Chlodowsky variant of Bernstein-type operators on the domain

Sevilay Kırcı Serenbay; Rabia Aktaş Karaman

doi:10.1515/math-2025-0201

Article Open Access

Chlodowsky variant of Bernstein-type operators on the domain

Sevilay Kırcı Serenbay and Rabia Aktaş Karaman

Published/Copyright: October 30, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

In the present paper, we deal with Bernstein-Chlodowsky type operators for approximating functions on the domain. We first present Bernstein-Chlodowsky type operators in two variables and then we discuss some examples of these operators under a domain transformation. Finally, we give bivarite shifted m th Bernstein-Chlodowsky-Stancu operators and we present some figures for approximation properties of our operator.

Keywords: Bernstein operators; Bernstein-Chlodowsky operators; modulus of continuity

MSC 2020: 41A25; 26A16; 41A10; 41A36

1 Introduction

The classical Bernstein operators were initially introduced by S. Bernstein [1] in 1912 as a way to approximate continuous functions on the interval [0, 1] using polynomials as

B m g , x = ∑ k = 0 m g k m p m , k x ,

where p m , k x = m k x k 1 − x m − k for 0 ≤ x ≤ 1, m ≥ 0, and they are well-known for their convergence properties. Bernstein operators are extensively used in approximating continuous functions. They offer a constructive proof of the Weierstrass approximation theorem [2], showing that any continuous function on a closed interval can be uniformly approximated by a sequence of Bernstein polynomials. Over the years, numerous modifications and extensions of Bernstein operators have been studied by many authors, refer to [3], [4], [5], [6], [7], [8]. Bernstein polynomials have several applications in approximation theory [2],9]. Besides this, Bernstein polynomials form the basis of Bézier curves [10] and surfaces, which are fundamental tools in computer graphics, animation, and computer aided geometric design [11]. These curves and surfaces are widely used for modeling smooth shapes. Furthermore, the multivariate extensions of Bernstein operators have been widely studied, with the most common form being defined on the unit simplex in higher dimensions [12], [13], [14], [15], [16]. In the recent paper [17], the authors present an extension of the Bernstein operator to approximate functions defined on the unit disk. The shifted mth Bernstein-Stancu operator and the shifted mth Bernstein-type operator are defined and their approximation properties are investigated. Several examples are discussed, comparing the approximation results of both Bernstein-type operators on the unit disk.

One of the generalization of Bernstein operators is Bernstein-Chlodowsky operators. These polynomials, introduced by Chlodowsky [18] in 1937, extend the concept of Bernstein polynomials (from 1912) to an unbounded set. mth Bernstein-Chlodowsky operator is defined as

(1) B m C l g , x = ∑ k = 0 m g k m b m q m , k x b m ,

where

(2) q m , k x b m = m k x b m k 1 − x b m m − k

for 0 ≤ x ≤ b _m, b m ≥ 0 , lim m → ∞ b m = ∞ , lim m → ∞ b m m = 0 . The classical Bernstein-Chlodowsky operators are linear and positive operators acting on the function g and they satisfy

(3) B m C l 1 , x = 1 , B m C l t , x = x and B m C l t 2 , x = x 2 + x b m − x m .

From the Korovkin’s theorem, the operator B m C l g , x converges to g uniformly on the interval 0 , b m .

The Bernstein-Chlodowsky operators and their extensions have been studied by many authors [19], [20], [21], [22], [23], [24]. The multivariable generalization of these operators has also been introduced [25], [26], [27], [28], [29]. In [25], two dimensional Bernstein-Stancu-Chlodowsky operators on triangle with mobile boundaries are investigated. In [28], Bernstein-Chlodowsky polynomials on a triangular domain are studied and the problem of weighted approximations of continuous functions of two variables by a sequences of linear positive operators is discussed.

This paper is motivated by a recent study [17] of Bernstein-type operators on the unit disk. In the present paper, we focus on extending the Bernstein-Chlodowsky operator for approximating functions on domains in R 2 , such as square region, triangular or elliptic region. We explore two modifications: transforming the argument of the target function and defining a suitable function basis as described in (2). We present two Bernstein-Chlodowsky type approximants, comparing them through some examples.

