(λ, ψ)-Bernstein-Kantorovich operators

Hüseyin Aktuğlu; Mustafa Kara; Erdem Baytunç; Saner Fidan

doi:10.1515/dema-2025-0126

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(λ, ψ)-Bernstein-Kantorovich operators

Hüseyin Aktuğlu , Mustafa Kara , Erdem Baytunç and Saner Fidan

Published/Copyright: June 6, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

In this article, we introduce a new family of ( λ , ψ ) -Bernstein-Kantorovich operators which depends on a parameter λ , derived from the basis functions of Bézier curves and an integrable function ψ . In this approach, all moments and central moments of the new operators can be obtained in terms of two numbers M 1 , ψ and M 2 , ψ , which are the integrals of ψ and ψ 2 , respectively. For operators L n ( f ; x ) with L n ( 1 ; x ) = 1 , the order of approximation to a function f by L n ( f ; x ) is more controlled by the term L n ( ( t − x ) 2 ; x ) . For our operators K n , λ , ψ ( f ; x ) , the second central moment K n , λ , ψ ( ( t − x ) 2 ; x ) depend on M 1 , ψ and M 2 , ψ . This means that in our new approach, it is possible to search for a function ψ with different values of M 1 , ψ and M 2 , ψ to make K n , λ , ψ ( ( t − x ) 2 ; x ) smaller. Using this new approach, we show that there exists a function ψ such that the order of approximation to a function f by our new ( λ , ψ ) -Bernstein-Kantorovich operators is better than the classical λ -Bernstein-Kantorovich operators on the interval [0, 1]. Moreover, we obtain some direct and local approximation properties of new operators. We also show that our operators preserve monotonicity properties. Furthermore, we illustrate the approximation results of our operators graphically and numerically.

Keywords: Bernstein operators; Bernstein-Kantorovich operators; polynomial approximation; rate of convergence; modulus of continuity; shape-preserving properties; uniform convergence

MSC 2010: 41A10; 41A25; 41A36; 41A60

1 Introduction

In 1912, Bernstein introduced the Bernstein polynomials [1] to prove the Weierstrass approximation theorem [2]. Bernstein polynomials have many noteworthy approximation properties, which have led to intensive research. The Bernstein operators are defined as

(1) B n ( f ; x ) = ∑ k = 0 n b n , k ( x ) f k n ,

where

(2) b n , k ( x ) = n k x k ( 1 − x ) n − k , x ∈ [ 0 , 1 ] .

The classical Bernstein operators are a widely used tool for approximating functions in the bounded function space on the interval [0, 1] (see [1,3–8]).

In 2010, Ye et al. [9] introduced and studied a new sort of basis functions of the Bézier curve in terms of shape parameter λ ∈ [ − 1, 1 ] . Based on this, Cai et al. [10] introduced and considered a new generalization of Bernstein polynomials depending on the parameter λ as follows:

(3) B n , λ ( f ; x ) = ∑ k = 0 n b ˜ n , k ( λ ; x ) f k n ,

where λ ∈ [ − 1, 1 ] and b ˜ n , k ( λ ; x ) , k = 0, 1 , … , n , are defined as

(4) b ˜ n , k ( λ ; x ) = b n , 0 ( x ) − λ n + 1 b n + 1,1 ( x ) , if k = 0 , b n , k ( x ) + λ n − 2 k + 1 n 2 − 1 b n + 1 , k ( x ) − λ n − 2 k − 1 n 2 − 1 b n + 1 , k + 1 ( x ) , if ( 1 ≤ k ≤ n − 1 ) , b n , n ( x ) − λ n + 1 b n + 1 , n ( x ) , if k = n .

Cai et al. [10] have extensively investigated various approximation properties of λ -Bernstein operators (3) such as uniform convergence and rate of convergence in terms of modulus of continuity and shape-preserving properties. For more details and recent published about these operators, we refer [11–15]. Bernstein operators have limitations when approximating functions have singularities or discontinuities. To address this limitation, Kantorovich introduced the Bernstein-Kantorovich operators for the Lebesgue integrable function space, accommodating functions that may have singularities or discontinuities [6]:

K n ( f ; x ) = ( n + 1 ) ∑ k = 0 n b n , k ( x ) ∫ k n + 1 k + 1 n + 1 f ( t ) d t ,

or equivalently,

(5) K n ( f ; x ) = ∑ k = 0 n b n , k ( x ) ∫ 0 1 f k + t n + 1 d t ,

for all x ∈ [ 0, 1 ] and n ∈ N .

Later, in [16] the following modification of Bernstein-Kantorovich operators was considered and studied,

(6) K n , α ( f ; x ) = ∑ k = 0 n b n , k ( x ) ∫ 0 1 f k + t α n + 1 d t ,

for all x ∈ [ 0, 1 ] , α > 0 and n ∈ N .

In recent years, the Bernstein-Kantorovich operators and their modifications have also been used by reasearchers in many research studies (see [6,7,16–23]). Two studies, in particular, have caught our attention. These are, respectively, [18] and [19] as follows:

Acu et al. [18] defined the Kantorovich modification of the λ -Bernstein operator (4) as follows:

(7) K n , λ ( f ; x ) = ( n + 1 ) ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ k n + 1 k + 1 n + 1 f ( t ) d t ,

where b ˜ n , k ( λ ; x ) are defined in equation (4).

Aktuğlu et al. [19] have introduced a new generalization of the Bernstein-Kantorovich operators as

(8) K n , ψ ( f ; x ) = ∑ k = 0 n b n , k ( x ) ∫ 0 1 f k + ψ ( t ) n + 1 d t ,

where b n , k ( x ) are defined in equation (2) and ψ is a function satisfying conditions given in (10). By choosing an appropriate function ψ , the order of approximation of the operators K n , ψ ( f ; x ) to a function f is at least as good as the classical Bernstein-Kantorovich operators K n ( f ; x ) on the interval [0, 1] [19].

