On a Family of Parameter-Based Bernstein Type Operators with Shape-Preserving Properties

Tis article aims to introduce a new linear positive operator with a parameter. Our focus lies in analyzing the distinct characteristics and inherent properties exhibited by this operator. Additionally, we provide a proof of the convergence rate and present a revised version of the Voronovskaja theorem specifcally tailored for this newly defned operator. Furthermore, we provide an upper bound for the error according to the modulus of continuity. Finally, the preservation of monotonicity and convexity by the operator is being investigated.


Introduction
Te Bernstein polynomials are widely regarded as one of the most well-known algebraic polynomials in approximation theory, and they are defned as Tese polynomials were frst introduced by Bernstein in 1912 to provide the frst constructive proof of Weierstrass' approximation theorem [1].Bernstein employed the Bernstein operators to approximate a given function on the interval [0, 1] by a polynomial of degree n, as it is understood by its formula the structure gets advantage of weighted sum of the function values at equidistant points on the interval.Many researchers have written books and papers dedicated to studying Bernstein polynomials and operators, with Lorentz's book being one of the most famous [2].Te signifcance of the Bernstein polynomials lies not only in their own properties but also in the fact that their form has inspired mathematicians to develop a wide range of other approximation operators.Since its introduction, many researchers have proposed modifcations and generalizations of Bernstein operators to improve its approximation properties and extend its applications.Some of the most famous cases are the Schurer polynomials, Kantorovich polynomials, Stancu polynomials, q-Bernstein polynomials, Durrmeyer polynomials, Favard-Szász-Mirakyan operators, Baskakov operators, and numerous others [3][4][5][6].Some of the recent advances could be traced in [7][8][9][10][11].
Te construction and analysis of Bernstein-type operators aimed to achieve two primary objectives: preserving the form of various functions, such as polynomials and exponentials, and upholding their shape-preserving properties [12,13].
Lately, attempts have been made to develop operators of the Bernstein type.In 2017, Chen et al. [22] proposed a new modifcation of Bernstein operators based on the so called α-Bernstein polynomials; this paper has garnered signifcant attention and inspired many researchers to further develop and extend their fndings for other families of Bernstein-type operators, see [23][24][25] and the references therein.Tere are also other simpler modifcations, like the one proposed by Usta in [26].
Tis paper introduces a fresh set of Bernstein-like operators that are based on a particular shape parameter.It has been verifed that the new operators are linear positive operators that preserve linearity, monotonicity, and convexity.By taking advantage of Korovkin's theorem, we provide an alternative proof for Weierstrass' approximation theorem.Furthermore, we provide in-depth proofs regarding the convergence rate and the Voronovskaja-type asymptotic estimation formula for these operators.
Overall, our work contributes to the ongoing research on Bernstein-like operators and shows that further modifcations and generalizations can lead to even more powerful approximation tools with wider applications.
Te paper is structured as follows: Section 2 introduces the new Bernstein-type operator and examines its fundamental properties.In Section 3, we focus on the shapepreserving aspects.Finally, Section 4 concludes the paper by highlighting key fndings of this study.

A Revised Version of the Bernstein Operator
In [27], the authors introduced a new set of Bernstein-like basis functions.Building on this set, we propose a unique variation of Bernstein-type operators and investigate their fundamental properties.
Te sq-Bernstein operator is a type of approximation operator that maps the given function f(x) defned on [0, 1] to R n,] (f; x) defned in (6).Te parameter ] controls the level of smoothness of the approximation.In comparison with the classical Bernstein operator, the sq-Bernstein operator provides greater fexibility for controlling the shape of the curve being approximated.By fne-adjusting the shape parameter, one can exert a better infuence over the curve's curvature, degree of smoothness, and other geometric attributes.Furthermore, the sq-Bernstein operator surpasses the constraints of polynomial approximation and accommodates more intricate curved representations.
Tere are several properties and results for these operators that we will discuss.

Lemma 3 (End point interpolation). Te sq-Bernstein operator applied to the function f(x) guarantees interpolation of f(x) at the endpoints of
Lemma 4 (Linearity).Te sq-Bernstein operator satisfes linearity, that is, for any functions f 1 (x) and f 2 (x) defned on the interval [0, 1], as well as any real values λ and μ.
Proof.From the defnition of the sq-Bernstein operator, for any functions f 1 (x) and f 2 (x) and any real values λ and μ, we have □ Lemma 5 (Positivity).Te sq-Bernstein operators for 0 < ] ≤ 1 form a collection of positive operators.Tis means that for x ∈ [0, 1], we have Proof.It is an obvious result of the nonnegativity of the sqbasis functions (see [27]).
Te above lemma leads to the following direct result: for the sq-Bernstein operators.