The organization of the paper is as follows: in the second section, we present basic properties of univariate Bernstein-Chlodowsky type operators. In the third section, we study Bernstein-Chlodowsky type operators in two variables and, in the next section we discuss some examples of these operators. Finally, we give bivarite shifted mth Bernstein-Chlodowsky-Stancu operators and discuss their approximation properties.

2 Univariate Bernstein-Chlodowsky type operators

This section deals with some basic properties of the shifted m th Bernstein-Chlodowsky operators which we will use in the next sections. We first start with the shifted Bernstein-Chlodowsky basis on the interval α b m , β b m .

From the change of variable

x = β − α b m t + α b m , b m ≥ 0 , α < β , 0 ≤ t ≤ 1 ,

we can define the univariate Bernstein-Chlodowsky basis on the interval α b m , β b m as follows

(4) q ̃ m , k x ; α b m , β b m = p m , k x b m − α β − α = 1 β − α m b m m m k x − α b m k β b m − x m − k

for αb _m ≤ x ≤ βb _m. For the set of the polynomials q ̃ m , k x ; α b m , β b m , the following properties hold:

(5) ∑ k = 0 m q ̃ m , k x ; α b m , β b m = 1 β − α m b m m ∑ k = 0 m m k x − α b m k β b m − x m − k = 1 β − α m b m m β b m − α b m m = 1 ,
q ̃ m , k x ; α b m , β b m ≥ 0 , α b m ≤ x ≤ β b m ,
q ̃ m , k α b m = p m , k 0 = δ k , 0 and q ̃ m , k β b m = p m , k 1 = δ k , m where δ _k,m denotes the Kronecker delta,
For m ≠ 0, the function q ̃ m , k x ; α b m , β b m takes maximum value at x = β − α k b m m + α b m and maximum value equals to
q ̃ m , k β − α k b m m + α b m ; α b m , β b m = p m , k k m = m k k k m m m − k m − k ,
β − α b m q ̃ m , k ′ x ; α b m , β b m = m q ̃ m − 1 , k − 1 x ; α b m , β b m − q ̃ m − 1 , k x ; α b m , β b m .

We now consider the shifted univariate mth Bernstein-Chlodowsky operator as

B ̃ m C l g x , J = ∑ k = 0 m g β − α k b m m + α b m q ̃ m , k x , J

for every function g defined on J = α b m , β b m . Here B ̃ m C l g x , J is a polynomial of degree at most m and it follows

B ̃ m C l g x , J = B m C l G s , 0 ≤ s ≤ b m ,

where G s = g β − α b m s + α b m .

Lemma 1.

For x ∈ α b m , β b m and m ∈ N , the shifted univariate mth Bernstein-Chlodowsky operator B ̃ m C l g x , J has the following results

B ̃ m C l 1 , J = 1 ,
B ̃ m C l x , J = x ,
B ̃ m C l x 2 , J = x 2 + 1 m x − α b m β b m − x .

Proof.

(i) It is clear from (5)

B ̃ m C l 1 , J = ∑ k = 0 m q ̃ m , k x , J = 1 .

It follows from (5)

B ̃ m C l x , J = ∑ k = 0 m β − α k b m m + α b m q ̃ m , k x , J = β − α b m ∑ k = 0 m k m q ̃ m , k x , J + α b m = x − α b m ∑ k = 0 m − 1 q ̃ m − 1 , k x , J + α b m = x − α b m + α b m = x .

Finally, we have

B ̃ m C l x 2 , J = ∑ k = 0 m β − α k b m m + α b m 2 q ̃ m , k x , J = β − α 2 b m 2 ∑ k = 0 m k 2 m 2 q ̃ m , k x , J + 2 α b m x − α b m ∑ k = 0 m − 1 q ̃ m − 1 , k x , J + α 2 b m 2 = m − 1 m x − α b m 2 ∑ k = 0 m − 2 q ̃ m − 2 , k x ; J + β − α b m x − α b m m ∑ k = 0 m − 1 q ̃ m − 1 , k x ; J + 2 α b m x − α b m + α 2 b m 2 = m − 1 m x − α b m 2 + β − α b m x − α b m m + 2 α b m x − α b m + α 2 b m 2 = x 2 + 1 m x − α b m β b m − x ,

which completes the proof. □

By taking into account the results in Lemma 1, it follows from Korovkin’s theorem [30]:

Theorem 2.