In this study, inspired by operators (7) and (8), we introduce the ( λ , ψ ) -Bernstein-Kantorovich operators as follows:

(9) K n , λ , ψ ( f ; x ) = ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 f k + ψ ( t ) n + 1 d t ,

for all x ∈ [0, 1], where b ˜ n , k ( λ ; x ) is defined in (4). The function ψ is any integrable function on [0, 1] such that

(10) 0 ≤ ψ ( t ) ≤ 1 , ψ ( 0 ) = 0 , ψ ( 1 ) = 1 , and M p , ψ ≔ ∫ 0 1 ψ p ( t ) d t ,

exist, where p is any non-negative integer. Obviously, 0 ≤ M p , ψ ≤ 1 for all p .

Remark 1

The operators K n , λ , ψ ( f ; x ) have the following special cases:

If ψ ( t ) = t , then K n , λ , ψ ( f ; x ) = K n , λ ( f ; x ) .
If λ = 0 , then K n , λ , ψ ( f ; x ) = K n , ψ ( f ; x ) .
If λ = 0 and ψ ( t ) = t , then K n , λ , ψ ( f ; x ) = K n ( f ; x ) .
If ψ ( t ) = t α and λ = 0 , then K n , λ , ψ ( f ; x ) = K n , α ( f ; x ) , which is given in [16].

Throughout this article, we examine the approximation properties of K n , λ , ψ ( f ; x ) . Our aim is to demonstrate that, for certain functions ψ , the modified operator K n , λ , ψ ( f ; x ) yields improved approximation results compared to the operators defined in (6) and (7).

2 Basic results

In this section, we present some results and properties of the operator K n , λ , ψ ( f ; x ) , which will be utilized in the following sections. Thanks to the next lemma, we can find the moments of the operators K n , λ , ψ ( f ; x ) by utilizing the moments of the classical Bernstein operators B n , λ ( f ; x ) , as given in equation (3).

Lemma 1

For each ψ , given in (10), we have the following relation:

(11) K n , λ , ψ ( t m ; x ) = 1 ( n + 1 ) m ∑ i = 0 m m i n i M m − i , ψ B n , λ ( t i ; x ) ,

where m ∈ N and B n , λ ( f ; x ) is given in (3).

Proof

For any m ∈ N ,

K n , λ , ψ ( t m ; x ) = ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 m d t = 1 ( n + 1 ) m ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 ( k + ψ ( t ) ) m d t = 1 ( n + 1 ) m ∑ k = 0 n b ˜ n , k ( λ ; x ) ∑ i = 0 m m i k i M m − i , ψ = 1 ( n + 1 ) m ∑ i = 0 m m i n i M m − i , ψ ∑ k = 0 n b ˜ n , k ( λ ; x ) k i n i

= 1 ( n + 1 ) m ∑ i = 0 m m i n i M m − i , ψ B n , λ ( t i ; x ) ,

where M m − i , ψ is given in (10).□

Lemma 2

[10] For any n ∈ N , and λ ∈ [ − 1, 1 ] , we have

B n , λ ( 1 ; x ) = 1 ,
B n , λ ( t ; x ) = x + λ 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n ( n − 1 ) ,
B n , λ ( t 2 ; x ) = x 2 + x ( 1 − x ) n + λ 2 x − 4 x 2 + 2 x n + 1 n ( n − 1 ) + x n + 1 + ( 1 − x ) n + 1 − 1 n 2 ( n − 1 ) .

Using Lemma 2, we can state the following lemma.

Lemma 3

For any function ψ , given in (10), we have

K n , λ , ψ ( 1 ; x ) = 1 .
K n , λ , ψ ( t ; x ) = n x n + 1 + M 1 , ψ n + 1 + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ .
K n , λ , ψ ( t 2 ; x ) = n 2 x 2 ( n + 1 ) 2 + n x ( 1 − x ) ( n + 1 ) 2 + λ n 2 ( n + 1 ) 2 2 x − 4 x 2 + 2 x n + 1 n ( n − 1 ) + x n + 1 + ( 1 − x ) n + 1 − 1 n 2 ( n − 1 ) + 2 n M 1 , ψ ( n + 1 ) 2 x + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n ( n − 1 ) λ + M 2 , ψ ( n + 1 ) 2 .

Corollary 1

We have the following special cases of the aforementioned lemma.

For ψ ( t ) = t , K n , λ , ψ ( t i ; x ) = K n , λ ( t i ; x ) with M 1 , ψ = 1 2 and M 2 , ψ = 1 3 , for i = 0, 1, 2.
For λ = 0 and ψ ( t ) = t , K n , λ , ψ ( t i ; x ) = K n ( t i ; x ) , with M 1 , ψ = 1 2 and M 2 , ψ = 1 3 for i = 0, 1, 2.

The following lemma establishes the relation between central moments of the operators K n , λ , ψ ( f , x ) and moments of B n , λ ( f ; x ) .

Lemma 4

For any ψ , given in (10), we have

K n , λ , ψ ( ( t − x ) m ; x ) = ∑ r = 0 m m r ( − 1 ) m − r x m − r ( n + 1 ) r ∑ i = 0 r r i n i M r − i , ψ B n , λ ( t i ; x ) ,

where m ∈ N and B n , λ ( f ; x ) are defined in (3).

Proof

Using the linearity property of K n , λ , ψ , we can write

K n , λ , ψ ( ( t − x ) m ; x ) = K n , ψ , λ ∑ r = 0 m m r t r ( − x ) m − r ; x = ∑ r = 0 m m r ( − x ) m − r K n , λ , ψ ( t r ; x ) = ∑ r = 0 m m r ( − 1 ) m − r x m − r K n , λ , ψ ( t r ; x ) .