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From the above result, we have

Corollary 7 (Boundedness preservation). If the function f(x) satisfes the inequality m
Theorem 8. Te sq-Bernstein operators fulfll equalities Proof.We use induction for verifying this case.Assuming (3) yields  2 i�0 b 2,i (x) � 1 and  2 i�0 i/2b 2,i (x) � x, which verifes the base case.Now, for the induction step, we suppose that the result holds true for m ∈ N ≥2 .By utilizing the recurrence relation (5), we can derive the following conclusions: .
Our next step involves analyzing the sq-Bernstein operators applied to the functions f(x) � x 2 , x 3 , x 4 .□ Lemma 10.Te sq-Bernstein operators satisfy the following equalities: Proof.To prove these relations, we will use mathematical induction.Te case n � 2 is simple, and the induction steps are straightforward.Te most efective criteria for determining the convergence of a positive linear operator to the identity operator is provided by the Bohman-Korovkin theorem, as follows: □ Lemma 11 (see [29,30]).Consider a sequence of linear positive operators, denoted as L n f; n � 0, 1, . . . the corresponding sequence of functions L n (f; x)   will also converge uniformly to f(x); i.e.,

􏼈 􏼉, operating on the interval
Based on Lemma 11, we can now present the key fnding of this paper, showcasing the convergence of the sequence comprised of sq-Bernstein operators.
Proof.From equation (15), we can deduce that the sq-Bernstein operators converge uniformly to x 2 for any ] ∈ (0, 1).Considering this fact along with equations ( 11) and ( 12), the convergence is a straightforward result of the Bohman-Korovkin theorem.
Te following lemma is a prerequisite for the Voronovskaja theorem.
We have the following equalities: Proof.Te necessary outcomes can be obtained by utilizing the binomial expansion of (i − nx) k , k � 1, 2, 3, along with Teorem 8 and Lemma 10.
Once we have established the convergence of the newly introduced Bernstein-type operators, the next crucial consideration is how quickly these operators approximate the function f(x).Voronovskaya (1932) answered this question for the Bernstein operator.In the next theorem, we present a new variant of Voronovskaja's result [31] for our newly defned operator in (6) and we introduce the asymptotic error for the sq-Bernstein operators.□ Theorem 14.Let f(x) be a bounded function on [0, 1], then for any x ∈ [0, 1], at which f ″ (x) exists, we have where 0 < ] ≤ 1.
Proof.Employing Taylor's expansion for i ≤ n where lim t ⟶ x r(t, x) � 0, at t � i/n results in Consequently, 4

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Ten, according to ( 20) and ( 21), one can write where To complete the proof, it is necessary to demonstrate that By utilizing the Cauchy-Schwarz inequality, it is straightforward to infer that Using Korovkin's theorem, we get Since r 2 (x, x) � 0 and continuity of the function r 2 (., x) in (0, 1), along with the fact that R n,] ((t − x) 4 ; x) does not increase faster than O(n − 2 ), we can conclude that equation ( 29) holds, which completes the proof.Now, we can analyze the rate of convergence of sq-Bernstein operators with respect to the modulus of continuity (ω), and by getting advantage of the characteristics of the modulus of continuity stated in [32], we can demonstrate the principal outcome regarding the upper bound of the approximation error.Te error is measured by the uniform norm, which is defned on the interval [0, 1] as follows: □ Theorem 15.For any value Proof.According to ( 8) and ( 11), for 0 < ] ≤ 1, one has Now, considering properties of modulus of continuity, one can deduce which results in Journal of Mathematics Next, we get advantage of the Cauchy-Schwarz's inequality to have For the last term, we can have an upper bound according to (21) as follows: which leads us to the fnal result and this completes the proof.According to Teorem 15, we provided an upper bound for the error f(x) − R n,] (f; x) in terms of the modulus of continuity.In addition, according to properties of the modulus of continuity and continuity of the function f(x) on [0, 1], we have In another manner, Teorem 12 can be proven.

□
Remark 16.Similar to the above proof process, another upper bound for the error can be presented in terms of the parameter ] as follows:

Shape-Preserving Properties
In this section, we will demonstrate that the sq-Bernstein operators preserve certain geometric properties such as monotonicity and convexity.