Let g be a continuous function on α b m , β b m . Then

lim m → ∞ B ̃ m C l g , J = g x

holds and the operators B ̃ m C l converge uniformly on α b m , β b m .

3 Bivariate Bernstein-Chlodowsky-Stancu operators

D.D. Stancu introduced a method for obtaining polynomials of Bernstein type of two variables in his paper [16]. This method can be applied to define a bivariate operator from univariate Bernstein-Chlodowsky operators B m C l g , x .

Assume that φ 1 = φ 1 x and φ 2 = φ 2 x are continuous functions satisfying φ ₁ < φ ₂ on 0 , b m . Let Δ ⊂ R 2 be a domain which is bounded by the curves y = φ 1 x , y = φ 2 x and the straight lines x = 0, x = b _m. By taking into account

y = φ 2 x − φ 1 x t b m + φ 1 x ,

for every function g x , y on the domain Δ, consider

G x , t = g x , φ 2 x − φ 1 x t b m + φ 1 x , 0 ≤ t ≤ b m .

We can define the mth Bernstein-Chlodowsky-Stancu operator as follows

(6) B m C l g x , y , Δ = ∑ k = 0 m ∑ j = 0 m k G k b m m , j b m m k q m , k x b m q m k , j t b m ,

where the integer m _k is nonnegative and corresponds to the kth node, given by x k = k b m m . In the explicit form, the operator B m C l can be rewritten as

B m C l g x , y , Δ = ∑ k = 0 m ∑ j = 0 m k G k b m m , j b m m k q m , k x b m q m k , j y − φ 1 x φ 2 x − φ 1 x ,

from which, it follows

B m C l g x , y , Δ = ∑ k = 0 m B m k C l ( t ) G k b m m , t q m , k x b m ,

where B m C l ( t ) denotes the univariate Bernstein-Chlodowsky operator acting on the variable t.

Notice that the partition step size along the x-axis is b m m . For a fixed node x k = k m b m , the partition step size along the t-axis is b m m k . Thus, the partition step size along the y-axis is b m n k , where

n k = m k φ 2 k m b m − φ 1 k m b m ,

from which, we have

G k m b m , j m k b m = g k m b m , j n k + φ 1 k m b m .

4 Examples of Bernstein-Chlodowsky-type operators under a domain transformation

In general, B m C l g x , y , Δ is not a polynomial. However, by selecting φ ₁, φ ₂ and m _k appropriately, one can obtain polynomials in the following examples. In this section, under a suitable domain transformation, we extend Bernstein-Chlodowsky-type operators on the square domain to another domain in R 2 .

Under the choices of φ ₁ = 0 and φ ₂ = b _m, we consider the square S = [0, b _m] × [0, b _m]. For a function g defined on S, we have the Bernstein-Chlodowsky-Stancu operator on the square as

B m C l g x , y , S = ∑ k = 0 m ∑ j = 0 m k g k b m m , j b m m k q m , k x b m q m k , j y b m .

4.1 The Bernstein-Chlodowsky-Stancu operators on the triangular domain

The transformation

x b m = 2 u − 1 and y b m = 1 − v ( 1 − | 2 u − 1 | ) , ( u , v ) ∈ [ 0,1 ] 2

maps the square domain [0, 1]² into the triangular domain

S b m = ( x , y ) ∈ R 2 ∣ − b m ≤ x ≤ b m , y ≥ x , y ≥ − x , y ≤ b m .

For every function g defined on S b m , we can define the function G : S → R in the form

G ( u , v ) = g ( ( 2 u − 1 ) b m , ( 1 − v ( 1 − | 2 u − 1 | ) ) b m ) , ( u , v ) ∈ S .