From Lemma 1, we obtain

K n , λ , ψ ( ( t − x ) m ; x ) = ∑ r = 0 m m r ( − 1 ) m − r x m − r ( n + 1 ) r ∑ i = 0 r r i n i M r − i , ψ B n , λ ( t i ; x ) ,

which completes the proof.□

From Lemma 4, we obtain the following lemma.

Lemma 5

For any ψ , given in (10), we have

( i ) K n , λ , ψ ( t − x ; x ) = − x n + 1 + M 1 , ψ n + 1 + λ 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 . ( i i ) K n , λ , ψ ( ( t − x ) 2 ; x ) = n 2 x 2 ( n + 1 ) 2 + n x ( 1 − x ) ( n + 1 ) 2 + λ n 2 ( n + 1 ) 2 2 x − 4 x 2 + 2 x n + 1 n ( n − 1 ) + x n + 1 + ( 1 − x ) n + 1 − 1 n 2 ( n − 1 ) + 2 n x M 1 , ψ ( n + 1 ) 2 + 2 n M 1 , ψ ( n + 1 ) 2 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n ( n − 1 ) λ + M 2 , ψ ( n + 1 ) 2 − 2 x n x n + 1 + λ 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 + M 1 , ψ n + 1 + x 2 .

Also, using Remark 1, the following central moments for K n , λ ( f ; x ) and K n ( f ; x ) can be obtained from Lemma 5.

Corollary 2

Taking M 1 , ψ = 1 2 and M 2 , ψ = 1 3 in Lemma 4 [6], we obtain

K n , λ ( t − x ; x ) = 1 − 2 x 2 ( n + 1 ) + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ .
K n , λ ( ( t − x ) 2 ; x ) = 3 x ( 1 − x ) ( n − 1 ) + 1 3 ( n + 1 ) 2 + 2 x λ ( 1 − x ) ( n − 1 ) ( n + 1 ) 2 ( [ ( 1 − x ) n + x n ] ( n + 1 ) − 2 ) .

Corollary 3

Taking λ = 0 , M 1 , ψ = 1 2 , and M 2 , ψ = 1 3 in Lemma 5 [6], we obtain

K n ( t − x ; x ) = − x n + 1 + 1 2 ( n + 1 ) ,
K n ( ( t − x ) 2 ; x ) = x 2 ( n + 1 ) 2 + n x ( 1 − x ) ( n + 1 ) 2 − x ( n + 1 ) 2 + 1 3 ( n + 1 ) 2 .

3 Direct and local approximation properties

In this section, we give direct and local approximation properties of the operators K n , λ , ψ ( f , x ) . First, we can state the following Bohman-Korovkin-type theorem [24] and [25].

Theorem 1

For any ψ , given in (10) and f ∈ C [0, 1], K n , λ , ψ ( f ; x ) converge uniformly to f on [0, 1], i.e.,

lim n → ∞ ‖ K n , λ , ψ ( f ) − f ‖ C [ 0,1 ] = 0 .

Proof

Let f ∈ C [0, 1]. Since M 1 , ψ and M 2 , ψ are the positive real numbers, we obtain from Lemma 3 that

lim n → ∞ K n , λ , ψ ( 1 ; x ) = 1 ,

lim n → ∞ K n , λ , ψ ( t ; x ) = x ,

and

lim n → ∞ K n , λ , ψ ( t 2 ; x ) = x 2 .

Hence, by the Korovkin theorem, we obtain

□ lim n → ∞ ‖ K n , λ , ψ ( f ) − f ‖ C [ 0,1 ] = 0 .

Since the operator K n , ψ satisfies the property K n , ψ ( 1 ; x ) = 1 , it also satisfies the following inequality:

(12) ∣ K n , λ , ψ ( f ; x ) − f ( x ) ∣ ≤ ε + 2 ‖ f ‖ C [ 0,1 ] δ 2 K n , λ , ψ ( ( t − x ) 2 ; x ) ,

where δ is determined by the uniform continuity of the function f that is being approximated, i.e., inequality (12) demonstrates that the order of approximation to a function f by K n , λ , ψ ( f ; x ) is more influenced by the term K n , λ , ψ ( ( t − x ) 2 ; x ) . Therefore, we can evaluate which of the operators K n , λ , ψ ( f ; x ) and K n , λ ( f ; x ) approximate the function f better, by examining their second central moments. In view of the aforementioned information, the following inequality can be used to compare the order of the approximation to a function f by operators K n , λ , ψ ( f ; x ) and K n , λ ( f ; x ) :

(13) K n , λ , ψ ( ( t − x ) 2 ; x ) < K n , λ ( ( t − x ) 2 ; x ) .

Since the terms K n , λ , ψ ( ( t − x ) 2 ; x ) depend on the parameters M 1 , ψ and M 2 , ψ , different values of K n , λ , ψ ( ( t − x ) 2 ; x ) are obtained for different functions ψ given in (10). This allows for the exploration of various values of M 1 , ψ and M 2 , ψ to achieve a better approximation. Therefore, to obtain a better approximation, we need to solve the following two problems.

Under which conditions on λ , M 1 , ψ , and M 2 , ψ , inequality (13) holds?
Is there any function ψ that gives these M 1 , ψ and M 2 , ψ values?

This study demonstrates that solutions exist for both problems.

Now, using Lemma 5 and Corollary 2, inequality (13) holds for n > 1 , if

(14) ( 1 − 2 M 1 , ψ ) x + M 2 , ψ − 1 3 + λ ( 1 − 2 M 1 , ψ ) ( − x n + 1 + ( 1 − x ) n + 1 + 2 x − 1 ) < 0 .

Lemma 6

For all x ∈ [ 0, 1 ] ,

(15) − 1 ≤ − x n + 1 + ( 1 − x ) n + 1 + 2 x − 1 ≤ 1 .

