ) ( t − x ) 2 ,

where ε ( t , x ) is the remainder term and ε ( t , x ) ⟶ 0 as t ⟶ x . By the linearity of the operators S m ∗ , we obtain

(16) S m ∗ ( f ; x , a ) − f ( x ) = f ′ ( x ) S m ∗ ( ( t − x ) ; x ) + 1 2 f ″ ( x ) S m ∗ ( ( t − x ) 2 ; x ) + S m ∗ ( ε ( t , x ) ( t − x ) 2 ; x , a ) .

Using the Cauchy-Schwarz inequality, we obtain the following result:

(17) S m ∗ ( ε ( t , x ) ( t − x ) 2 ; x , a ) ≤ S m ∗ ( ε 2 ( t , x ) ; x , a ) S m ∗ ( ( t − x ) 4 ; x , a ) .

Since ε 2 ( x , x ) = 0 and ε 2 ( x , x ) ∈ C 2 ∗ [ 0 , ∞ ) , we obtain

lim m → ∞ S m ∗ ( ε 2 ( t , x ) ; x , a ) = ε 2 ( x , x ) = 0 ,

uniformly with concerning [ 0 , A ] . By (17) and (2.1), we have the following result:

(18) lim m → ∞ γ m S m ∗ ( ε 2 ( t , x ) ; x , a ) = 0 .

Taking the limit of both sides of (5) and (6), one can see that

(19) lim m → ∞ γ m ( S m ∗ ( ( t − x ) ; x , a ) ) = 3 2

(20) lim m → ∞ γ m ( S m ∗ ( ( t − x ) 2 ; x , a ) ) = 1 + 1 1 − a x .

Considering (18), (19), and (20), and taking the limit on both sides of (16), we obtain the following result:

lim m → ∞ γ m ( S m ∗ ( f ; x , a ) − f ( x ) ) = lim m → ∞ γ m S m ∗ ( ( t − x ) ; x , a ) f ′ ( x ) + 1 2 f ″ ( x ) S m ∗ ( ( t − x ) 2 ; x , a ) + S m ∗ ( ε ( t , x ) ( t − x ) 2 ; x , a ) = 3 2 f ′ ( x ) + 1 2 1 + 1 1 − a x f ″ ( x ) .

This step completes the proof.□

3 Concluding remarks

Szàsz [1] first described operators in the literature that generalize Bernstein operators to an infinite interval. These operators and generalizations have been studied by many researcher [19–25]. Moreover, Varma and Taşdelen [10] introduced and studied the Szàsz-type operators and its Kantorovich variant that involves Charlier polynomials.

In this article for an approximation integrable function, we define the Kantorovich variant of the modified Sz‘asz-Charlier operator. For the special case, a ⟶ ∞ and x − ( 1 ∕ m ) instead of x , the operators S m ∗ reduce to the modified Szàsz operators studied by Walczak [14]. If we choose β m = γ m = m , S m ∗ reduces to the operators studied by Varma and Taşdelen [10]. We introduce Kantorovich-type modified Szàsz–Charlier operators, and we give approximation properties of our operators in the weighted space of continuous functions. Also, we estimate the rate of convergence using the weighted modulus of continuity and present the Voronovskaja-type asymptotic formula for these operators.

Finally, we can give the following results for the operator we have defined:

Let S m ∗ be a sequence of linear positive operators defined (3), then for each f ∈ C x 2 ∗ [ 0 , ∞ )
lim n → ∞ ‖ S m ∗ ( f ; x , a ) − f ( x ) ‖ x 2 = 0 .
For the S m ∗ operator, the rate of convergence using the weighted modulus of continuity:
sup x ≥ 0 ∣ S m ∗ ( f ; x , a ) − f ( x ) ∣ ( 1 + x 2 ) 3 ≤ C 1 Ω f ; 1 γ m ,
For the S m ∗ operator, the rate of convergence by means of Peetre’s K -functional. Let f ∈ C B [ 0 , ∞ ) . Then, for every x ≥ 0 ,
∣ S m ∗ ( f ; x , a ) − f ( x ) ∣ ≤ C ω 2 ( f , ϕ m ( x ) ) + ω f ; β m − γ m γ m x + 3 2 γ m
For this operator, the Voronovskaja-type asymptotic formula is given as follows:

Let f be a bounded function on [ 0 , ∞ ) that has a second derivative at a point x ∈ [ 0 , ∞ ) . Then

lim m → ∞ γ m ( S m ∗ ( f ; x , a ) − f ( x ) ) = 3 2 f ′ ( x ) + 1 2 1 + 1 1 − a x f ″ ( x ) .