Under this transformation, the Bernstein-Chlodowsky-Stancu operator on the triangular domain S b m is in the following form

B ̂ m C l g x , y , S b m = ∑ k = 0 m ∑ j = 0 m k g b m ( 2 u − 1 ) , b m 1 − v ( 1 − | 2 u − 1 | ) q m , k ( 2 u − 1 ) × q m k , j 1 − v ( 1 − | 2 u − 1 | ) = ∑ k = 0 m ∑ j = 0 m − k g b m ( 2 u − 1 ) , b m 1 − v 1 − | 2 u − 1 | × m k ( 2 u − 1 ) k 1 − ( 2 u − 1 ) m − k m − k j × 1 − v 1 − | 2 u − 1 | j v 1 − | 2 u − 1 | m − k − j

where m _k = m − k (Figure 1).

$Figure 1: The transformation of a square domain into a triangular domain.$

Figure 1:

The transformation of a square domain into a triangular domain.

4.2 The Bernstein-Chlodowsky-Stancu operators on the elliptic domain

In this subsection, we consider two different transformations which map the square domain into an elliptic domain.

Let

B b m = x , y ∈ R 2 : x 2 + y 2 b m 2 ≤ 1 .

The transformation x = 2 u − b m , y = ( 2 v − b m ) 1 − ( 2 u − b m ) 2 , ( u , v ) ∈ S maps the square S = [0, b _m] × [0, b _m] into B b m . For every function g defined on B b m , we can define the function G : S → R as

G ( u , v ) = g 2 u − b m , ( 2 v − b m ) 1 − ( 2 u − b m ) 2 .

Under this map, the operator transforms to the following operator

B ̂ m C l g x , y , B b m = ∑ k = 0 m ∑ j = 0 m k g 2 k − m m b m , 2 j − m k m k m 2 − b m 2 ( 2 k − m ) 2 m b m × q m , k x + b m 2 b m q m k , j y 1 − x 2 + b m 2 b m ,

where

q m , k x + b m 2 b m q m k , j y 1 − x 2 + b m 2 b m = m k m k j x + b m k b m − x m − k ( 2 b m ) m + m k 1 − x 2 m k × y + b m 1 − x 2 j b m 1 − x 2 − y m k − j .

Since for y = 0 it follows

q m , k x + b m 2 b m q m k , j 1 2 = 1 2 m + m k b m m m k m k j x + b m k b m − x m − k

and for x = 0

q m , k 1 2 q m k , j y + b m 2 b m = 1 2 m + m k b m m k m k m k j y + b m j b m − y m k − j ,

it is seen that B ̂ m C l g x , y , B b m is a polynomial for any m _k on the x-axis and y-axis.

We now consider another map to obtain another bivariate Bernstein-Chlodowsky-Stancu operator on the elliptic domain B b m .

If we apply the transformation

x = u b m b m 2 − v 2 , y = v ,

where (u, v) ∈ S = [0, b _m] × [0, b _m], the square S maps into the first quadrant B b m 1 . Similarly, for every function g defined on B b m , we can define the function G : S → R 2 which maps each quadrant to S. The corresponding bivariate Bernstein-Chlodowsky-Stancu operators on the four quadrants of B b m are given as follows (Figure 2):

B ̂ m C l g x , y , B b m 1 = ∑ k = 0 m ∑ j = 0 m k g k b m m k 2 − j 2 m k m , j m k b m q m , k x b m 2 − y 2 q m k , j y b m , B ̂ m C l g x , y , B b m 2 = ∑ k = 0 m ∑ j = 0 m k g − k b m m k 2 − j 2 m k m , j m k b m q m , k x b m 2 − y 2 q m k , j y b m , B ̂ m C l g x , y , B b m 3 = ∑ k = 0 m ∑ j = 0 m k g − k b m m k 2 − j 2 m k m , − j m k b m q m , k x b m 2 − y 2 q m k , j y b m , B ̂ m C l g x , y , B b m 4 = ∑ k = 0 m ∑ j = 0 m k g k b m m k 2 − j 2 m k m , − j m k b m q m , k x b m 2 − y 2 q m k , j y b m ,

where the functions G _i(u, v) (i = 1, 2, 3, 4) on S are as follows for every function g on B b m

G 1 ( u , v ) = g u b m b m 2 − v 2 , v , G 2 ( u , v ) = g − u b m b m 2 − v 2 , v , G 3 ( u , v ) = g − u b m b m 2 − v 2 , − v , G 4 ( u , v ) = g u b m b m 2 − v 2 , − v .