Theorem 2

Let ψ be a function satisfying (10), the approximation order of f by K n , λ , ψ ( f ; x ) is at least as good as the approximation order of f by K n , λ ( f ; x ) for the following cases:

For all x satisfying,
x < 1 3 − M 2 , ψ ( 1 − 2 M 1 , ψ ) − λ ,
when λ > 0 and 1 − 2 M 1 , ψ > 0 .
For all x satisfying,
x > 1 3 − M 2 , ψ ( 1 − 2 M 1 , ψ ) − λ ,
when λ < 0 and 1 − 2 M 1 , ψ < 0 .
For all x satisfying,
x > 1 3 − M 2 , ψ ( 1 − 2 M 1 , ψ ) + λ ,
when λ > 0 and 1 − 2 M 1 , ψ < 0 .
For all x satisfying,
x < 1 3 − M 2 , ψ ( 1 − 2 M 1 , ψ ) + λ ,
when λ < 0 and 1 − 2 M 1 , ψ > 0 .

Proof

Each item can be proved by using (14) and the following inequalities:

If λ ( 1 − 2 M 1 , ψ ) > 0 , then from (15), we have
λ ( 1 − 2 M 1 , ψ ) ( − x n + 1 + ( 1 − x ) n + 1 + 2 x − 1 ) ≤ λ ( 1 − 2 M 1 , ψ ) .
If λ ( 1 − 2 M 1 , ψ ) < 0 , then
□ λ ( 1 − 2 M 1 , ψ ) ( − x n + 1 + ( 1 − x ) n + 1 + 2 x − 1 ) ≤ − λ ( 1 − 2 M 1 , ψ ) .

In Theorem 2, the solutions of the first problem were obtained. Now we will move on to the second problem which is, “is there at least one function ψ α with values of M 1 , ψ α and M 2 , ψ α and λ satisfying each item of Theorem 2.”

Now, consider the the following functions:

(16) ψ α * ( t ) ≔ a α − 1 α t 1 α , 0 ≤ t ≤ a t α , a ≤ t ≤ 1 ,

where a ∈ [ 0, 1 ] and α > 0 . Obviously, ψ α * ( t ) is continuous for all a ∈ [ 0, 1 ] , α > 0 , and satisfies (10), with

M 1 , ψ α * = α − 1 α + 1 a α + 1 + 1 α + 1 and M 2 , ψ α * = 2 ( α − 1 ) ( α + 1 ) ( α + 2 ) ( 2 α + 1 ) a 2 α + 1 + 1 2 α + 1 .

Remark 2

Using the function ψ α * given in (16) and Theorem 2, we have the following cases, where K n , λ , ψ α * ( f ; x ) has better error estimation than λ -Bernstein-Kantorovich operators K n , λ ( f ; x ) :

For λ = 0.1 , α = 1.1 and a = 0.65 , M 1 , ψ α * = 0.4955 , M 2 , ψ α * = 0.3232 , and 1 − 2 M 1 , ψ α * = 0.0091 therefore from Theorem 2, i) the order of approximation to f by K n , λ , ψ α * ( f : x ) is as good as K n , λ ( f ; x ) on the interval:
x < 1 3 − M 2 , ψ α * 1 − 2 M 1 , ψ α * − λ = 1.0199 ,
which includes [0, 1].
For λ = − 0.8 , α = 1.1 and a = 0.75 , M 1 , ψ α * = 0.5022 , M 2 , ψ α * = 0.3294 , and 1 − 2 M 1 , ψ α * = − 0.0044 , therefore from Theorem 2, ii) the order of approximation to f by K n , λ , ψ α * ( f : x ) is as good as K n , λ ( f ; x ) on the interval:
x > 1 3 − M 2 , ψ α * 1 − 2 M 1 , ψ α * − λ = − 0.0956 ,
which includes [0, 1].
For λ = 0.8 , α = 1.1 , and a = 0.75 , M 1 , ψ α * = 0.5022 , M 2 , ψ α * = 0.3294 , and 1 − 2 M 1 , ψ α * = − 0.0044 , therefore from Theorem 2, ii) the order of approximation to f by K n , λ , ψ α * ( f : x ) is as good as K n , λ ( f ; x ) on the interval:
x > 1 3 − M 2 , ψ α * 1 − 2 M 1 , ψ α * + λ = − 0.0956 ,
which includes [0, 1].
For λ = − 0.1 , α = 1.1 , and a = 0.65 , M 1 , ψ α * = 0.4955 , M 2 , ψ α * = 0.3232 , and 1 − 2 M 1 , ψ α * = 0.0091 , therefore from Theorem 2, (ii) the order of approximation to f by K n , λ , ψ α * ( f : x ) is as good as K n , λ ( f ; x ) on the interval:
x < 1 3 − M 2 , ψ α * 1 − 2 M 1 , ψ α * + λ = 1.0199 ,
which includes [0, 1].

x	K 10,0.1 , ψ α * ( ( t − x ) 2 ; x )	K 10,0.1 ( ( t − x ) 2 ; x )
0	0.002671	0.002755
0.1	0.009402	0.009479
0.2	0.014562	0.014632
0.3	0.018248	0.018310
0.4	0.020467	0.020521
0.5	0.021213	0.021259
0.6	0.020482	0.020521
0.7	0.018278	0.018310
0.8	0.014608	0.014632
0.9	0.009463	0.009479
1	0.002746	0.002755

Recall that the first-order modulus of continuity,

w ( f ; δ ) = sup t 1 , t 2 ∈ [ 0,1 ] sup ∣ t 1 − t 2 ∣ ≤ δ ∣ f ( t 1 ) − f ( t 2 ) ∣ ,

where f ∈ C [0, 1], has the following properties [26,27]:

∣ f ( t 1 ) − f ( t 2 ) ∣ ≤ ω ( f ; ∣ t 1 − t 2 ∣ ) ,
ω ( f ; λ δ ) ≤ ( 1 + λ ) ω ( f ; δ ) , where λ > 0 .