Monotonicity Preservation.
For verifying the monotonypreserving property of the sq-Bernstein operators, we frst state the following lemma: , as a linear combination of the elements of A, is monotonically increasing.
Proof.We use induction to verify the result.Te base of induction is n � 2, so for any set of real values α 0 ≤ α 1 ≤ α 2 , we show that the function G(x, ]) �  2 i�0 α i b 2,i (x) is monotonically increasing on [0, 1], and by rewriting G(x, ]) in a suitable way, one has 6

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We need to show that G(x, ]) has a nonnegative derivative with respect to x: As the coefcients are nonnegative in (43), it is sufcient to verify the nonnegativity of the corresponding functions which we verify term by term.
(i) From the partition of unity of the sq-basis functions, one has (ii) Te second term could be treated as follows: From the convexity of the function φ(x) in [0, 1], we conclude that the function −d/dxφ(x) takes its minimum at x � 1, so one has  2 j�1 d/dx b 2,j (x)| x�1 � ] which has a nonnegative value.(iii) For the frst term, we have which by taking advantage of the convexity of φ(x), one observes that d/dxφ(x) has its minimum at x � 0. Since d/dxb 2,2 (x)| x�0 � ], the nonnegativity of the frst term is verifed.
As the induction hypothesis, we assume that for any set of monotone real values is monotonically increasing.Now, by considering the set of increasing values According to the induction hypothesis, we have  n i�0 β i d/dxb n,i (x),  n i�0 β i+1 d/dxb n,i (x) ≥ 0 and β i+1 − β i , x, 1 − x ≥ 0, so d/dxG(x, ]) ≥ 0. Tis completes the proof.
Te preceding lemma is more than needed and illustrates a more general result, and we employ a special case to prove the monotonicity preservation of the sq-Bernstein bases.
Proof.Let f(x) be a monotonically increasing function, so one has Proof.We can employ the method of induction to demonstrate the validity of the result.Our base case for induction will be when n � 2. Consider a set of real values α 0 , α 1 , α 2 and assume that they form a convex data set, satisfying the condition α 2 − 2α 1 + α 0 ≥ 0. We aim to prove that the function S(x, ]) �  2 i�0 α i b 2,i (x) is convex on the interval [0, 1].We show that the function's second derivative is nonnegative over the interval [0, 1]: Given that φ(x) is a convex function, we can establish that d 2 /dx 2 φ(x) ≥ 0. Consequently, based on this fact and α 2 − 2α 1 + α 0 ≥ 0, we can conclude that d 2 /dx 2 S(x, ]) ≥ 0.
Assuming that a set of real values α i   n i�0 is convex, we can consider the expression  n i�0 α i b n,i (x).As the induction hypothesis, we assume this expression to be convex.Now, let us consider a new set of convex values β i   n+1 i�0 .Our goal is to demonstrate that the function  n+1 i�0 β i b n+1,i (x) is also convex.To do so, we consider and show that the function d/dxS(x, ]) is increasing.
By assuming the induction hypothesis, since represents convex data, it follows that both  n i�0 β i b n,i (x) and  n i�0 β i+1 b n,i (x) are convex functions.Tis implies that their respective derivatives,  n i�0 β i d/dxb n,i (x) and  n i�0 β i+1 d/dxb n,i (x), are increasing functions.
We observe that if f is a convex function, then f(0/n), f(1/n), . . ., f(n/n) forms a set of convex data.Based on the obtained results, we can conclude that R n,] (f; x) �  n i�0 f(i/n)b n,i (x) is also a convex function.

Conclusion
We have introduced a novel class of linear positive operators characterized by shape parameters.Tese operators not only share several properties with the Bernstein operators but also possess the ability to preserve important shape characteristics such as monotonicity and convexity of the underlying data.Te operators converge uniformly for any value of the parameter, and an upper bound for the approximation error has been provided based on the modulus of continuity.By altering the parameter value, it becomes possible to adjust the shape of the resulting approximate curve produced by the operator.Te sq-Bernstein operators may provide an appropriate basis for solving functional equations from diferent disciplines.Tis certainly could be a proposition for future studies.Moreover, the structure used in equation (3) may be generalized by using some novel auxiliary parameter-based functions, φ(t), from diferent families of functions such as trigonometrics, exponentials, etc. Tis will result in an operator that is not only linear and preserves the properties of the Bernstein operator but also preserves the monotonicity and convexity of the data.Te operator defned by these bases surpasses the constraints of polynomial approximation and accommodates more intricate curved representations.
[a, b] from the space C[a, b] to itself.If this sequence of operators converges uniformly to f(x) for the functions f(x) � 1, x, x 2 on [a, b], then for any function f(x) belonging to C[a, b] and any x within the interval [a, b], If f(x) is a convex function in C[0, 1], then all its sq-Bernstein operators are convex.