$Figure 2: T h e o p e r a t o r d o m a i n t r a n s f o r m a t i o n f o r b m = m , m = 20 . $Theoperatordomaintransformationfor{b}_{m}=\sqrt{m}, m=20.$ .$

Figure 2:

T h e o p e r a t o r d o m a i n t r a n s f o r m a t i o n f o r b m = m , m = 20 . .

If we choose m _k = m − k, we have

B ̄ ̂ m C l g x , y , B b m = B ̂ m C l g x , y , B b m 1 , x , y ∈ B b m 1 , B ̂ m C l g x , y , B b m 2 , x , y ∈ B b m 2 , B ̂ m C l g x , y , B b m 3 , x , y ∈ B b m 3 , B ̂ m C l g x , y , B b m 4 , x , y ∈ B b m 4 .

5 Bivariate shifted Bernstein-Chlodowsky-Stancu operators

In this section, we introduce bivariate shifted mth Bernstein-Chlodowsky-Stancu operators and discuss some their approximation properties.

Assume that φ 1 x and φ 2 x are continuous functions satisfying φ ₁ < φ ₂ on the interval J = α b m , β b m . Let Δ ⊂ R 2 be a domain which is bounded by the curves y = φ 1 x , y = φ 2 x and the straight lines x = αb _m, x = βb _m. For a fixed x ∈ J, the polynomial q ̃ m , k y ; φ 1 x , φ 2 x , 0 ≤ k ≤ m , m ≥ 0 denotes a univariate shifted Bernstein-Chlodowsky basis on the interval φ 1 x , φ 2 x .

For every function g x , y on Δ, we consider the function

G ̃ u , v ; Δ = g β b m − α b m u + α b m , φ ̃ 2 u − φ ̃ 1 u v + φ ̃ 1 u ,

where 0 ≤ u, v ≤ 1 and

φ ̃ i u = φ i β b m − α b m u + α b m ; i = 1,2 .

The shifted mth Bernstein-Chlodowsky-Stancu operator is as follows

B ̃ m C l g x , y ; Δ = ∑ k = 0 m ∑ j = 0 m k G ̃ k m , j m k ; Δ q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 ,

where

q ̃ m , k x ; J = p m , k x − α b m β − α b m and q ̃ m k , j y ; φ 1 , φ 2 = p m k , j y − φ 1 x φ 2 x − φ 1 x

and m _k = m − k or m _k = k for 0 ≤ k ≤ m. In the explicit form, the univariate Bernstein-Chlodowsky is defined as

B ̃ m C l g x , y ; Δ = ∑ k = 0 m ∑ j = 0 m k G ̃ k m , j m k ; Δ p m , k x − α b m β − α b m × p m k , j y − φ 1 x φ 2 x − φ 1 x .

For the operator B ̃ m C l g x , y ; Δ , the following results hold true.

Lemma 3.

B ̃ m C l 1 ; Δ = 1
B ̃ m C l x ; Δ = x
B ̃ m C l y ; Δ → y uniformly on the interval J when m → ∞
B ̃ m C l x 2 ; Δ = x 2 + 1 m x − α b m β b m − x
B ̃ m C l y 2 ; Δ → y 2 uniformly on the interval J when m → ∞.

Proof.

(i) It follows from (4) and (5)

B ̃ m C l 1 ; Δ = ∑ k = 0 m ∑ j = 0 m k p m , k x − α b m β − α b m p m k , j y − φ 1 x φ 2 x − φ 1 x = ∑ k = 0 m ∑ j = 0 m k q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 = ∑ j = 0 m k q ̃ m k , j y ; φ 1 , φ 2 = ∑ j = 0 m k m k j y − φ 1 x φ 2 x − φ 1 x j 1 − y − φ 1 x φ 2 x − φ 1 x m k − j = ∑ j = 0 m k m k j y − φ 1 x j φ 2 x − y m k − j 1 φ 2 x − φ 1 x m k = φ 2 x − φ 1 x m k 1 φ 2 x − φ 1 x m k = 1 .