Theorem 3

If f ∈ C [0, 1], for any ψ , given in (10), then we have

∣ K n , ψ , λ ( f ; x ) − f ( x ) ∣ ≤ 2 ω ( f , K n , λ , ψ ( ( t − x ) 2 ; x ) ) ,

where x ∈ [ 0,1 ] and n ∈ N .

Proof

Since K n , λ , ψ ( 1 ; x ) = 1 and b ˜ n , k ( λ ; x ) ≥ 0 on [0, 1], we can write

(17) ∣ K n , ψ , λ ( f ; x ) − f ( x ) ∣ ≤ ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 f k + ψ ( t ) n + 1 − f ( x ) d t .

Additionally, by applying the properties of the modulus of continuity, we obtain

(18) f k + ψ ( t ) n + 1 − f ( x ) ≤ ω f ; 1 δ k + ψ ( t ) n + 1 − x δ ≤ 1 + 1 δ k + ψ ( t ) n + 1 − x ω ( f ; δ ) .

By using (18) into (17), we obtain

(19) ∣ K n , λ , ψ ( f ; x ) − f ( x ) ∣ ≤ ∑ k = 0 n b ˜ n , k ( x ) ∫ 0 1 1 + 1 δ k + ψ n + 1 − x ω ( f ; δ ) d t = 1 + 1 δ ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x d t ω ( f ; δ ) .

Using the Cauchy-Schwartz inequality, we can state the following inequality:

(20) ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x d t ≤ ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x 2 d t 1 2 = K n , λ , ψ ( ( t − x ) 2 ; x ) .

Combining (19) and (20), we obtain the following inequality:

∣ K n , λ , ψ ( f ; x ) − f ( x ) ∣ ≤ 1 + K n , λ , ψ ( ( t − x ) 2 ; x ) δ ω ( f ; δ ) .

Hence, Theorem 3 can be obtained by choosing δ = K n , λ , ψ ( ( t − x ) 2 ; x ) .□

Let W 2 [ 0, 1 ] = { g ∈ C [ 0, 1 ] : g ′ , g ″ ∈ C [ 0, 1 ] } . For δ > 0 and f ∈ C [0, 1], the Peetre’s K -functional is defined by

K 2 ( f ; δ ) ≔ inf g ∈ W 2 [ 0,1 ] { ‖ f − g ‖ C [ 0,1 ] + δ ‖ g ″ ‖ C [ 0,1 ] } .

Moreover, there exists R > 0 [4] such that

K 2 ( f ; δ ) ≤ R ω 2 ( f ; δ ) ,

where ω 2 ( f ; δ ) is called the second-order modulus of continuity of f ∈ C [0, 1], i.e.,

ω 2 ( f ; δ ) = sup 0 < h ≤ δ sup x , x + h , x + 2 h ∈ [ 0,1 ] ∣ f ( x + 2 h ) − 2 f ( x + h ) + f ( x ) ∣ .

To prove the local approximation properties of the operators K n , λ , ψ based on the second-order modulus of continuity, we need to state the following lemma.

Lemma 7

For any f ∈ C [0, 1] and ψ , given in (10), we have

(21) ∣ K n , λ , ψ ( f ; x ) ∣ ≤ ‖ f ‖ C [ 0,1 ] ,

where ‖ . ‖ C [ 0,1 ] is the uniform norm of C [0, 1].

Theorem 4

Let f ∈ C [0, 1], and ψ be a function satisfying (10), then there exists R ∈ R + such that

∣ K n , λ , ψ ( f ; x ) − f ( x ) ∣ ≤ R ω 2 f ; 1 2 K n , λ , ψ ( ( t − x ) 2 ; x ) + ( K n , λ , ψ ( ( t − x ) ; x ) ) 2 + ω ( f ; K n , λ , ψ ( ( t − x ) ; x ) ) .

Proof

First, we define the auxiliary operator as

(22) K n , λ , ψ * ( f ; x ) ≔ K n , λ , ψ ( f ; x ) + f ( x ) − f n x n + 1 + M 1 , ψ n + 1 + [ 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 ] n 2 − 1 λ .

From Lemma 3, we have

K n , λ , ψ * ( 1 ; x ) = 1

and

(23) K n , λ , ψ * ( t − x ; x ) = 0 ,

where x ∈ [ 0, 1 ] and n ∈ N .

Let g ∈ W 2 [0, 1], by Taylor’s expansion,

(24) g ( t ) = g ( x ) + ( t − x ) g ′ ( x ) + ∫ x t ( t − u ) g ″ ( u ) d u ,

and (23), we have

K n , λ , ψ * ( g ; x ) = g ( x ) + K n , λ , ψ * ∫ x t ( t − u ) g ″ ( u ) d u ; x = g ( x ) + K n , λ , ψ ∫ x t ( t − u ) g ″ ( u ) d u ; x − ∫ x n x n + 1 + M 1 , ψ n + 1 + [ 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 ] n 2 − 1 λ n x n + 1 + M 1 , ψ n + 1 + [ 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 ] n 2 − 1 λ − u g ″ ( u ) d u .

Therefore,

(25) K n , ψ α , λ * ( g ; x ) − g ( x ) = K n , ψ α , λ ∫ x t ( t − u ) g ″ ( u ) d u ; x − ∫ x n x n + 1 + M 1 , ψ n + 1 + [ 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 ] n 2 − 1 λ n x n + 1 + M 1 , ψ α n + 1 + [ 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 ] n 2 − 1 λ − u g ″ ( u ) d u .