With the help of Maple 2020 (Maplesoft, a division of Waterloo Maple Inc.), illustrative graphics showing the rate of convergence of S m ∗ ( f ; x , a ) operators to certain functions (Figure 1).

$Figure 1 The comparison convergence of S 2 ∗ ( f ; x , 2 ) {S}_{2}^{\ast }\left(f;\hspace{0.33em}x,2) (red), S 4 ∗ ( f ; x , 2 ) {S}_{4}^{\ast }\left(f;\hspace{0.33em}x,2) (green) with γ m = m + m 6 {\gamma }_{m}=m+\sqrt[6]{m} , β m = m {\beta }_{m}=m and f ( x ) = x 2 + 2 x f\left(x)={x}^{2}+2x (blue).$

Figure 1

The comparison convergence of S 2 ∗ ( f ; x , 2 ) (red), S 4 ∗ ( f ; x , 2 ) (green) with γ m = m + m 6 , β m = m and f ( x ) = x 2 + 2 x (blue).

The presence of values in the fourth quadrant in Figure 2 can be attributed to the behavior of the given function and its approximation operators. The function f ( x ) = sin x 2 + cos x 2 exhibits periodic oscillations and assumes negative values for certain values of x . Consequently, its output falls within the fourth quadrant, where x is positive and y is negative. Furthermore, the approximation operators S 2 * ( f ; x , 2 ) and S 4 * ( f ; x , 2 ) , represented by the red and green curves, respectively, are designed to approximate the behavior of the original function. However, these operators also attain negative values in specific regions, thereby reinforcing their presence in the fourth quadrant. In summary, the values appear in the fourth quadrant due to the negative function outputs combined with positive x -values, resulting in their placement in this region of the coordinate plane.

$Figure 2 The comparison convergence of S 2 ∗ ( f ; x , 2 ) {S}_{2}^{\ast }\left(f;\hspace{0.33em}x,2) (red), S 4 ∗ ( f ; x , 2 ) {S}_{4}^{\ast }\left(f;\hspace{0.33em}x,2) (green) with γ m = m + m 6 {\gamma }_{m}=m+\sqrt[6]{m} , β m = m {\beta }_{m}=m and f ( x ) = sin x 2 + cos x 2 f\left(x)=\sin \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{x}{2}\right)+\cos \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{x}{2}\right) (blue).$

Figure 2

The comparison convergence of S 2 ∗ ( f ; x , 2 ) (red), S 4 ∗ ( f ; x , 2 ) (green) with γ m = m + m 6 , β m = m and f ( x ) = sin x 2 + cos x 2 (blue).

Acknowledgments

This article is derived from a master’s thesis, and the authors would like to thank everyone who contributed during this process.

Funding information: This research received no external funding.
Author contributions: Conceptualization – ÜK; validation – AA; formal analysis – ÜK; and writing – AA. All authors have read and agreed to the published version of the manuscript.
Conflict of interest: The authors declare no competing interests.
Ethical approval: Ethical approval was not required for this study, as it did not involve any procedures that would necessitate review by an ethical committee or internal review board.
Data availabilitiy statement: Not applicable.

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Received: 2024-10-16

Revised: 2025-03-19

Accepted: 2025-06-06

Published Online: 2025-07-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0144

Keywords for this article

Kantorovich-type Szàsz-Charlier operators; the weighted modulus of continuity; local approximation; weighted space; Voronovskaja-type asymptotic formula

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