Since ∑ k = 0 m q ̃ m , k x ; J = 1 , we have

B ̃ m C l x ; Δ = ∑ k = 0 m ∑ j = 0 m k β − α b m k m + α b m q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 = ∑ k = 0 m β − α b m k m + α b m q ̃ m , k x ; J ∑ j = 0 m k q ̃ m k , j y ; φ 1 , φ 2 = β − α b m ∑ k = 0 m k m q ̃ m , k x ; J + α b m ∑ k = 0 m q ̃ m , k x ; J = β − α b m ∑ k = 0 m k m q ̃ m , k x ; J + α b m = β − α b m ∑ k = 0 m k m 1 β − α m b m m m k x − α b m k β b m − x m − k + α b m = x − α b m ∑ k = 0 m − 1 q ̃ m − 1 , k x ; J + α b m = x − α b m + α b m = x .

We obtain

(7) B ̃ m C l y ; Δ = ∑ k = 0 m ∑ j = 0 m k φ ̃ 2 k m − φ ̃ 1 k m j m k + φ ̃ 1 k m q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 = ∑ k = 0 m φ ̃ 2 k m − φ ̃ 1 k m q ̃ m , k x ; J ∑ j = 0 m k q ̃ m k , j y ; φ 1 , φ 2 j m k + ∑ k = 0 m φ ̃ 1 k m q ̃ m , k x ; J .

Since ∑ j = 0 m k q ̃ m k , j y ; φ 1 , φ 2 j m k = y − φ 1 x φ 2 x − φ 1 x ∑ j = 0 m k − 1 q ̃ m k − 1 , j y ; φ 1 , φ 2 = y − φ 1 x φ 2 x − φ 1 x , the equation (7) reduces to

B ̃ m C l y ; Δ = y − φ 1 x φ 2 x − φ 1 x ∑ k = 0 m φ ̃ 2 k m − φ ̃ 1 k m q ̃ m , k x ; J + ∑ k = 0 m φ ̃ 1 k m q ̃ m , k x ; J = y − φ 1 x φ 2 x − φ 1 x B ̃ m C l φ 2 − φ 1 ; J + B ̃ m C l φ 1 ; J ,

where B ̃ m C l is the univariate shifted Bernstein-Chlodowsky operators. From Theorem 2, since

B ̃ m C l φ 2 − φ 1 ; J → φ 2 − φ 1 and B ̃ m C l φ 1 ; J → φ 1 ,

we arrive at

lim m → ∞ B ̃ m C l y ; Δ = y − φ 1 x φ 2 x − φ 1 x φ 2 x − φ 1 x + φ 1 x = y .

B ̃ m C l x 2 ; Δ = ∑ k = 0 m ∑ j = 0 m k β − α b m k m + α b m 2 q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 = β − α 2 b m 2 ∑ k = 0 m k m 2 q ̃ m , k x ; J + 2 β − α α b m 2 ∑ k = 0 m k m q ̃ m , k x ; J + α 2 b m 2 = m − 1 m x − α b m 2 ∑ k = 0 m − 2 q ̃ m − 2 , k x ; J + β − α b m x − α b m m ∑ k = 0 m − 1 q ̃ m − 1 , k x ; J + 2 α b m x − α b m ∑ k = 0 m − 1 q ̃ m − 1 , k x ; J + α 2 b m 2 = m − 1 m x − α b m 2 + β − α b m x − α b m m + 2 α b m x − α b m + α 2 b m 2 = x 2 + 1 m x − α b m β b m − x .