By applying the absolute value to both sides of (25) and using the triangle inequality, we obtain

(26) ∣ K n , λ , ψ * ( g ; x ) − g ( x ) ∣ ≤ K n , λ , ψ ∫ x t ( t − u ) g ″ ( u ) d u ; x + ∫ x n x n + 1 + M 1 , ψ n + 1 + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ n x n + 1 + M 1 , ψ n + 1 + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ − u g ″ ( u ) d u ≤ K n , λ , ψ ∫ x t ( t − u ) g ″ ( u ) d u ; x + ∫ x n x n + 1 + M 1 , ψ n + 1 + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ n x n + 1 + M 1 , ψ n + 1 + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ − u ∣ g ″ ( u ) ∣ d u ≤ K n , λ , ψ ∫ x t ∣ ( t − u ) ∣ d u ; x ‖ g ″ ‖ + ∫ x n x n + 1 + M 1 , ψ n + 1 + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ n x n + 1 + M 1 , ψ n + 1 + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ − u d u ‖ g ″ ‖ C [ 0,1 ] ≤ K n , λ , ψ ( ( t − x ) 2 ; x ) ‖ g ″ ‖ C [ 0,1 ] + n x n + 1 + M 1 , ψ n + 1 + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ − x 2 ‖ g ″ ‖ C [ 0,1 ] ≤ [ K n , λ , ψ ( ( t − x ) 2 ; x ) + ( K n , λ , ψ ( ( t − x ) ; x ) ) 2 ] ‖ g ″ ‖ C [ 0,1 ] .

So,

∣ K n , λ , ψ * ( g ; x ) − g ( x ) ∣ ≤ [ K n , λ , ψ ( ( t − x ) 2 ; x ) + ( K n , λ , ψ ( ( t − x ) ; x ) ) 2 ] ‖ g ″ ‖ C [ 0,1 ] .

Combining (21) and (22), we obtain

(27) ∣ K n , λ , ψ * ( f ; x ) ∣ ≤ ∣ K n , λ , ψ ( f ; x ) ∣ + ∣ f ( x ) ∣ + f n x n + 1 + M 1 , ψ n + 1 + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ ≤ ‖ f ‖ C [ 0,1 ] ∣ K n , λ , ψ ( 1 ; x ) ∣ + 2 ‖ f ‖ C [ 0,1 ] ≤ 3 ‖ f ‖ C [ 0,1 ] ,

for all f ∈ C [0, 1] and x ∈ [ 0, 1 ] .

Equations (26) and (27) imply

∣ K n , λ , ψ ( f ; x ) − f ( x ) ∣ = ∣ K n , λ , ψ * ( f ; x ) − f ( x ) + f n x n + 1 + M 1 , ψ n + 1 + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ − f ( x ) = ∣ K n , λ , ψ * ( f ; x ) − K n , λ , ψ * ( g ; x ) + K n , λ , ψ * ( g ; x ) − g ( x ) + g ( x ) − f ( x ) + f n x n + 1 + M 1 , ψ n + 1 + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ − f ( x ) ≤ ∣ K n , λ , ψ * ( f ; x ) − K n , λ , ψ * ( g ; x ) ∣ + ∣ K n , λ , ψ * ( g ; x ) − g ( x ) ∣ + ∣ g ( x ) − f ( x ) ∣ + f n x n + 1 + M 1 , ψ n + 1 + 1 − 2 x + x n + 1 − ( 1 − x ) n + 1 n 2 − 1 λ − f ( x ) ≤ 4 ‖ f − g ‖ + [ K n , λ , ψ ( ( t − x ) 2 ; x ) + ( K n , λ , ψ ( ( t − x ) ; x ) ) 2 ] ‖ g ″ ‖ C [ 0,1 ] + ω ( f ; K n , λ , ψ ( ( t − x ) ; x ) ) .

Taking infimum on the right-hand side over all g ∈ W 2 [0, 1], we obtain

∣ K n , λ , ψ ( f ; x ) − f ( x ) ∣ ≤ 4 K 2 f ; K n , λ , ψ ( ( t − x ) 2 ; x ) + ( K n , λ , ψ ( ( t − x ) ; x ) ) 2 4 + ω ( f ; K n , λ , ψ ( ( t − x ) ; x ) ) ,

which completes the proof.□

The following theorem examines the rate of convergence of the operators K n , λ , ψ ( f ; x ) in terms of the Lipschitz class functions. For R > 0 and 0 < μ ≤ 1 , the usual Lipschitz class is defined by

Lip R ( μ ) ≔ { f ∈ C [ 0, 1 ] : ∣ f ( x ) − f ( t ) ∣ ≤ R ∣ x − t ∣ μ } ,

where x , t ∈ [ 0, 1 ] .

Theorem 5

If f ∈ Lip R ( μ ) , for any ψ , given in (10), we have

∣ K n , λ , ψ ( f ; x ) − f ( x ) ∣ ≤ R ( K n , λ , ψ ( ( t − x ) 2 ; x ) μ ) ,

where x ∈ [ 0, 1 ] , R > 0 , and 0 < μ ≤ 1 .

Proof

Using linearity and positivity properties of K n , λ , ψ ( f ; x ) , we obtain

∣ K n , λ , ψ ( f ; x ) − f ( x ) ∣ ≤ K n , λ , ψ ( ∣ f ( t ) − f ( x ) ∣ ; x ) = ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 f k + ψ ( t ) n + 1 − f ( x ) d t ≤ R ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x μ d t .

Applying Hölder’s inequality with p = 2 μ and q = 2 2 − μ ,

∣ K n , λ , ψ ( f ; x ) − f ( x ) ∣ ≤ R ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x 2 d t μ 2 = R [ K n , λ , ψ ( ( t − x ) 2 ; x ) ] μ 2 ,

which gives the desired result.□

Now, we investigate the rate of convergence of the operators K n , ψ , λ ( f ; x ) for a function space C 1 [0, 1].

Theorem 6

For all f ∈ C 1 [0, 1] and ψ , given in (10), we have

∣ K n , λ , ψ ( f ; x ) − f ( x ) ∣ ≤ ∣ K n , λ , ψ ( t − x ; x ) ∣ ∣ f ′ ( x ) ∣ + 2 K n , λ , ψ ( ( t − x ) 2 ; x ) × ω ( f ′ ; K n , λ , ψ ( ( t − x ) 2 ; x ) ) .