Finally, it follows

B ̃ m C l y 2 ; Δ = ∑ k = 0 m ∑ j = 0 m k φ ̃ 2 k m − φ ̃ 1 k m j m k + φ ̃ 1 k m 2 × q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 = ∑ k = 0 m φ ̃ 2 k m − φ ̃ 1 k m 2 q ̃ m , k x ; J ∑ j = 0 m k q ̃ m k , j y ; φ 1 , φ 2 j m k 2 + 2 ∑ k = 0 m ∑ j = 0 m k φ ̃ 2 k m − φ ̃ 1 k m j m k φ ̃ 1 k m q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 + ∑ k = 0 m ∑ j = 0 m k φ ̃ 1 k m 2 q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 = B ̃ m C l ( φ 2 − φ 1 ) 2 ; J y − φ 1 x φ 2 x − φ 1 x 2 + y − φ 1 x φ 2 x − y φ 2 x − φ 1 x 2 ∑ k = 0 m 1 m k φ ̃ 2 k m − φ ̃ 1 k m 2 q ̃ m , k x ; J + 2 y − φ 1 x φ 2 x − φ 1 x ∑ k = 0 m φ ̃ 2 k m − φ ̃ 1 k m φ ̃ 1 k m q ̃ m , k x ; J + B ̃ m C l φ 1 2 ; J = B ̃ m C l ( φ 2 − φ 1 ) 2 ; J y − φ 1 x φ 2 x − φ 1 x 2 + 2 y − φ 1 x φ 2 x − φ 1 x B ̃ m C l φ 2 − φ 1 φ 1 ; J + B ̃ m C l φ 1 2 ; J + Q m x ,

where

Q m x = y − φ 1 x φ 2 x − y φ 2 x − φ 1 x 2 ∑ k = 0 m 1 m k φ ̃ 2 k m − φ ̃ 1 k m 2 q ̃ m , k x ; J .

When we choose m _k = m − k, we have

Q m x = y − φ 1 x φ 2 x − y m φ 2 x − φ 1 x 2 ∑ k = 0 m 1 1 − k / m φ ̃ 2 k m − φ ̃ 1 k m 2 q ̃ m , k x ; J = y − φ 1 x φ 2 x − y m φ 2 x − φ 1 x 2 B ̃ m C l ( φ 2 − φ 1 ) 2 1 − x − α b m β − α b m ; J

and in the case of m _k = k, then

Q m x = y − φ 1 x φ 2 x − y m φ 2 x − φ 1 x 2 B ̃ m C l ( φ 2 − φ 1 ) 2 x − α b m β − α b m ; J .

Since

B ̃ m C l ( φ 2 − φ 1 ) 2 ; J → ( φ 2 − φ 1 ) 2 , B ̃ m C l φ 2 − φ 1 φ 1 ; J → φ 2 − φ 1 φ 1 , and B ̃ m C l φ 1 2 ; J → φ 1 2 ,

from Theorem 2 and Q m x → 0 as m → ∞, it follows lim m → ∞ B ̃ m C l y 2 ; Δ = y 2 .

□

In order to give approximation properties of the operator B ̃ m C l g x , y ; Δ , we first give the modulus of continuity of the function g.

Definition 4.

[15] For a continuous function g on the domain Δ, if δ ₁ and δ ₂ are positive real numbers, the modulus of continuity of g is given by

w δ 1 , δ 2 = sup g x 2 , y 2 − g x 1 , y 1 .

It is also well known that, for each (x ₁, y ₁), (x ₂, y ₂) ∈ Δ such that |x ₂ − x ₁| ≤ δ ₁ and |y ₂ − y ₁| ≤ δ ₂,

(8) g x 2 , y 2 − g x 1 , y 1 ≤ w x 2 − x 1 , y 2 − y 1 ≤ w δ 1 , δ 2 .

Also, the inequality

(9) w a δ 1 , b δ 2 ≤ a + b + 1 w δ 1 , δ 2

is satisfied for a, b > 0 [15],16].

Theorem 5.

For a continuous function g on the domain Δ, we have

B ̃ m C l g x , y ; Δ → g x , y

as m → ∞ uniformly on Δ.

Proof.

From (8) and (9), we can write

(10) g x , y − G ̃ k m , j m k ; Δ ≤ w x − β − α b m k m − α b m , y − φ ̃ 2 k m − φ ̃ 1 k m j m k − φ ̃ 1 k m ≤ θ 1 + θ 2 + 1 w δ 1 , δ 2 ,

where

θ 1 = 1 δ 1 x − β − α b m k m − α b m and θ 2 = 1 δ 2 y − φ ̃ 2 k m − φ ̃ 1 k m j m k − φ ̃ 1 k m .

By using the fact that B ̃ m C l 1 ; Δ = 1 and q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 ≥ 0 , it follows from (10)

g x , y − B ̃ m C l g x , y ; Δ ≤ ∑ k = 0 m ∑ j = 0 m k q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 g x , y − G ̃ k m , j m k ; Δ ≤ ∑ k = 0 m ∑ j = 0 m k q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 1 + θ 1 + θ 2 w δ 1 , δ 2 = I ( 0 ) x , y , k , j , m , m k , δ 1 , δ 2 + I ( 1 ) x , y , k , j , m , m k , δ 1 , δ 2 + I ( 2 ) x , y , k , j , m , m k , δ 1 , δ 2 ,

where

I ( i ) x , y , k , j , m , m k , δ 1 , δ 2 = ∑ k = 0 m ∑ j = 0 m k q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 θ i w δ 1 , δ 2 ,

for i = 0, 1, 2 and θ ₀ = 1. We now compute each I ( i ) x , y , k , j , m , m k , δ 1 , δ 2 for i = 0, 1, 2.

For the first term I ( 1 ) x , y , k , j , m , m k , δ 1 , δ 2 , since g x = x is a concave function, we can write from Cauchy–Schwarz inequality

∑ k = 0 m ∑ j = 0 m k q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 x − β − α b m k m − α b m ≤ ∑ k = 0 m ∑ j = 0 m k q ̃ m , k x ; J q ̃ m k , j y ; φ 1 , φ 2 x − β − α b m k m − α b m 2 1 / 2 = x 2 B ̃ m C l 1 ; Δ − 2 x B ̃ m C l x ; Δ + B ̃ m C l x 2 ; Δ 1 / 2 ,

which converges to zero when m → ∞ uniformly from the results in Lemma 3. Similarly, the second term converges to zero when m → ∞ uniformly. Finally, if we get δ 1 = δ 2 = 1 m , it follows w 1 m , 1 m → 0 when m → ∞. Hence, the proof is completed. □

Example 1.

Let

φ 1 ( x ) = 0.5 ⁡ sin 2 π x β − α ,

φ 2 ( x ) = 0.5 + 0.5 ⁡ cos 2 π x β − α ,

φ ̃ 1 ( u ) = φ 1 ( ( β − α ) u b m + α b m ) ,

φ ̃ 2 ( u ) = φ 2 ( ( β − α ) u b m + α b m ) ,

G ( u , v ) = exp − x − 0.5 ( α + β ) 2 + y − 0.5 2 ,

x = ( β − α ) u b m + α b m , y = ( φ ̃ 2 ( u ) − φ ̃ 1 ( u ) ) v + φ ̃ 1 ( u ) .

Figure 3 presents the convergence of the operator B ̃ m C l G u , v ; Δ to the function G for b m = m and, m = 50 and m = 40, respectively.

$Figure 3: Approximation of the operator B ̃ m C l G u , v ; Δ ${\tilde {\mathcal{B}}}_{m}^{Cl}\left[G\left(u,v\right);{\Delta}\right]$ for b m = m ${b}_{m}=\sqrt{m}$ and, m = 50 and m = 40, respectively.$

Figure 3:

Approximation of the operator B ̃ m C l G u , v ; Δ for b m = m and, m = 50 and m = 40, respectively.

Corresponding author: Rabia Aktaş Karaman, Faculty of Science, Department of Mathematics, Ankara University, 06100 Tandoğan, Ankara, Türkiye, E-mail: raktas@science.ankara.edu.tr

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments which led us to improve this paper.

Research ethics: Not applied.
Informed consent: Not applied.
Author contributions: Both authors contributed equally to this work. Both authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: The authors state no conflicts of interest.
Research funding: None declared.
Data availability: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

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Received: 2025-03-08

Accepted: 2025-08-10

Published Online: 2025-10-30

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Keywords for this article

Bernstein operators; Bernstein-Chlodowsky operators; modulus of continuity

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