Proof

Let f ∈ C 1 [0, 1]. By the mean value theorem of differential calculus, we have

f k + ψ ( t ) n + 1 − f ( x ) = k + ψ ( t ) n + 1 − x f ′ ( x ) + k + ψ ( t ) n + 1 − x [ f ′ ( c ) − f ′ ( x ) ] ,

where c ∈ x , k + ψ ( t ) n + 1 .

Multiplying both sides of the aforementioned equality by b ˜ n , k and summing from 0 to n , we obtain

∑ k = 0 n b ˜ n , k ( λ ; x ) f k + ψ ( t ) n + 1 − f ( x ) = ∑ k = 0 n b ˜ n , k ( λ ; x ) k + ψ ( t ) n + 1 − x f ′ ( x ) + ∑ k = 0 n b ˜ n , k ( λ ; x ) k + ψ ( t ) n + 1 − x ( f ′ ( c ) − f ′ ( x ) ) .

Taking integral from 0 to 1, we obtain

∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 f k + ψ ( t ) n + 1 − f ( x ) d t = ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x f ′ ( x ) d t + ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x ( f ′ ( c ) − f ′ ( x ) ) d t ,

i.e.,

K n , λ , ψ ( f ; x ) − f ( x ) = K n , λ , ψ ( t − x ; x ) f ′ ( x ) + ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x ( f ′ ( c ) − f ′ ( x ) ) d t .

Thus,

(28) ∣ K n , ψ α , λ ( f ; x ) − f ( x ) ∣ ≤ ∣ K n , ψ ( t − x ; x ) ∣ ∣ f ′ ( x ) ∣ + ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x ∣ f ′ ( c ) − f ′ ( x ) ∣ d t .

On the other hand, we have

(29) ∣ f ′ ( c ) − f ′ ( x ) ∣ = 1 + 1 δ ∣ c − x ∣ ω ( f ′ ; δ ) ≤ 1 + 1 δ k + ψ ( t ) n + 1 − x ω ( f ′ ; δ ) ,

where δ is any positive number that does not depend on k .

Combining (28) and (29), we obtain

∣ K n , λ , ψ ( f ; x ) − f ( x ) ∣ ≤ ∣ K n , λ , ψ ( t − x ; x ) ∣ ∣ f ′ ( x ) ∣ + ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x d t ω ( f ′ ; δ ) + 1 δ ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x 2 d t ω ( f ′ ; δ ) ≤ ∣ K n , λ , ψ ( t − x ; x ) ∣ ∣ f ′ ( x ) ∣ + ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x d t ω ( f ′ ; δ ) + 1 δ K n , λ , ψ ( ( t − x ) 2 ; x ) ω ( f ′ ; δ ) .

Using the Cauchy-Schwarz inequality, we obtain

∣ K n , λ , ψ ( f ; x ) − f ( x ) ∣ ≤ ∣ K n , λ , ψ ( t − x ; x ) ∣ ∣ f ′ ( x ) ∣ + ∑ k = 0 n b ˜ n , k ( λ ; x ) ∫ 0 1 k + ψ ( t ) n + 1 − x 2 d t 1 2 ω ( f ′ ; δ ) + 1 δ K n , λ , ψ ( ( t − x ) 2 ; x ) ω ( f ′ ; δ ) = ∣ K n , λ , ψ ( t − x ; x ) ∣ ∣ f ′ ( x ) ∣ + [ K n , λ , ψ ( ( t − x ) 2 ; x ) ] 1 2 ω ( f ′ ; δ ) + 1 δ K n , λ , ψ ( ( t − x ) 2 ; x ) ω ( f ′ ; δ ) = ∣ K n , λ , ψ ( t − x ; x ) ∣ ∣ f ′ ( x ) ∣ + 1 + [ K n , λ , ψ ( ( t − x ) 2 ; x ) ] 1 2 δ [ K n , λ , ψ ( ( t − x ) 2 ; x ) ] 1 2 ω ( f ′ ; δ ) .

Choosing δ = K n , λ , ψ ( ( t − x ) 2 ; x ) completes the proof.□

4 Monotonicity-preserving properties

This section is devoted to the shape-preserving properties of the operators K n , λ , ψ ( f ; x ) , which can be written in the following form [28]:

(30) K n , λ , ψ ( f ; x ) = B n , ψ ( f ; x ) + λ ∑ k = 1 n n − 2 k + 1 n 2 − 1 b n + 1 , k ( x ) h k , ψ ( f ) − λ ∑ k = 1 n n − 2 k + 1 n 2 − 1 b n + 1 , k ( x ) h k − 1 , ψ ( f ) ,

where

h k , ψ ( f ) = ∫ 0 1 f k + ψ ( t ) n + 1 d t

and

B n , ψ ( f ; x ) = ∑ k = 0 n b n , k ( x ) h k , ψ ( f ) .

Moreover, by applying derivative operators d d x and d 2 d x 2 to the operators (30), for each λ ∈ [ − 1, 1 ] and f ∈ C [0, 1], we have the following equation:

(31) d d x K n , λ , ψ ( f ; x ) = ∑ k = 0 n − 1 n − k + λ n − 2 k − 1 n − 1 b n , k ( x ) [ h k + 1 , ψ ( f ) − h k , ψ ( f ) ] + ∑ k = 0 n − 1 k + 1 − λ n − 2 k − 1 n − 1 b n , k + 1 ( x ) [ h k + 1 , ψ ( f ) − h k , ψ ( f ) ] .

Theorem 7

For all n ∈ N and λ ∈ [ − 1, 1 ] , then K n , λ , ψ ( f ; x ) preserves the monotonocity property. In other words, if f is increasing (decreasing) on the interval [0, 1], then K n , λ , ψ ( f ; x ) is increasing (decreasing) on [0, 1].

Proof

Let f be an increasing function on the interval [0, 1]. Then, obviously, h k + 1 , ψ ( f ) − h k , ψ ( f ) ≥ 0 . On the other hand, using the following inequalities [28]:

(32) 0 ≤ n − k − 1 ≤ n − k + λ 1 − 2 k n − 1

and

(33) 0 ≤ ( k + 1 ) − 1 ≤ ( k + 1 ) − λ 1 − 2 k n − 1 ,

we have d d x K n , λ , ψ ( f ; x ) ≥ 0 . The aforementioned proof can be easily modified for decreasing function.□

5 Graphical analysis and error estimation

In this section, we give some graphics to illustrate the approximation of K n , λ , ψ ( f ; x ) to certain function f in Figure 1 when f ( x ) = 2 x 3 − 9 4 x 2 + 13 6 x − 3 32 for α = 1.1 , λ = 0.1 , and a = 0.65 . Similarly, shape-preserving properties of K n , λ , ψ ( f ; x ) are illustrated in Figures 2 and 3 for functions f ( x ) = − x 3 + 3 x 2 + 9 and function f ( x ) = 2 x 2 − 4 x − 3 , respectively, for α = 1.1 , λ = 0.1 , and a = 0.65 . For the following figures, ψ α * ( t ) is the function given in (16).

$Figure 1 Approximation of K n , λ , ψ α * ( f ; x ) {K}_{n,\lambda ,{\psi }_{\alpha }^{* }}\left(f;\hspace{0.33em}x) to f ( x ) = 2 x 3 − 9 4 x 2 + 13 6 x − 3 32 f\left(x)=2{x}^{3}-\frac{9}{4}{x}^{2}+\frac{13}{6}x-\frac{3}{32} for α = 1.1 \alpha =1.1 , λ = 0.1 \lambda =0.1 , and a = 0.65 a=0.65 .$

Figure 1

Approximation of K n , λ , ψ α * ( f ; x ) to f ( x ) = 2 x 3 − 9 4 x 2 + 13 6 x − 3 32 for α = 1.1 , λ = 0.1 , and a = 0.65 .

$Figure 2 K n , λ , ψ α * ( f ; x ) {K}_{n,\lambda ,{\psi }_{\alpha }^{* }}\left(f;\hspace{0.33em}x) is increasing for an increasing function f ( x ) = − x 3 + 3 x 2 + 9 f\left(x)=-{x}^{3}+3{x}^{2}+9 , α = 1.1 , λ = 0.1 \alpha =1.1,\lambda =0.1 , and a = 0.65 a=0.65 .$

Figure 2

K n , λ , ψ α * ( f ; x ) is increasing for an increasing function f ( x ) = − x 3 + 3 x 2 + 9 , α = 1.1 , λ = 0.1 , and a = 0.65 .

$Figure 3 K n , λ , ψ α * ( f ; x ) {K}_{n,\lambda ,{\psi }_{\alpha }^{* }}\left(f;\hspace{0.33em}x) is decreasing for a decreasing function f ( x ) = 2 x 2 − 4 x − 3 f\left(x)=2{x}^{2}-4x-3 , α = 1.1 , λ = 0.1 \alpha =1.1,\lambda =0.1 , and a = 0.65 a=0.65 .$

Figure 3

K n , λ , ψ α * ( f ; x ) is decreasing for a decreasing function f ( x ) = 2 x 2 − 4 x − 3 , α = 1.1 , λ = 0.1 , and a = 0.65 .

6 Conclusion

In this article, we introduce a family of λ -Bernstein-Kantorovich-type operators that depend on a function ψ , which satisfies specific conditions as given in Equation (10). We derive all moments and central moments of these new operators in terms of two variables, M 1 , ψ and M 2 , ψ , which represent the integrals of ψ and ψ 2 over [0, 1], respectively. Using this approach, and recognizing that the approximation order of a function f by L n ( f ; x ) is closely tied to the term L n ( ( t − x ) 2 ; x ) , we address two main questions: are there values of M 1 , ψ and M 2 , ψ for which K n , λ , ψ ( ( t − x ) 2 ; x ) < K n , λ ( ( t − x ) 2 ; x ) , and is there a function ψ with these values of M 1 , ψ and M 2 , ψ ? We establish that both questions have affirmative answers (Figure 4). Additionally, we identify operators with superior approximation properties compared to the classical Bernstein-Kantorovich operators across [0, 1], as well as operators from [18] on [0, 1]. This article also provides results on the uniform convergence and rate of convergence of these new operators, characterized by the first- and second-order modulus of continuity, and demonstrates that the operators exhibit monotonicity-preserving property (Figures 2 and 3). Finally, numerical examples are presented to support our findings.

$Figure 4 Graphical representations of K n , λ , ψ α * ( ( t − x ) 2 ; x ) {K}_{n,\lambda ,{\psi }_{\alpha }^{* }}\left({\left(t-x)}^{2};\hspace{0.33em}x) and K n , λ ( ( t − x ) 2 ; x ) {K}_{n,\lambda }\left({\left(t-x)}^{2};\hspace{0.33em}x) for α = 1.1 \alpha =1.1 , a = 0.65 a=0.65 , λ = 0.1 \lambda =0.1 , and n = 10 n=10 .$

Figure 4

Graphical representations of K n , λ , ψ α * ( ( t − x ) 2 ; x ) and K n , λ ( ( t − x ) 2 ; x ) for α = 1.1 , a = 0.65 , λ = 0.1 , and n = 10 .

Acknowledgements

The authors would like to thank the referee for his/her careful reading and valuable suggestions for improving this article.

Funding information: The authors state no funding involved.
Author contributions: All authors of the manuscript have made equal contributions.
Conflict of interest: The authors have no conflict of interest.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2024-11-08

Revised: 2025-02-05

Accepted: 2025-02-28

Published Online: 2025-06-06

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

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https://doi.org/10.1515/dema-2025-0126

Keywords for this article

Bernstein operators; Bernstein-Kantorovich operators; polynomial approximation; rate of convergence; modulus of continuity; shape-preserving properties; uniform convergence

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